The Homoclinic Tangle of Slightly Dissipative, Forced, Two Dimensional Systems

We analyze the homoclinic tangle of forced, damped, two dimensional dynamical systems, which are nearly Hamiltonian. This paper consists of three parts; In the rst part, we construct homoclinic bifurcation diagrams and curves in the near integrable case, using perturbational methods, and supply the geometrical interpretation of these homoclinic bifurcations. In the second part, we construct symbolic dynamics of segments of the unstable manifold, based on the geometrical interpretation of the homoclinic bifurcation points. This symbolic dynamics gives the minimal development of the homoclinic tangle, and a lower bound on the topological entropy. We implement the methods, constructed in these two parts, on the asymmetrically forced, damped, Du ng oscillator (AFDO). This model serves as a prototype for analyzing systems with symmetry breaking disturbances. We show that even a slight asymmetry may cause a substantial change in the asymptotic behavior of the system. In the third part we present numerical results regarding the existence of strange attractors in the model, and discuss these results with respect to the analysis of the structure of the homoclinic tangles. We observe that for a xed asymmetry parameter, and varying some values of the perturbation parameters, the AFDO exhibit transition from "two sided" to "one sided" strange attractors. Contents 1 Introduction 2 1.1 Dissipative, forced, two dimensional systems . . . . . . . . . . . . 2 1.2 Strange attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Du ng's equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 The topological approximation method the TAM . . . . . . . . . 7 1.5 The Whisker map . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 The homoclinic tangle of slightly dissipative, forced, two dimensional systems 9 2.1 General assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 The Asymmetrically Forced Du ng Oscillator (AFDO) . . . . . . 10 2.2.1 P (H) the period function of the unperturbed periodic orbits 11 2.2.2 The Melnikov function . . . . . . . . . . . . . . . . . . . . 12 2.3 Behavior of the stable and unstable manifolds . . . . . . . . . . . 14 2.3.1 Primary homoclinic intersection points . . . . . . . . . . . 14 2.3.2 Secondary homoclinic intersection points . . . . . . . . . . 17 2.3.3 Secondary homoclinic bifurcation diagrams . . . . . . . . . 20 2.3.4 Secondary homoclinic bifurcation points and curves . . . . 26 2.4 Symbolic dynamics, constructed of segments of the unstable manifold 36 2.5 Strange attractors in the AFDO . . . . . . . . . . . . . . . . . . . 45 3 Discussion and conclusion 52 A Secondary homoclinic bifurcation diagrams for the AFDO with 6= 0 or/and 6= 0 55 B Secondary homoclinic bifurcation points for the AFDO 65 C Closed and open systems 69 1 1. Introduction 1.1. Dissipative, forced, two dimensional systems Nearly Hamiltonian, forced, two dimensional systems with a weak damping arise in many subjects of research in physics. For example, the behavior of a particle in a double well potential (Du ng's equation) or in a single well potential (the Cubic potential equation), the Pendulum, etc. In such systems, the ratio between the forcing amplitude and the dissipation rate determines much of the dynamics. While the value of the dissipation is small in these near Hamiltonian ows, the ratio of the dissipation parameter and the amplitude of the forcing may be large (strongly dissipative case), which implies that the system is strongly area contracting. For the methods that we present here, both slightly dissipative (weak area contracting), and strongly dissipative (strong area contracting) systems may be considered. We note these near conservative systems since they frequently arise in applications. But, there are some rigorous results regarding strange attractors (see [10], [11], [18] and [12]), for strongly dissipative systems, such as the H enon map. We present a short summary of these results in section 2.5. For the development of methods of analysis, which are presented here, we use analytical and numerical tools, such as the Melnikov function, perturbation theory, the TAM (topological approximation method), Lyapunov exponents and some other features from ergodic theory. Some of these tools are described in more detail below. We implement our work on the asymmetric, forced, damped, Du ng oscillator, which we call: the Asymmetrically Forced Du ng OscillatorAFDO, and which is introduced below. Investigating the AFDO not only serves us as an example and a test case for our methods, but also enables us to emphasize some new phenomenons, as discussed below. 2 1.2. Strange attractors There has been vast numerical evidence indicating the existence of strange attractors for some parameter values in two dimensional maps and forced ows. The numerics indicate that in many cases the existence of the attractor is sensitive to changes in the parameter values; Small changes may lead to the destruction of a strange attractor, and the appearance of periodic attracting points. These observations re ect the di culties in proving the existence of a strange attractor in such systems. Recently, some general results, regarding strange attractors were proved for the H enon map, and for H enon-like families of di eomorphisms on a surface. The H enon map is de ned to be: (x; y) 7! (1 ax2 + y; bx); (1.1) a and b are real parameters. M. Benedicks and L. Carleson [10] proved that for b > 0 su ciently small, the H enon map possesses a strange attractor for parameters set of positive Lebesgue measure, near a = 2 (where the H enon map is strongly area contracting), and made some conjectures regarding this attractor. L. Mora and M. Viana [11] extended the methods and results of [10], and proved Palis's conjecture, that C1 one-parameter families of di eomorphisms on a surface, (f ) , unfolding a homoclinic tangency associated to an hyperbolic xed (or periodic) point p0 of f0, exhibit a strange attractor (or a repeller) for positive Lebesgue measure set of parameters near = 0, near the orbit of tangency, under some generic assumptions, including a strong dissipation of f0 at p0. They proved this conjecture, by proving it for H enon-like families, ('a)a, near a = 2 and small b, and by proving that considering the families, (f ) , under some assumptions, one can nd a renormalizations by H enon-like families, ('a)a, which are arbitrarily Cr close (1 r < 1 xed) to the family of endomorphisms: a(x; y) = (1 ax2; 0). Also they mention in [11] that although the strange attractors related to H enon and H enon-like maps are non hyperbolic, since the H enon-like maps are strongly area contracting, they are fairly structured and seem to share some of the properties of hyperbolic attractors. Benedicks and Young [18] have constructed a SRB-measure for the strange attractors in [10] (Benedicks and Carleson suggested in [10] that such measure can be constructed), by the use of the "hyperbolic features" of such strange attractors, and proved that this measure is unique. For de nition and some details about SRB-measures, see [17]. 3 See Cao [12] for results regarding the basin of attraction of the strange attractors in [10]. By strange attractor, we mean an attractor with sensitive dependence on initial conditions, i.e. an attractor which has a dense orbit with positive Lyapunov exponent (see [17]). Mora and Viana [11] proved the existence of an attractor in H enon-like families, ('a)a, near a = 2 and small b, by proving the existence of a compact invariant set, , having a dense orbit and whose stable set W s( ) has non empty interior. 1.3. Du ng's equation Du ng [1918] introduced a nonlinear oscillator with a cubic sti ness term to describe the hardening spring e ect observed in many mechanical problems. Since then, the Du ng equation has become one of the widespread examples in nonlinear oscillation texts and research articles. While it describes a speci c physical (mechanical) phenomena, its importance stems from the fact that it serves as a representative model for investigating the e ects of perturbations on Hamiltonian dynamical systems with homoclinic orbits. Moreover, as it can be analyzed analytically, it serves as a test case to new ideas and methods in this eld. The summary presented in this section is based on the reviews of this subject in [1] and [2]. Moon and Holmes [1979,1980] introduced the damped, forced Du ng oscillator with a negative linear sti ness term, in the form: :: x + : x x+ x3 = cos(!t); (x; t) 2 <1 <1; (1.2) which describes the dynamics of a buckled beam or plate when only one mode of vibration is considered. Physically, represents the dissipation (the dumping), the amplitude of the forcing, and ! the frequency. Since ; and ! are physical parameters, they may be regarded as real, and using symmetries one can show that they may be taken to be non negative. Introducing the phase space coordinates (x; y) 2 <2; one rewrites (1.2) as: : x = y; : y = x x3 + "( cos(!t) y); (1.3) 4 where " is a "perturbation scaling parameter", assumed to be small. The unperturbed system, given by (1.3) with " = 0; is an integrable Hamiltonian system, with a potential: V (x) = x2 2 + x4 4 ; (1.4) which is symmetric, and with the Hamiltonian function (energy): H(x; y) = y2 2 + V (x) = y2 2 x2 2 + x4 4 : (1.5) The unperturbed system has three equilibrium ( xed) points: two centers at (x; y) = ( 1; 0), and a saddle at (x; y) = (0; 0). The saddle point is connected to itself by two homoclinic orbits, with periodic orbits nested within and around them (see gure (1.1)). -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -1.5 -1 -0.5 0 0.5 1 1.5 x y Figure 1.1: The phase space portrait of the unperturbed (symmetric) system. In the unperturbed system the stable and the unstable manifolds of the saddle point (0; 0) coincide. For > 0, and = 0; the unstable manifold of the saddle point near the origin falls into the two sinks created near ( 1; 0). It has been proved that for su ciently small values of (when the stable and the unstable manifolds do not intersect) the closure of the unstable manifold (which contains the saddle and the sinks) is an attracting set of (1.3). 5 A Poincar e map in time is used to simplify the phase space portrait for the time dependent system ( 6= 0). Keeping > 0, and increasing , the following scenario occurs; for small values of , the Poincar e map is topologically equivalent to the Poincar e map with = 0, which is structurally stable. As increases, resonance bands of higher period and higher amplitudes are created (the interaction between these is not completely understood). As continues to increase, in addition to the resonances, a homoclinic bifurcation occurs, after which the stable and the unstable manifolds of the saddle point of the Poincar e map intersect in transversal homoclinic orbits. The presence of these orbits implies the existence of a complicated nonwandering Cantor set which possesses in nitely many unstable periodic orbits of arbitrary long period as well as bounded nonperiodic motions. The Smale Birkho Homoclinic Theorem implies that in this case system (1.3) has a chaotic dynamics. Numerical evidences for the existence of strange attractors in the Du ng's equation (1.3) were presented more than a decade ago (see for example [1]), but there is not yet an analytical proof for their existence. It is easy to show that there exists a trapping region for (1.3), and that this ow possesses a compact invariant set, , with a sensitive dependence on initial conditions (the invariant sets of the horseshoes near the homoclinic points). In fact, it can be shown that is contained in the trapping region, and hence is contained in the attracting set, i.e. this system possesses a chaotic attracting set (for details, see [1], [2]). Therefore, to show that this equation exhibits a strange attractor, one needs "only" to show that the sensitive dependence on initial conditions extends to the whole attracting set, and that the attracting set is topologically transitive. Or, in other words, the main challenge in proving the existence of strange attractor in (1.3), is to show that the attracting set possesses a dense orbit with a positive Lyapunov exponent. The di culty to show that generically is due to the fact that periodic sinks are always associated with quadratic homoclinic tangencies, and Newhouse proved, that for two dimensional dissipative maps, these tangencies will persist (if one is ruined, another one is created in the homoclinic tangle). The issue of the structure and parameter dependence of the fractal basin boundaries and the interaction between attractors in systems with homoclinic tangle is currently being investigated in numerous papers, such as [13], [14], [15]. 6 1.4. The topological approximation method the TAM The TAM supplies a description of the mixing and transport processes in chaotic two-dimensional time-periodic Hamiltonian ows, [7], [8] . It is based upon the structure of the homoclinic tangle, and supplies a detailed solution to a transport problem for this class of systems. As a result of applying the TAM one gets a classi cation of the parameter space, which provides a description of the changes in the homoclinic tangle as the physical parameters are varied, and indicates a special regions in the parameter space, in which the escape rates from the vicinity of the homoclinic tangle can be approximated. The TAM may be divided to two main parts: perturbational and topological. The rst part consist of perturbation methods, and hence requires the system to be near integrable. The second part consist of geometrical analysis of the homoclinic tangle, hence applies for systems which are not nearly integrable as well. The perturbational part of the TAM uses the Secondary Melnikov Function [9] (see section 2.3.2), which was constructed as a derivation from the Whisker map (see section 1.5 and [3], [4], [5]). Till now, TAM was applied only to "open", non dissipative systems. Here, we modify and generalize the TAM, to be suitable for analysis of two dimensional, forced systems with dissipation, which satisfy the assumptions, listed in section 2.1. 1.5. The Whisker map Consider the separatrix map, W , which is constructed by the geometrical interpretation of the Whisker Map (see [3], [4], [5], [7], [8] and [9]), and de ned as the return map of the energy and time variables (Hn; n) to the cross sections H and respectively, where H is composed of a segment of the x axis and a segment of the y axis, centered at the origin, and consist of a segment of the x axis centered at q0 (0) and a segment of the x axis centered at q0 +(0) (see gure (1.2)): W : (Hn; n)! (Hn+1; n+1); q( n; 0) 2 ; (1.6) Hn = H(q(t n; 0); t n); q(t n; 0) 2 H ; n 1 < t n < n; where H( ; ) is the energy, q0 (t) are the homoclinic orbits of the system, q(t; 0) is a solution to the system, q( 0; 0) 2 . In gure (1.2) we draw a solution qu(t; 0) belonging to the unstable manifold of the left side of the origin, so limt! 1 qu(t; 0) = (0; 0). 7 Figure 1.2: The Separatrix map. In the neighborhood of the separatrix the cross sections H and are transverse to the unperturbed trajectories of the system. Therefore, for " su ciently small, the separatrix map is well de ned there and so is the parametrization of the unstable manifolds by 0. For jHnj 1 near the separatrix, the Whisker map is de ned to be the leading order approximation in " to the separatrix map: Hn+1 = Hn + "M n(tn) +O("2); tn+1 = ( tn + P ( Hn); Hn < 0 tn + 1 2P ( Hn); Hn > 0 ; (1.7) n+1 = n sign( Hn); 1 2 f+1; 1g; whereM 1(t) is the Melnikov function, calculated at q0 (t) respectively, and P ( H) is the period of the unperturbed periodic orbit of the system with energy H at H . In [9] it is shown that the behavior of stable and unstable manifolds can be analyzed in view of (1.7) for " su ciently small. Since q0 +(t) (respectively q0 (t)) refers to the right side (respectively, left side) of the hyperbolic xed point, we use the notations: M+1(t) Mr(t) and M 1(t) Ml(t). 8 2. The homoclinic tangle of slightly dissipative, forced, two dimensional systems 2.1. General assumptions The methods developed here hold for near-integrable near-Hamiltonian systems, which satisfy the assumptions stated below. Consider: dx dt = f1(x; y) + "g1(x; y; t; "; ; ) dy dt = f2(x; y) + "g2(x; y; t; "; ; ) (2.1) Where, f = (f1; f2); g = (g1; g2); is the dissipation parameter, and " is su ciently small (so that perturbation analysis can be applied). When the dissipation parameter, , equals to zero, (2.1) is Hamiltonian of the form: H"(x; y; t) = H0(x; y) + "H1(x; y; t; "; ); (x; y) 2M; 2 0 for H < 0 and P 0(H) < 0 for H > 0 (when is surrounded by periodic orbits qH(t), with period P (H), as well), and P 0(H) = c H (1 + o(H)) as H ! 0, with c = 1 when H < 0, and c = 2 when H > 0. A4. g(0; 0; t; "; ; ) = 0, and g is analytic in x; y near the origin. A5. The Melnikov function M(t0), which measures the signed distance between the stable and the unstable manifolds of p, de ned by: M(t0) = M(t0; ; ) = R1 1 f ^ g j(q0(t t0);t) dt = = R1 1(f1g2 f2g1) j(q0(t t0);t) dt (2.4) has at least two simple zeros in [0; T ), (each simple zero of the Melnikov function indicates a transverse intersection of the stable and unstable manifolds of p). Moreover, M(t0) = c has a nite number of solutions (possibly zero), M 1;i(c) for t0 2 [0; T ) for all real c. Also, M(t0) has a nite number of extrema in a period, and M 0(t0) is bounded away from zero on intervals which are bounded away from these extrema. 2.2. The Asymmetrically Forced Du ng Oscillator (AFDO) A model satisfying conditions A0-A5 is the asymmetrically forced, damped, Du ing oscillator 1Depending on the behavior of the other (WNLG right) branches of the manifolds, the system may be either "open" or "closed". See section 2.4 and appendix C, for de nitions and more details. 10 : x = y; : y = x x3 + (x x2)" cos(!t) " y : (2.5) There are two di erences between (2.5) and (1.3). The important di erence is that (2.5) includes the "asymmetry scaling parameter" <, which by symmetry can be considered as non negative. The second di erence is that for convenience, with no loss of generality, the origin is xed for all ": As in the case of (1.3), for parameter values for which the Melnikov function (2.12) has simple zeros, the Smale Birkho Homoclinic Theorem implies that (2.5) has a chaotic behavior. 6= 0 corresponds to symmetry breaking disturbances, which will be shown to produce new dynamical phenomena. As an example, consider again the dynamics of a buckled beam, described in section 1.3: if the distance between the magnets and the beam is not exactly symmetric, or if the magnets do not activate exactly the same force on the beam, one needs a slightly asymmetric system to analyze this dynamics. It is easily seen from (2.5), that AFDO satis es the assumptions A0-A4 as they appear in section 2.1. Where H0 is as in (1.5), H" = H0 "(x2 2 x3 3 ) cos(!t) = H0 + "H1; f(x; y) = (@H0 @y ; @H0 @x ); g(x; y; t; "; ; ) = (0; @H1 @x y); and the hyperbolic xed point p being the origin. Since the origin is an hyperbolic xed point of (2.5), it follows that A3 is satis ed. We compute P (H) here for future references. The conditions needed for the AFDO to satisfy assumption A5 will be stated in section 2.2.2. 2.2.1. P (H) the period function of the unperturbed periodic orbits The Period P (H) of the unperturbed periodic orbits for this system is given by: P (H) = 8>>>><>>>: 2K(k)p2 k2; H(k) = k2 1 (2 k2)2 ; H < 0; 0 < k < 1 4K(k)p2k2 1; H(k) = 1 k2 1 (2 1 k2 )2 ; H > 0; 1 p2 < k < 1 ; (2.6) where K(k) is the complete elliptic integral of the rst kind. 11 Inverting (2.6), the following formulae for k(H) are found: k(H) = 8>>>><>>>>: s4H + 1 p4H + 1 2H ; 0:25 < H < 0 s12(1 + 1 p4H + 1); H > 0 ; (2.7) hence, by substituting (2.7) in P (H) of (2.6), we get: P (H) = 8>>>><>>>>: 2K(s4H + 1 p4H + 1 2H )s2 4H + 1 p4H + 1 2H ; 0:25 < H < 0 4K(s12(1 + 1 p4H + 1))s 1 p4H + 1 ; H > 0 : (2.8) And by asymptotically expanding (2.8) nearH = 0, we obtain the approximations: P (H) = 8>>><>>>: ln( 16 H )(1 +O(H)); H ! 0 2 ln(16 H )(1 +O(H)); H ! 0+ : (2.9) And thus the asymptotic expansion of the derivative of P (H) is: P 0(H) 8><>: 1 H ; H ! 0 2 H ; H ! 0+ : (2.10) Note that (2.10) meets the requirements of A3 in section 2.1 above. 2.2.2. The Melnikov function To see if A5 is satis ed, we calculate the integral of the Melnikov function of (2.5), along the homoclinic orbits : q0(t) = (p2 sech t; p2 sech t tanh t); (2.11) and get: "Mr;l(t0; ; !; ; ) = " sin(!t0)Fr;l(!; ) 43" ; (2.12) 12 with: Fr(!; ) = [F1(!) F2(!)]; (2.13) Fl(!; ) = [F1(!) + F2(!)]; (2.14) F1(!) = !2 csch( ! 2 ); (2.15) F2(!) = p2 3 !(1 + !2) sech( ! 2 ); (2.16) where Mr(t0; ; !; ; ) is referring to the right side of the saddle point, and Ml(t0; ; !; ; ) to the left side. See gure (2.1) of F1(!); F2(!), and the rela0 1 2 3 4 5 0 0.5 1 1.5 : F : F F ω 1 (ω) 2(ω) 0 1 2 3 4 5 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 F2/F1 ω Figure 2.1: Plot of F1(!); F2(!) fuctions of the Melnikov function, and of the relation F2(!) F1(!). tion F2(!) F1(!) . Notice that F1(!) and F2(!) are non-negative for all !. WNLG, for the convenience of notation the Melnikov function (2.12) was calculated at the Poincar e cross section of 0 = 0, therefore the notion of 0 was omitted. From (2.12) we conclude that assumptionA5 is satis ed if the parameter is su ciently small; If 3 jF1(!) F2(!)j 4 < 3 [F1(!) + F2(!)] 4 ; (2.17) then the left branches of the stable and the unstable manifolds intersect, while the right branches do not. See for example gure (2.2), which shows that primary 13 homoclinic intersections of the stable and the unstable manifolds exist on left side of the origin, but not on right side of the origin. If 3 jF1(!) F2(!)j 4 ; (2.18) then the stable and unstable manifolds intersect on both left and right sides of the hyperbolic xed point (see for example gure (2.13)). −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 x y ω=1.03, γ=1, β=0.1, δ=0.95, ε=0.4 Figure 2.2: Intersections of stable and unstable manifolds. -: stable manifold, |: unstable manifold. 2.3. Behavior of the stable and unstable manifolds 2.3.1. Primary homoclinic intersection points General formulation For " > 0 su ciently small, simple zeros (respectively quadratic zeros) in t0 of the Melnikov function imply the existence of transverse homoclinic intersections (respectively quadratic tangencies), called primary intersection points PIP (respectively primary tangencies). By our assumptionA5 (section 2.1), the Melnikov function has simple zeros in t0 for some region of the parameter space. On the other hand, for su ciently large values of (with all other parameters held xed), 14 the Melnikov function has no zeros in t0. Therefore, since the Melnikov function is a smooth function of its arguments (by assumption A0 and the de nition of the Melnikov function), M(t0) has quadratic zeros in t0. Solving: Mc(t0; ; ) = 0; 2 R+0 (!; ) primary intersections of the stable and the unstable manifolds occur both on the left and the right sides of the saddle point (like in (1.3), but in an asymmetric manner for 6= 0). The relation between R 0 (!; ) and R+0 (!; ) is: R 0 (!; ) R+0 (!; ) = 1 F2(!) F1(!) 1 + F2(!) F1(!) l r (!; ); (2.22) hence the size of region II depends on the values of F2(!) F1(!) , and may occupy a large section of the parameter space. Setting F2(!) F1(!) x, the behavior of the function r(x) = 1 x 1 + x (see gure (2.4)), will indicate the size of region II for di erent values of the parameters and . For large !, F2(!) F1(!) grows linearly with ! (see gure (2.1)). Therefore, since for x < 1, r(x) strictly decreases with x, 16 the relation R 0 (!; ) R+0 (!; ) linearly decreases with !, and the size of region II linearly increases with ! for xed, small values of and for ! large. Notice that for small , region II is of nonvanishing measure even for small values of ! (see for example gures (2.4) and (2.5)). For x > 1, r(x) strictly decreases with x, hence for xed, large and for ! 1, R 0 (!; ) R+0 (!; ) will linearly increase with !, and the region II will decrease as ! will increase. At x = 1, r(x) = 0, which is equivalent to having = F1(!) F2(!), hence for any xed 0 < < max!(F1(!) F2(!)), there exists an ! such that R+0 (!; )!1 as ! ! ! . For these values, region II occupies most of the region above region I, and region III shrinks till it disappears at ! = ! , to order ". 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ω γ /γ l r 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x |(1 -x )/( 1+ x) | Figure 2.4: The relation l r (!; ) for small ( = 0:1), and the function r(x) = 1 x 1 + x . The same homoclinic bifurcation is plotted in gure (2.5), in parameter space ( ; !), with xed . From this plot the relations between ; and ! needed for prime intersections of the stable and unstable manifolds on the left and/or right sides of the saddle may be read o , for each !. In gure (2.5) we also see the division of the parameter space to the three regions described above. 2.3.2. Secondary homoclinic intersection points It is necessary to nd the secondary homoclinic intersection points in order to understand the minimal development of the homoclinic tangle. If the dissipation parameter belongs to the interval of values for which the Melnikov function Ml(t0; ; ) (respectivelyMr(t0; ; )) has two simple zeros (see section (2.3.1)), we denote the corresponding PIP's, ordered by the direction of the unstable manifold, by pl0 (respectively pr0) and ql0 (respectively qr0). Also, 17 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 δ/γ ω β=0.1 : ω : ω r l I I II III Figure 2.5: Bifurcation curves in the parameter space ( ; !). let us denote their ordered images by the Poincar e map F by pli; qli (respectively pri; qri), i = 0; 1; 2; 3; :::, i.e. F i(pl0) = pli and so on. The areas enclosed by the segments of the stable and the unstable manifolds connecting two successive PIP's are called lobes. Denote the lobes enclosed by segments of the stable and the unstable manifolds connecting pli; qli by Dli, and ones that between qli; pli+1, by Eli, when again, Dli = F i(Dl0); Eli = F i(El0) (we use the equivalent notations for the right side). See gure (2.6) as an example. If Elj \Dl0 6= ; (respectively Erj \Dr0 6= ;) for some non negative integer j, or if Dlk+1\Er0 6= ; (respectivelyDrk+1\El0 6= ;) for some non negative integer k, then there exist secondary intersection points SIP in these intersections. The minimal integers j; k for which this happens on the left side of the hyperbolic xed point are called the structural indices `ll; `lr (respectively `rr; `rl). Each such structural index imposes minimal complexity for the structure of the homoclinic tangle, and the parameter space may be divided to areas, according to the values of these indices. Next we describe how secondary homoclinic points and those indices may be calculated for systems which meet our assumptions, perturbationally. Then we discuss the scenarios they induce in di erent regions of the parameters space. We will illustrate the methods we use by performing the calculations for the AFDO. Consider the Secondary Melnikov Function SMF (see section 2.4, and [7], 18 −1.5 −1 −0.5 0 0.5 1 1.5 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 x y Dl0 El SIP 0 Dr Er0 0 Dl1 El -1 Er -1 Dr 1 Dl -1 El 1 Er 1 Dr -1 Lobe qr0 SIP pr0 Figure 2.6: Example of an homoclinic tangle of AFDO. [8]): hcd 2 (t0; ") = Mc(t0) +Md(t1cd(t0; ")); c; d 2 fl; rg (2.23) (see [7], [8], [9]), where Mc(t) is the Melnikov function, t1cd(t0; ") = ( t0 + P ("Mc(t0)); Mc(t0) < 0 t0 + 12P ("Mc(t0)); Mc(t0) > 0 ; c; d 2 fl; rg; (2.24) and P (H) is the period of the unperturbed periodic orbits of the system, where H = 0 on the separatrix. If a system satis es the assumptions of section 2.1, then for su ciently small ", simple zeros (respectively degenerate zeros) of (2.23) imply transverse secondary homoclinic intersections (respectively tangencies) with a transition number: jcd(t0; ") = "t1cd(t0; ") T # s(t0); 0 t0 < T; (2.25) where [x] is the integer part of x, T is the period of the perturbation and s(t0) is either 0 or 1, depending on the interval to which t0 belongs (as described below). The structural index `cd (c; d 2 fl; rg) is de ned to be the minimal transition number jcd(t0; "). This analytical de nition of the structural index meets the geometrical one. 19 It was also proved (see [9]) that under some generic conditions on hcd 2 (t0; "; ; ), the structural indices satisfy: `cd = jcd(ti0cd; "i0cd); c; d 2 fl; rg; (2.26) where (ti0cd; "i0cd) ("i0cd small, i = 1; 2) are the solutions to the equations: hcd 2 (t0; ") = 0; (2.27) @hcd 2 (t0; ") @t0 = 0; (2.28) and ti0cd 2 [0; T ) ; i = 1; 2; :::;m; c; d 2 fl; rg; "10cd < "20cd (2.29) ti1cd(ti0cd; "i0cd) 2 [(jcd(ti0cd; "i0cd) + s(ti0cd))T; (jcd(ti0cd; "i0cd) + s(ti0cd) + 1)T ); s(ti0cd) = 8<: 0; ti0cd 2 h0; T2 1; ti0cd 2 hT2 ; T : (2.30) Typically, for "i0cd su ciently small, the solutions (ti0cd; "i0cd) of (2.27) and (2.28), divide the parameter space to three regions: below the surface " = "10cd(t10cd; ; ; `cd) there are no SIP's, between the surfaces " = "10cd(t10cd; ; ; `cd) and " = "20cd(t20cd; ; ; `cd) two SIP's occur (see for example gures (2.13)-(2.15)), and above the surface " = "20cd(t20cd; ; ; `cd) additional two SIP's, to the existing ones, occur (see for example gure (2.16)). 2.3.3. Secondary homoclinic bifurcation diagrams General construction The secondary homoclinic bifurcation types related to SIP's of symmetric and asymmetric, dissipative and non dissipative systems are essentially di erent. We discuss these di erences below for the general systems described in section 2.1, and demonstrate them by presenting the formulae and bifurcation diagrams in (t0; ") space for di erent values of the asymmetry and dissipation parameters, and , of the AFDO. If one of the perturbation parameters, i0 ; 1 i0 k, in a system corresponding to the description in section 2.1 is referring to symmetry breaking 20 disturbances2, we denote it by i0 . When this asymmetry parameter, , and the dissipation parameter, , are both equal to zero, then we assume that the bifurcation diagram in (t0; ") space, corresponding to a pitchfork bifurcation diagram (see, for example, gure (2.7)) (this assumption is satis ed, for example, if M(t0) is odd function). However, if 6= 0 or/and 6= 0, the pitchfork bifurcation diagram typically splits to bifurcation diagrams of other forms, described below, and these may also appear for = = 0 and non odd Melnikov function. These splitting of the bifurcation diagrams in (t0; ") space, indicate us how the SIP's should be calculated (see section 2.3.4 and appendices A,B for details), and some of the spliting forms refer to changes in the geometry of the lobes intersections (see below). From (2.23) (2.27) we construct the equations: ti1cd(t0) = M 1;i d ( Mc(t0)) + jcdT; (2.31) "icd(t0) = 8>><>>: P 1(ti1cd(t0) t0) Mc(t0) ; c = d P 1(2(ti1cd(t0) t0)) Mc(t0) ; c 6= d (2.32) where c; d 2 fl; rg; i = 1; 2; jcd 2 N. From (2.26), (2.29) and (2.30) we get that: t0 2 [0; T ); jcd = `cd + s(t0), where s(t0) = 0 for t0 2 [0; T2 ), and s(t0) = 1, for t0 2 [T2 ; T ). And, we assume the above equations ((2.31) and (2.32)), to be dependent on the perturbation parameters: (asymmetry), (dissipation) and 2 <>: 0; t0 2 [0; ! ) 1; t0 2 [ !; 2 ! ) ; c; d 2 fl; rg; (2.39) "icd(t0) 8>>>><>>>: 16 exp(t0 ti1cd(t0)) Fc(!) sin(!t0) 4 3 ; c = d; "icc(t0)[ Fc(!) sin(!t0) 4 3 ]! 0 16 exp(t0 ti1cd(t0)) Fc(!) sin(!t0) 4 3 ; c 6= d; "icd(t0)[ Fc(!) sin(!t0) 4 3 ]! 0+ ; i = 1; 2: Remark 1. Fc; Fd are as in (2.13), (2.14) of section 2.2, for c; d 2 fl; rg, and "icd(t0) was calculated here with the use of the approximated value of the period function of AFDO, P (H), from (2.9) (see section 2.2.1). Since the inverse function of the period function of the AFDO, P 1(x), cannot be found analytically, to nd "icd(t0); i = 1; 2 by use of the actual value of P (H) in (2.32), one can solve for "(t0), with Newton's method, the equations: P ("icc(t0)Mc(t0)) = ti1cc(t0) t0; Mc(t0) < 0 (2.40) P ("icd(t0)Mc(t0)) = 2(ti1cd(t0) t0); Mc(t0) > 0; c 6= d; (2.41) using "icd(t0) of (2.39) as the initial guess. The indices c; d in table 1 for the AFDO are: c = l; d = r. For details, additional gures and explicit description of intervals of de nition for the solution curves "icd(t0); i = 1; 2 to equation (2.27), as they appear in (2.39), for the AFDO, see appendix A. As another illustration for the geometrical behavior of the lobes when the "gap scenario" occurs in (t0; ") space, see gure (2.10). In this gure the intersections of El1 \Dl0 and Er1 \Dr0 at two SIP's are shown, where the "tip" of the lobe El1 is outside the lobe Dl0, and the "tip" of the lobe Er1 is outside the lobe Dr0. 2.3.4. Secondary homoclinic bifurcation points and curves In this section, we describe how the secondary homoclinic bifurcation points may be found perturbationally in ", for systems that satisfy our assumptions. We apply the methods to the AFDO, calculating the secondary homoclinic bifurcation points for di erent values of the asymmetry parameter, , and of the dissipation 26 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x y ω=1.63, γ=1, β=0.01, δ=0.95, ε=0.21 Figure 2.10: Secondary homoclinic intersections of the stable and the unstable manifolds of the origin for the AFDO. |: unstable manifold, -: stable manifold. parameter, . The calculation supplies bifurcation co-dimension-one surfaces in the parameter space, labeled by the structural indices `cd (c; d 2 fl; rg). Thus, for each structural index `cd, the parameter space is divided to regions, as was described at the beginning of section 2.3.2. General construction It follows from section 2.3.3, that the secondary homoclinic bifurcation points ((2.37) and (2.38)) are the minimal solutions (t0; "icd) for which the SMF vanishes (equation (2.27)), when t1cd(t0; ") (de ned in (2.24)) is kept in a given period (see (2.29), (2.30) and (2.26)). Therefore, if "icd(t0) from equation (2.32) is di erentiable in an open neighborhood of the minimizing t0, then the minimum may be found using a Newton's method, combined with a continuation method. We rewrite equations (2.27) and (2.28) as: P ("Mc(t0)) = i cd(t0) (2.42) @P ("Mc(t0)) @t0 = @ i cd(t0) @t0 ; (2.43) 27 where: i cd(t0) = ( M 1;i d ( Mc(t0)) + (`cd + s(t0))T t0; Mc(t0) < 0 2(M 1;i d ( Mc(t0)) + (`cd + s(t0))T t0); Mc(t0) > 0 ; c; d 2 fl; rg; i 2 f1; 2g; t0;M 1;i d ( Mc(t0)) 2 [0; T ); (2.44) s(t0) = ( 0; t0 2 [0; T2 ) 1; t0 2 [T2 ; 0) : In addition, by de nition of the SMF and assumption A3, for jHj = j"Mc(t0)j 1 we get (see equations (2.27), (2.28) and section 2.1): @hcd 2 (t0; ") @t0 M 0 c(t0) +M 0 d(t1cd)(1 M 0 c(t0) Mc(t0)) = 0; c; d 2 fl; rg: (2.45) And substituting ti1cd(t0) (from equation (2.31)) gives: M 0 c(t0) +M 0 d(M 1;i d ( Mc(t0)))(1 M 0 c(t0) Mc(t0)) = 0; i = 1; 2: (2.46) Substituting the solutions ti0cd of equation (2.46) into (2.32) with j"icdMc(ti0cd)j 1 we conclude: "icd(ti0cd) 8>>>><>>>: C exp(ti0cd M 1;i d ( Mc(ti0cd)) (`cd + s(ti0cd))T ) Mc(ti0cd) ; c = d; C exp(ti0cd M 1;i d ( Mc(ti0cd)) (`cd + s(ti0cd))T ) Mc(ti0cd) ; c 6= d; (2.47) s(ti0cd) = ( 0; ti0cd 2 [0; T2 ) 1; ti0cd 2 [T2 ; T ) ; C = constant : By the construction of equations (2.31), (2.32), such (ti0cd; "icd(ti0cd)) approximate the secondary homoclinic bifurcation points ((2.37) for i = 1, and (2.38) for i = 2). These approximations are used as the initial guesses in a Newton's method, which is applied with a continuation scheme, for solving equations (2.42) and (2.43) simultaneously. If "icd(t0) cannot be di erentiated in an open neighborhood of the minimizing t0, then the secondary homoclinic bifurcation point, (2.37), can be found by substituting into "1cd(t0) the smallest bounding value, t0, of the region of its de nition 28 in [0; T ) (equation (2.32), with t11cd(t0) from equation (2.31)). Similarly, (2.38) can be found by substituting into "2cd(t0) the largest bounding value, t0, of the region of its de nition in [0; T ). These values of t0 make the equality in (2.33) with c = d, or respectively, in (2.34) with c 6= d, to hold, and they are found to be: t10cc = M 1;1 c ( max t02[0;T )Mc(t0)); (2.48) t20cc = M 1;2 c ( max t02[0;T )Mc(t0)); (2.49) t10cc < t20cc; and for c 6= d: t10cd = M 1;1 c ( min t02[0;T )Md(t0)); (2.50) t20cd = M 1;2 c ( min t02[0;T )Md(t0)); (2.51) t10cd < t20cd: Keeping j"icdMc(ti0cd)j 1; c; d 2 fl; rg, and substituting (2.48) or (2.50) into "1cd(t10cd) from (2.47), gives the approximated secondary homoclinic bifurcation points (t10cd; "1cd(t10cd)), and similarly, substituting (2.49) or (2.51) into "2cd(t20cd), gives the approximated second secondary homoclinic bifurcation points (t20cd; "2cd(t20cd)). The approximated bifurcation points (t10cd; "1cd(t10cd)) and (t20cd; "2cd(t20cd)), with ti0 from (2.48) (2.51), are used as the initial guesses in a Newton's method, which is applied with a continuation scheme, to solve (only) equation (2.42). Application to AFDO We use here our conclusions from section 2.3.3 and appendix A, regarding the AFDO, to nd approximations to the secondary homoclinic bifurcation points. For j"Ml;r(t0)j 1, equation (2.46) for the AFDO (equation (2.5)) is: Fc(!; ) Fd(!; ) sin(!t0) + cos(!ti1cd(t0))[tan(!t0) ! sin(!t0) sin(!t0) 4 3 Fc(!; ) ] = 0; (2.52) 29 where c; d 2 fl; rg; i = 1; 2 and ti1cd(t0) is from (2.39). For the AFDO with = 0 and = 0, equation (2.52) is solved analytically (for details see appendix B), and its solutions are: t10cd(!) = 1 ! arctan(!2 ); c 6= d; t10cc(!) = 1 ! [ + arctan(!2 )]; t20cd(!) = 2! ; c 6= d; t20cc(!) = 3 2! : (2.53) Noticing that substitution of t10cd and t10cc (respectively t20cd and t20cc) into "icd(t0) from (2.39) gives the same values of "1cd (respectively "2cd), we get the approximated secondary homoclinic bifurcation values: "1cc(!; ; `cc) 16 exp( 2 ! arctan(!2 ) (`cd + 1)2 ! ) !F1(!) p4 + !2 (2.54) "1cd(!; ; `cd) "1cc(!; ; `cc); c 6= d; "2cc(!; ; `cc) 16 exp( (`cd + 1 2)2 ! ) F1(!) (2.55) "2cd(!; ; `cd) "2cc(!; ; `cc); c 6= d 2 fl; rg: And since = 0, "ill(!; ; `ll) "irr(!; ; `rr) and "ilr(!; ; `lr) "irl(!; ; `rl) for i = 1; 2. Using (ti0cd(!); "icd(!; ; `cd)) (c; d 2 fl; rg, i = 1; 2) as the initial guesses for solving by a Newton's method with a linear prolongation method equations (2.42) and (2.43), secondary homoclinic bifurcation surfaces, "icd(ti0cd(!); !; ; `cd), are obtained in parameter space. In gure (2.11), we show the secondary homoclinic bifurcation curves ( xed) for the AFDO with = 0 and = 0 in parameter space (!; "). Remark 2. "icc(!; ; `cc) and "icd(!; ; `cd) are identical in (2.54) and (2.55) because only leading term in H of the period function, P (H), has been used in the calculation (see equation (2.9)). In fact, using the elliptic function for P (H) (equation (2.8)) shows that "icc(!; ; `cc) 6= "icd(!; ; `cd). See gure (2.11). 30 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ=1, β=0, δ=0 ω ε l=0 l=1 l=2 Figure 2.11: Secondary homoclinic bifurcation curves for the AFDO. |: "1lr, -: "2lr, : "1ll, : "2ll. For = 0 and c = d, equation (2.52) is the same for 6= 0 as for = 0. Hence, in this case, the approximated bifurcation points can be found in similar way to the way described above for = 0; = 0. However, in all other cases, equation (2.52) cannot be solved explicitly if 6= 0 or/and 6= 0. If "icd(t0); c; d 2 fl; rg; i = 1; 2 from (2.39), can be di erentiated in an open neighborhood of the minimizing t0, equation (2.46) may be solved for t0 near t10cd(!) (from equations (2.53)) to some order in near zero or/and near zero, and near t20cd(!) to some order in 1 3 near zero or/and 1 3 near zero (see appendix B for details). The obtained approximations may be used as the initial guesses for a Newton's method, applied with a linear prolongation method, to solve equations (2.42) and (2.43) simultaneously. Then the obtained solutions may be plotted in the parameter space (!; "), to get secondary homoclinic bifurcation curves, labeled by the structural indices, `cd. See, for example, gures (2.13) (2.16). To nd the secondary homoclinic bifurcation points, (t10cd; "1cd(t10cd)); c; d 2 fl; rg, we use the approximated solutions to equation (2.52), t10cd(!; ; ; ), found by perturbational expansions of equation (2.52) near = 0; = 0 (see appendix B), with "1cd(t10cd;!; ; ; ; `cd) from (2.39), as initial guesses for Newton's method. Near t20cd(!) from (2.53), regular perturbation expansions fails, and expansions in 13 ; 1 3 need to be used to nd approximations to the second secondary homo31 clinic bifurcation points, (t20cd; "2cd(t20cd)). We used as initial guesses for Newton's method, the approximations in 1 3 or/and 13 small, which are listed in appendix B, only in cases which correspond to occurrence of the "discontinuity point scenario". In cases which correspond to occurrence of the "gap scenario", we used as the initial guesses for t20cd in Newton's method, the boundary values of the "gap", near which the bifurcation points (2.38) occur (see appendices A,B): t20ll(!; ; ; ) = 1 ! (2 arcsin(1 8 3 Fl(!; ))); (2.56) t20rr(!; ; ; ) = 1 ! ((1 + v) arcsin(1 8 3 jFr(!; )j)); (2.57) v = ( 1; Fr(!; ) > 0 0; Fr(!; ) < 0 ; and t20lr(!; ; ; ) = 1 ! ( arcsin( jFr(!; )j Fl(!; ) + 8 3 Fl(!; ))): (2.58) We preferred to use t20cd(!; ; ; ) from equations (2.56) (2.58), as initial guesses for Newton's method (with "2cd(t20cd) from (2.39)), since they may be used for large values of and , and they present good approximations. If 8 3 jFc(!; )j > 1,then "icc(t0); i = 1; 2 from (2.39) is not di erentiable in t0, in an open neighborhood of the minimizing t0 (see gure (A.7) in appendix A). Hence, by inequalities (A.3), presented in appendix A, case I.2, the secondary homoclinic bifurcation points, (t10cc; "1cc(t0)), occur at: t10cc(!; ; ; ) = 1 ! (v + arcsin( 1 + 8 3 jFc(!; )j)); (2.59) v = ( 0; Fc(!; ) > 0 1; Fc(!; ) < 0 ; and the second secondary homoclinic bifurcation points, (t20cc; "2cc(t0)), occur at: t20cc(!; ; ; ) = 1 ! ((v + 1) arcsin( 1 + 8 3 jFc(!; )j)); (2.60) v = ( 0; Fc(!; ) > 0 1; Fc(!; ) < 0 : 32 Since in this case ti0cc(!; ; ; ) (i = 1; 2), from (2.59) and (2.60), do not present solutions to equation (2.52), we solve only equation (2.42), by a Newton's method with a linear prolongation method, using these approximations and "icc(ti0cc;!; ; ; ; `cc) from (2.39) as initial guesses. Following these steps, secondary homoclinic bifurcation curves, "icc(ti0cc; !; ; ; ; `cc); c 2 fl; rg, i = 1; 2, may be obtained in parameter space (!; "), as shown for example in gure (2.12). 0.8 1 1.2 1.4 1.6 1.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ=1, β=0.01, δ=0.95 ω ε ll l=0 l=1 l=2 0.8 1 1.2 1.4 1.6 1.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ=1, β=0.01, δ=0.95 ω ε rr l=0 l=1 l=2 Figure 2.12: Secondary homoclinic bifurcation curves for the AFDO. |: "1ll, -: "2ll, : "1rr, : "2rr. In this special case (i.e. when "icc(ti0cc) is not di erentiable), for "1cc(t10cc; !; ; ; ; `cc) " < "2cc(t20cc; !; ; ; ; `cc), hcc 2 (t0; ") has only one zero in [0; T ). Since two SIP's correspond to two zeros of hcc 2 (t0; ") in [0; T ), in this case only for " > "2cc(t20cc; !; ; ; ; `cc), two SIP's may occur (compare, for example, the secondary homoclinic bifurcation curves, presented in gure (2.12) and the numerical calculation of the manifolds, presented in gure (2.10)). As " grows, additional SIP's may occur as well (as can be seen from gure (A.7)). We do not calculate here for which values of the parameters this happens, but it may be calculated by similar methods to the ones presented here. Comparison between numerical and analytical results We get satisfying agreement between our analytical results, and numerical calculations (using Runge-Kutta method) of the stable and unstable manifolds for the AFDO, for " su ciently small and `cd 1; c; d 2 fl; rg. See for example gures (2.13) 33 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ=1, β=0.1, δ=0.05 ω εlr l=0 l=1 l=2 −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y ω=2.1, γ=1, β=0.1, δ=0.05, ε=0.108 Figure 2.13: Secondary homoclinic bifurcation curves and manifolds intersections for the AFDO. |: "1lr, -: "2lr, -: stable manifold, |: unstable manifold. (2.16). In these gures we show the analytical prediction for occurrence of SIP's on left and right sides of the saddle outside and inside the "separatrix", versus numerical calculations of the stable and unstable manifolds (which show the geometrical behavior of the tangle). The at the (!; ") parameter space, indicates the parameters values for which the manifolds were calculated. The Whisker map (see section 1.5) was used for the construction of the above described methods, and hence these methods fail if the manifolds pass through a 1 : 1 resonance relation. The 1 : m resonance relation for the periodic orbits of equation (2.5), as for equation (1.3), is given by: P (H) = 2 m ! (see [1], [2]), where m is a prime integer and P (H) is from equation (2.8) in section 2.2.1. Since by de nition of t1cc from (2.24), P ("ccMc(t0cc)) = t1cc t0cc, and by conditions (2.29) and (2.30), t1cc t0cc 2 [`cc 2 ! ; (`cc + 1)2 ! ), the manifolds might pass trough 1 : 1 resonance for `cc = 0. Hence, the results that we can get by the analytical method for `cc = 0 might be inaccurate even for small values of ". To avoid passage of the manifolds trough 1 : 1 resonance, " should satisfy the condition: "cc < P 1( 2 ! ) Mc(t0cc) . This condition holds for `cc 1. Comparing between our analytical predictions for `ll = 0, in gure (2.11) and the numerical calculation of the manifolds, shown in gure (2.6), one can see that our analytical prediction fails in this case. However, 34 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ=1, β=0.1, δ=0.05 ω εrl l=0 l=1 l=2 −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y ω=2.1, γ=1, β=0.1, δ=0.05, ε=0.115 Figure 2.14: Secondary homoclinic bifurcation curves and manifolds intersections for the AFDO. |: "1rl, -: "2rl, -: stable manifold, |: unstable manifold. 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ=1, β=0.1, δ=0.05 ω ε l=0 l=1 l=2 ll −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y ω=2.1, γ=1, β=0.1, δ=0.05, ε=0.13 Figure 2.15: Secondary homoclinic bifurcation curves and manifolds intersections for the AFDO. |: "1ll, -: "2ll, -: stable manifold, |: unstable manifold. 35 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ=1, β=0.1, δ=0.05 ω εrr l=0 l=1 l=2 −1.5 −1 −0.5 0 0.5 1 1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x y ω=2.1, γ=1, β=0.1, δ=0.05, ε=0.16 Figure 2.16: Secondary homoclinic bifurcation curves and manifolds intersections for the AFDO. |: "1rr, -: "2rr, -: stable manifold, |: unstable manifold. for small values of ! (for example, ! = 0:55), where "cc is very small ("1cc 0:00185), our analytical predictions for `cc = 0 improve. 2.4. Symbolic dynamics, constructed of segments of the unstable manifold In this section we construct symbolic dynamics of the lobes for mappings which contain homoclinic tangles. This construction of the symbolic dynamics follows that of the TAM (see [6], [7] and [8]). The assumption that the system is nearly integrable is not required here. Here, the assumptions on the mapping, F , are topological: We assume there exist a nite `ll (WNLG), such that F j(El0)\Dl0 6= ; only for j `ll, we then distinguish between "open" and "closed" systems (see de nitions below and appendix C). In our discussion here, we refer to the PIP's pli; qli; pri; qri, the lobes Eli;Dli; Eri;Dri; i 0, that were de ned at the beginning of section 2.3.2 (see gure (2.6)). If the branches of the stable and the unstable manifolds on each side intersect each other in PIP's (e.g. in the near integrable case both Melnikov functions related to the system, Ml(t0) and Mr(t0), have simple zeros), we de ne three regions: region 36 Rl and region Rr, which are the regions enclosed by the segments of the stable and unstable manifolds, connecting pl0 (respectively pr0) and the hyperbolic xed point, and region Ro, which is the complimentary region to Rl [ Rr and has a common boundary with Rl [Rr (see gure (2.17)). Rl; Rr are "unigate" regions, and Ro is a "multigate" region. An "unigate" region, is a region that contains only one turnstile lobe, and a "multigate" region may contain more than one turnstile lobes. A turnstile lobe is an area that satis es the condition: Lij Rj and F 1(Lij) Ri, where F denotes the mapping and Rk denotes the de ned regions (see [7]). Figure 2.17: Schematic sketch of the regions Rl; Rr; Ro and the states f cd; gcd; c; d 2 fl; rg. If only the left branches of the stable and the unstable manifolds intersect in a PIP (e.g. only one of the Melnikov functions, Ml(t0), has simple zeros), yet `lr is nite, then the region Rr is de ned to be the region enclosed by segments of the stable and the unstable manifolds, connecting the rst created secondary 37 homoclinic intersection point (see section 2.3.2), slr 0 , with the hyperbolic xed point (see gure (2.18)). Figure 2.18: Schematic sketch of the regions Rl; Rr; Ro for a "closed" system with the Melnikov function Mr(t0) having no zeros. And nally, if the right branches of the stable and the unstable manifolds do not intersect (e.g. Mr(t0) has no zeros), and `lr is in nite, then regionRr is de ned to be the region enclosed by the y-axis and a segment of the stable manifold on the right side of the hyperbolic xed point3. De nition 1. If there exist a segment, U , of the left branch of the unstable manifold which has left region Rl at some iteration, n (F n(U) \ Rl = ;), and which returns to this region after n + m iterations of F (F n+m(U) \ Rl 6= ;; n;m 2 N), then the system is called "closed". De nition 2. If all segments of the left branch of the unstable manifold which have left region Rl, never return to this region, the system is called "open". Remark 3. A su cient condition for a system to be "open" is: In the unperturbed system, the branches of the stable and unstable manifolds coincide on the 3In case of the one sided tangle (for example, the homoclinic tangle of the Cubic equation), Rr is not de ned. 38 left side of the hyperbolic xed point, creating a homoclinic loop, and extend to in nity on the right side of the hyperbolic xed point (without ever folding back to the left side of the xed point). This is true since the unstable manifold may not intersect itself. See, for example, gure (2.19). This is the de nition taken in [8] for "open" ow. If the branch of the stable manifold on the right side of the hyperbolic xed point folds back and extend to the left side of the hyperbolic xed point, then the system may be either "closed" or "open". Figure 2.19: A schematic phase space portrait of an unperturbed ow which satis es the su cient condition of an "open" ow. A type-f`cdg trellis (c; d 2 fl; rg) is de ned to be the homoclinic tangle with structural indices `lr; `ll; `rl and `rr, near the rst corresponding bifurcation curves, "1cd (see section 2.3.4). If the system is "open", `rr; `rl and `lr are not de ned (in nite). The type(`cd;mcd; kcd; 0) trellis is de ned to be the homoclinic tangle with a structural index `cd, after the second secondary homoclinic bifurcation has occurred, which approach the type-`cd trellis 'from above' as mcd; kcd !1. The indices mcd refer to lobes of type Ec mcd (respectively Dc mcc), which contain the "tips" of the lobes Dc`cd+1 (respectively Ec`cc). And, the indices kcd refer to the "tips" of lobes of type Ec kcd (respectively Dc kcc ), which contained in the lobes Dc`cd+1 (respectively Ec`cc). Finally, the type-(`cd 1; ucd; vcd; 1) trellis is de ned to be the homoclinic tangle with a structural index `cd 1, after the second secondary homoclinic bifurcation has occurred, which approach the type`cd 1 trellis 'from below' as ucd; vcd !1. The indices ucd refer to lobes of type Ec ucd (respectively Dc ucc), which contain the "tip" of lobes Dc`cd (respectively Ec`cc 1), and the indices vcd refer to "tips" of lobes Ec vcd (respectively Dc vcc), which contained in the "tip" of lobes Dc`cd (respectively Ec`cc 1). In the TAM, 39 symbolic dynamics of the lobes was constructed for structural indices `cd 1, with the assumption that the trellises do not develop any spontaneous homoclinic intersections points. With these assumptions, the symbolic dynamics of the lobes enables the description of the minimal development of the homoclinic tangle with structural indices `cd 1. However, for `cc = 0 we have observed that the basic templet is di erent then the standard one, assumed in [8] for `cc 1. The origin of this new structure can be understood topologically by analyzing the type(0; ucc; vcc; 1) trellis. Here, we generalize the symbolic dynamics of the lobes, to t the cases in which `cd = 0; c; d 2 fl; rg. The scenario for the type-(0; ucc; vcc; 1) trellis, as the forcing parameter is increased, is as follows: after the lobe Ec1, c 2 fl; rg intersected the lobe Dc0 at four SIP's, the "tip" of the lobe Ec0 grows, intersecting the "tips" of Dc ucc , with a growing index ucc. The growing tip of Ec1 is forced to fold (since it may not intersect the unstable boundary of Ec0, and the space in which it grows, shrinks), until it intersects the lobe Dc0 at a fth homoclinic point. As the "tip" of Ec0, the "tip" of Ec1 and the index ucc continue to grow, sixth, seventh and eighth homoclinic points are created by intersections of the growing and folding "tip" of Ec1 and the lobe Dc0. Until nally, as ucc ! 1, the "tip" of Ec0 intersects the lobe Dc0 and `cc decreases from one to zero. Since this scenario occurs, the typef`cdg trellis with `cc = 0 contains new states, f cc 0a and f cc 0b , created by the four additional homoclinic intersections of the "tip" of lobe Ec1 and the lobe Dc0. The states f cc 0a and f cc 0b are a splitting of the state f cc 0 f cc `cc . See gure (2.20). Hence, the symbolic dynamics of the lobes, which gives a lower bound for the development of a type-f`cdg trellis with `cc 0 and `cd 0, under the assumption that the lobe Ec`cc (respectivelyDc`cd+1) is tangent to the lobe Dc0 (respectively Ed0), is: f cc j ! f cc j 1; 2 j `cc; and `cc 1 f cc 1 % 2f cc 0 & gcd `cd f cc 0 % f cc 0 ! f cc `cc & gcd `cd `cc = 0; 40 Figure 2.20: Symbolic dynamics for structural index `cc = 0; c 2 fl; rg. f cc 0a % 2f cc 0a ! f cc 0b & 2gcd `cd f cc 0b % 2f cc 0a & 2gcd `cd For the state f cc 0b , we assumed the minimal necessary behavior, and its actual behavior is: f cc 0b % 2f cc 0a ! 2gcd `cd &g; f cc 0b : Where, topologically g f cc 0b = f cc 0b , but the states g f cc 0b are always "below" to the "original" state f cc 0b , hence their size is decreasing, until some iteration of f cc 0b , g f cc 0b , will be tangent to the lobe Dc0, and the next iteration will give us the behavior that we assumed. gcd `cd ! gcd `cd 1 ! ! gcd 0 ! 2f cd 0 f cd k ! f cd k 1; 2 k `cd; 41 f cd 1 % 2f cd 0 & gdd `dd f cd 0 ! 8>>>>>>><>>>>>>>>: % f cd 0 ! fdc `dc ; `dc 1 & gdd `dd % gcc `cc ! f cd 0 ; `dc = 0 & 2fdc 0 & gdd `dd ; gcc `cc ! gcc `cc 1 ! ! gcc 0 gcc 0 ! 8><>: 2f cc 0 ; `cc 1 2f cc 0a; `cc = 0 : Where states gcc 0 ; gcd 0 assumed to be tangent to the lobes Dc 1; Ed 1, respectively. However, generally, only some iteration of them is tangent to Dc 1; Ed 1. For more accurate approximation, more indices should be added, to describe the precise behavior of the states gcc i; gcd j ; 0 i z; 0 j w . Where z;w are some integers for which gcc z ; gcd w are tangent to Dc 1; Ed 1, respectively, and gcc i ! gcc i 1; gcd j ! gcd j 1. If one is interested to obtain a lower bound on the stretching, these states g should be omitted from the symbolic dynamics of the lobes. If Mr(t0) has no zeros, yet `lr is nite, the states f rr, grr and grl are not de ned. And, the states f rl are replaced by the states f rl, in the following way: The state f rl `rl is replaced by the state f rl m , for some delay index m, where f rl m ! f rl m 1 ! ! f rl 1 ! 2f rl 0 . In our approximation here, we assume that f rl m f rl m and m `lr. The above described relations between the states de ne a square transfer matrix, Tfg, between vectors of states. This transfer matrix has three diagonal blocks, each corresponding to one of the regions Rl; Rr or Ro. The number of columns in each such diagonal block is the number of all the f and g states related to the structural indices `cd; c; d 2 fl; rg in the corresponding region. Most of the other 42 entries of the transfer matrix are zeros. Tfg = 26664 h f ll gll i h glr f lr f rl grl i h grr f rr i 37775 (2 (`ll + `lr + `rl + `rr) + 3)2 dim(Tfg) (2 (`ll + `lr + `rl + `rr) + 10)2, and vTfg = u, where v; u are horizontal vectors of states. For example, the transfer matrix for the AFDO with `cd = 0 for all c; d 2 fl; rg is: T 0 fg = 66666666666666666666664 f ll 0a f ll 0b gll 0 glr 0 f lr 0 f rl 0 grl 0 grr 0 f rr 0a f rr 0b 264 2 1 0 2 0 0 2 0 0 375 2 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 26664 0 2 0 0 0 1 2 0 0 2 1 0 0 0 2 0 37775 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 264 0 2 0 0 2 1 0 2 0 375 77777777777777777777775 : (2.61) If some of the indices `rd; d 2 fl; rg are not de ned (or in nite), the corresponding states, which are not de ned, should be omitted from the transfer matrix. For example, in the case when Mr(t0) of the AFDO has no zeros, and `ll = 0, `lr = 0, since by our previous assumption on m `rl = 0, the transfer matrix, T 0 fg, is: T 0 fg = 266666666666664 f ll 0a f ll 0b gll 0 glr 0 f lr 0 f rl 0 264 2 1 0 2 0 0 2 0 0 375 2 0 0 2 0 0 0 0 0 0 0 0 0 0 1 0 0 1 264 0 2 0 0 1 2 0 2 1 375 377777777777775 : (2.62) If the system is "open", then the escape rates from an unigate region Rc can be calculated, as shown in [8]. It follows from [16] and [8], that log( ), where 43 is the modulus of the largest eigenvalue of the transfer matrix, gives a lower bound on the topological entropy with respect to the map F (WNLG Poincar e map). If the system is "closed", should be taken as an eigenvalue of Tfg, and if the system is "open", should be taken as an eigenvalue of Tf , where Tf is the transfer matrix with the states g omitted. Therefore, the lower bound on the topological entropy for the AFDO with `cd = 0; c; d 2 fl; rg (calculated from matrix (2.61)) is log(3:9231). The lower bound on the topological entropy for the AFDO with no PIP's of the right branches of the stable and the unstable manifolds (e.g. Mr(t0) nonvanishing), and with `ld = 0; d 2 fl; rg (calculated from matrix (2.62)) is log(3:6709). In [8], lower bounds on the topological entropy are given for the Cubic equation (which is an "open" system, hence Tf was considered), for 1 `ll 10. For example, for `ll = 1, log( ) = log(2), which is the largest value of the approximated topological entropy, stated in [8] for the Cubic equation. All the states f cd i (respectively gcd j ) for a xed i (respectively j), c; d 2 fl; rg, are topologically equivalent. But, they are of di erent lengths and widths, since a state may split in transition to several states. Therefore, weights, wn, may be assigned to each transition from one state to others, as shown in [8]. For example, the weighted transfer matrix for the AFDO, for `cd = 0; 8c; d 2 fl; rg, is: T 0 wfg = 66666666666666666666664 f ll 0a f ll 0b gll 0 glr 0 f lr 0 f rl 0 grl 0 grr 0 f rr 0a f rr 0b 264 2w5 w6 0 2w8 0 0 2 0 0 375 2w7 0 0 0 2w9 0 0 0 0 0 0 0 0 0 0 0 0 0 w3 0 0 w4 0 0 0 26664 0 2 0 0 0 w1 2w2 0 0 2w2 w1 0 0 0 2 0 37775 0 0 0 w4 0 0 w3 0 0 0 0 0 0 0 0 0 0 0 0 0 2w7 0 0 0 2w9 264 0 2 0 0 2w5 w6 0 2w8 0 375 77777777777777777777775 ; (2.63) where w1 + 2w2 + w3 + w4 = 1; 2w5 + w6 + 2w7 = 1 and 2w8 + 2w9 = 1 and = exp( 2 ! ). These weights are di erent, for di erent type-f`cdg trellises of the same system, but = det(DxF ), hence unique for each system. If the system is de ned by _ x = f(x; t) = f(x; t+ T ), and F is the Poincar e map, then: = exp0 Z T tr(Dxf(x(t); t) 44 (T is the period of the perturbation). See [1] and [2] for details. indicates the rate of area contraction of the system. Notice that dim(Twfg) = dim(Tfg). 2.5. Strange attractors in the AFDO As for equation (1.3), it is easy to show that the AFDO (equation (2.5)) possesses a chaotic attracting set (see section 1.3 and [1], [2]). Yet, the same di culties arise regarding a construction of an analytical proof for the existence of strange attractors in the AFDO, as for equation (1.3). FromNumerical experiments that we performed for the AFDO, using DSTOOLS [21], we observed, that strange attractors appear in the area of the parameter space related to the structural indices `cd = 0; c; d 2 fl; rg. To investigate this subject more thoroughly, we constructed a computer program in Matlab, which computes the Lyapunov exponents of orbits of (2.5), based on the algorithm given in [19]. The logarithmic function that we used for computation of the Lyapunov exponents in our computer program is log2 (as in the original algorithm of [19]). Viewing (2.5) as an autonomous system, each orbit will have three Lyapunov exponents, where one of them must be zero and one must be negative. The third Lyapunov exponent may be either positive or negative, where positive Lyapunov exponent indicates in this case the existence of a strange attractor (see [17]), and negative Lyapunov exponent indicates that the orbit is periodic sink. Since a Lyapunov exponent of an orbit need to be calculated for time t!1, one needs to provide a stopping criteria. We have developed the following criteria: after performing Nin iterations of the Poincar e map (Nin = 200 was found experimentally as the minimal number of iterations needed for checking convergence), and then every Nit iterations (Nit = 100), it nds (by a least square tting) the order one polynomial coe cients (a line) corresponding to the logarithm of the modulus of the Lyapunov exponent values obtained in the last 100 iterations. If the last obtained Lyapunov exponent's value is positive and the slope of the obtained line is nearly zero (up to an error of 1e 6), the program stops, and the last obtained Lyapunov exponent's value (which is nearly equivalent to the value of 2a0, where a0 is the free coe cient of the line), considered as a good approximation. In cases where the third Lyapunov exponent is negative, we didn't run the program until it's value converged, but stopped the program, as soon as the value was negative and bounded away from zero. Also, this program provides the data points of the iterations of the Poincar e map of the orbit, which can be plotted in Matlab, and provide a geometrical picture of the strange attractor, or of the periodic orbit. 45 We performed all our numerical experiments for the asymmetry parameter , being zero, or near zero (small), and weak dissipation of the orbit (" small). For various values of xed and " < 1 we obtained numerical evidences for the existence of strange attractors in the region of the parameter space (!; "), where `cd = 0 or close to it. See for example gures (2.21). Moreover, we observed that strange attractors appear in the region of the parameter space (!; "), which is above the second secondary bifurcation curve "2ll(!; ; ; ; `ll = 1) (see section 2.3.4), as can be seen in these gures. Notice that in gure (2.21,a), "ill "irr and "ilr "irl, since it was plotted for = 0, in gure (2.21,b) "ill 6= "irr and "ilr 6= "irl, but "irr, "irl are not speci ed in this gure, and in gure (2.21,c) "irr, "irl are not de ned for the speci ed ! values, since the Melnikov function,Mr(t0) has no zeros for this parameters values. In fact, in gures (2.21) it appears that there are windows in parameter space, where the strange attractors appear near the secondary homoclinic bifurcation curves, which refer to existence of two SIP's corresponding to `ll = 0 and `lr = 0. We magnify this windows in gures (2.22). The puzzle is, that as we described in section 2.3.4, our methods are not accurate for the structural indices `ll = 0, especially for such large values of ". Hence, it is unclear what is the signi cance of these approximate curves. We are currently examining the resonance structure and the manifolds structure along these approximate bifurcation curves. The structure of the strange attractors that we obtain, is di erent for di erent values of the parameters, and exhibit three main forms, described below. The "two sided" strange attractors If the parameters values are such that both Melnikov functions,Ml(t0) andMr(t0) have simple zeros (see section 2.3.1), then: for parameters values, corresponding to weak area contraction ( small), we obtain non hyperbolic "two sided" strange attractors with similar structure to the attractors that were obtained for (1.3) (and can be seen, for example in [1]). See for example gure (2.23). The strange attractors presented in gure (2.21,a) are of such structure. For parameters values, corresponding to strong area contraction ( large), we obtain more structured "two sided" strange attractors, created from the folding of the unstable manifold on both sides of the origin, and which look like union of Cantor sets and a line (see gure (2.24)). The strange attractors presented in gure (2.21,b) are of such structure. The positive Lyapunov exponents values of the "two sided" strange attractors that we get, are near log2( 1) 0:2 0:03. 46 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 γ=1, β=0.0, δ=0.05 ω ε l=0 l=1 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 l=0 l=1 ω ε γ=1, β=0.01, δ=0.95 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ=1, β=0.1, δ=0.95 ω ε l=0 l=1 Figure 2.21: Secondary homoclinic bifurcation curves and strange attractors for the AFDO. indicates a strange attractor (positive Lyapunov exponent), o indicates a periodic orbit (negative Lyapunov exponent), |: "1lr, -: "2lr, : "1ll, : "2ll. a) = 0, = 0:05. b) = 0:01, = 0:95. c) = 0:1, = 0:95. 47 0.5 1 1.5 2 0.2 0.25 0.3 0.35 0.4 γ=1, β=0.0, δ=0.05 ω ε l=0 l=1 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 l=0 l=1