One Parameter Bifurcation Diagram for Chua’s Circuit

The objective of this letter is to report on the usefulness of a l-dimensional map for characterizing the dynamics of a third-order piecewise-linear circuit. The map is used to reproduce period-doubling birfurcations and particularly to compute Feigenbaum’s number. The l-dimensional map gives qualitatively identical results as the circuit equations and proved to be far more computationally efficient. A l-dimensional map was applied to performing an approximate analysis of a third-order piecewise-linear chaotic circuit. Several positive results made this approach an appealing form of analysis: 1) the l-dimensional map correctly predicted all qualitative behavior for the circuit, namely periodic windows, onset of the double scroll attractor, disappearance of the attractor, existence of a saddle-type periodic orbit, and conditions for a homoclinic orbit [l]; 2) it dramatically reduced the computation time of diagrams depicting dynamic bifurcations; and 3) it made possible an efficient and simple means to compute Feigenbaum’s constant. With these reasons motivating a more in-depth discussion of the l-dimensional map, the framework in which this map Fig. 1. Eigenspaces for the three regions of Chua’s circuit. is formulated, some of its special properties, and results obtained by it will be presented. We will emphasize the map’s interpretation as representing the behavior of a physical circuit. attractor. For continuity, the real eigenvalues of adjacent regions The autonomous differential equations in dimensionless form must have opposite stability properties. To obtain the double for Chua’s circuit are scroll, we must have the real eigenvalue in the middle region unstable and the complex eigenvalues in the top region unstable. n=a(y-h(x)) Note that the vector field is symmetric so it suffices to discuss j=x-ysz only the top and middle regions. It is easy to prove that the i=-/3y (1) points where the vector field in the x =l plane are parallel to that plane compose a line, labeled L, in Fig. 1. It also logically where the piecewise-linear term is follows that this line represents points of tangency of trajectories mO-ml to the x = 1 plane. Moreover, enough information had been given h(x) :=m,x+2 (Ix+lI-1x-Y). (4 to convince oneself that all trajectories to the right of L, pass the x = 1 plane in an increasing-x direction, while trajectories to the In what follows, m0 = -l/7 and mi = 2/7, while (Y and /3 are left of L, pass “down” through the boundary plane. the bifurcation parameters. The following three facts are now apparent. 1) Trajectories The important feature of this equation is that the state space is originating in the triangle ABE pass to the middle region and divided into three linear (or affine) regions, each containing an reenter the top region without entering the lower region; trajectoequilibrium point. One can identify a complex eigenplane and ries passing through the wedge 0 A, AEE, will continue to the real eigenvector corresponding to a pair of complex eigenvalues bottom region. 2) Since the double scroll has been observed to and a real eigenvalue, respectively, for each region of the vector span all three regions, it must have trajectories to the left and to field. The vector field under consideration is constrained so that the right of line L,. 3) Trajectories emanating from q A, BE, the eigenspaces of either the real or complex eigenvalues are not will eventually return to it; hence, it forms a natural boundary parallel to the x = 1 or x = 1 boundary planes. A possible for defining a Poincart map. geometry of the eigenspaces is depicted in Fig. 1. With one more piece of information, the setting will be comThe l-dimensional map is based on observations of the special plete for defining the l-dimensional map: for the parameter properties of the chaotic attractor (called the double scroll) in values of interest (for which the double scroll attractor has been Chua’s circuit. Hence, it is insightful to examine the dynamics of observed), the real stable eigenvalue in the upper region is the circuit. A good understanding can be derived simply from a relatively large in absolute value compared to the other eigenvalcloser inspection of Fig. 1. Keep in mind that the eigenspaces are ues. So trajectories are rapidly contracted to the complex eigeninvariant sets, so trajectories cannot pass through them. plane of P + and will transfer to the middle region very close to A restricted set of eigenvalues (or parameter values (Y, /I, the line BA. m,, m,) is allowed to ensure continuity of the vector field at the The l-dimensional map is defined as boundaries and to guarantee the existence of the double scroll P:BA,+K. To ensure that the l-dimensional map is continuous and maps to Manuscript received August 14, 1986; revised October 7, 1986. The author is with the Department of Electrical Engineering and Computer unique points, we require that trajectories emanating from Science, University of California, Berkeley, CA 94720. 0 A, AEE, , upon returning to the top region, do not hit the line IEEE Log Number 8611949. L, (i.e., the trajectory will transfer to the top region, as opposed 0098-4094/87/0200-0208$01.00 01987 IEEE 208 IEEE TRANSACTlONS ON CIRCUITS AND SYSTEMS. VOL. CAS-34, NO. 2, FEBRUARY 1987 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-34, NO. 2, FEBRUARY 1987 209 Fig. 2. Bifurcation diagram using the l-dimensional map, showing the period-doubling route to chaos. to being tangent to x =l and remaining in the middle region). Secondly, we assume that trajectories emanating from 0 A, AEE, return to the x =l plane to the right of point D in Fig. 1. Considering the plane W, the second assumption guarantees that trajectories will pass through W. Moreover, they are compressed closely to the ray m Hence, we can define an equivalent but simpler l-dimensional map