Nonlinear Dynamics of a Modified Autonomous Van Der Pol-duffing Chaotic Circuit

In this work, we investigate the dynamical behaviors of the modi…ed autonomous Van der Pol-Du¢ ng chaotic circuit (MADVP). Some stability conditions of the equilibrium points are discussed. The existence of pitchfork bifurcation is veri…ed by using the bifurcation theory and the center manifold theorem. The occurrences of Hopf bifurcation about the equilibrium points are proved. Conditions for supercritical and subcritical Hopf bifurcations are also derived. A route to chaos in this system is shown via period-doubling bifurcations. Furthermore, the analytical conditions of the existence of homoclinic orbits and Smale horseshoe chaos in this system are obtained. Numerical simulations are used to support the theoretical predictions. 1. Introduction Chaos is one of the most fascinating phenomena which has been extensively studied and developed by scientists since the pioneering work of Lorenz in 1963 [1]-[7]. The chaotic system has complex dynamical behaviors such as the unpredictability of the long-term future behavior and irregularity. Thus, chaos has great potential applications in many disciplines such as encryption, cryptography [8]-[9], chaos control and synchronization [10]-[17], secure communications [18], neuroscience [19], and mathematical biology [20]. As a matter of fact, nonlinear electronic circuits play an important role in studying various phenomena that undergo complex dynamical behaviors and chaos. Thus, nonlinear electronic circuits are widely used as an experimental vehicle to study nonlinear phenomena. This …eld of research was initiated by L.O. Chua who developed a nonlinear circuit with a piecewise nonlinear term called Chua’s circuit [21], however the simplest autonomous nonlinear circuit which generates chaotic signals was presented in [22]. Recently, Chen circuit [23] and Lü circuit [24] have been implemented with quadratic nonlinear terms. Thus, our objective is to study the nonlinear dynamics of MADVP circuit. We show that the circuit’s system has three equilibrium points E0, E+; and E , then we study their stability conditions. The conditions of existence of pitchfork bifurcation are derived by using center 1991 Mathematics Subject Classi…cation. 34C15, 34C28, 37M05. Key words and phrases. Circuit implementation; Center manifold; Pitchfork bifurcation; Hopf bifurcation; Homoclinic orbits; Chaos. Submitted April 6, 2014. 199 200 A. M. A. EL-SAYED, ET AL. EJMAA-2014/2(2) manifold theorem and the bifurcation theory [25]. The occurrences of Hopf bifurcation near the equilibrium points are also discussed. The stability conditions of the periodic solutions are obtained by using Hopf bifurcation theorems [26]-[27]. Also, we investigate the analytical conditions for the existence of the homoclinic orbits in this system by using the theorem given in [28]. The paper is organized as follows: In Section 2, a circuit realization of MADVP system is proposed. In Section 3, some stability conditions of the equilibrium points are investigated. In Section 4, pitchfork bifurcation analysis is demonstrated. In Section 5, Hopf bifurcation analysis of the equilibrium points is discussed. In Section 6, the existence of homoclinic orbits is analytically obtained. Finally, in Section 7, conclusions are drawn. 2. The circuit realization of MADVP system The circuit implementation of MADVP circuit is shown in …gure 1. The MADVP system is given as follows [13]: _ x = (x x y); _ y = x y z; (1) _ z = y; where ; ; are positive real numbers and 2 R. The equilibrium points of system (1) are: E0 = (0; 0; 0); E+ = ( p ; 0; p ); and E = ( p ; 0; p ) (2) where E+ and E exist if > 0: 3. Some stability conditions of the equilibrium points Consider the three-dimensional autonomous system dX dt = F (X); X 2 R (3) where the vector …eld F (X) : R ! R belongs to the class C(r 2) and the …xed point Xe 2 R is a hyperbolic saddle focus, i.e., the eigenvalues of the Jacobian matrix J have the form: 1 = ; 2;3 = iw; < 0; w 6= 0; and i = p 1 (4) where ; ; and w are real constants. The eigenvalues equation of the equilibrium point is given by the following polynomial: P ( ) = 3 + a1 2 + a2 + a3 = 0 (5) and its discriminant D(P ) is given by: D(P ) = 18a1a2a3 + (a1a2) 2 4(a1)a3 4(a2) 27(a3): (6) If D(P ) < 0, then the characteristic equation (5) has one real root 0 and two complex-conjugate roots = i!. Hence, the characteristic equation (5) can be written as: 3 (2 + 0) 2 + (j +j + 2 0) j +j 0 = 0: (7) EJMAA-2014/2(2) NONLINEAR DYNAMICS OF MADVP CIRCUIT 201