Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

The paper proposes an approach to constructing feasible examples of dynamical systems with hyperbolic chaotic attractors based on the successive transfer of excitation between two pairs of self-oscillators that are alternately active. An angular variable that measures the relations of the current amplitudes for the two oscillators of each pair undergoes a transformation in accordance with the expanding circle map during each cycle of the process. We start with equations describing the dynamics in terms of complex or real amplitudes and then examine two models based on van der Pol oscillators. One model corresponds to the situation of equality of natural frequencies of the partial oscillators, and another to a nonresonant ratio of the oscillation frequencies relating to each of the two pairs. Dynamics of all models are illustrated with diagrams indicating the transformation of the angular variables, portraits of attractors, Lyapunov exponents, etc. The uniformly hyperbolic nature of the attractor in the stroboscopic Poincaré map is confirmed for a real-amplitude version of the equations by computations of statistical distribution of angles between stable and unstable manifolds at a representative set of points on the attractor. In other versions of the equations the attractors relate presumably to the partially hyperbolic class.

Figure 1

Typical example of time dependencies for the amplitude variables of four oscillators obtained from numerical solution of Eq. (4) with $T=10$, $A=3$, $\varepsilon =0.06$: (a) pairwise sums of squared amplitudes ${\rho }_a=a_1^2+a_2^2$ and ${\rho }_b=b_1^2+b_2^2$, and the amplitude variables for (b) the first and (c) the second pair of the oscillators.

Figure 2

Attractor in the Poincaré section in two-dimensional projection on the plane of real amplitudes of (a) the first two oscillators and (b) diagram for the angular variables at successive instants $t=nT$ for the system in Eq. (4) with $T=10$, $\varepsilon =0.06$, $A=3$.

Figure 3

Lyapunov exponents of the Poincaré map for the system in Eq. (4) plotted versus parameters: (a) the dependence on the amplitude of the slow modulation $A$ at $T=10$, $\varepsilon =0.06$; (b) the dependence on the coupling parameter $\varepsilon$ at $A=3$, $T=10$; (c) the dependence on the period of the slow modulation $T$ at $A=3$, $\varepsilon =0.06$.

Figure 4

An estimate of the density function for the distribution of angles between the local stable and unstable manifolds for the attractor in the stroboscopic Poincaré map obtained numerically at $T=10$, $A=3$, $\varepsilon =0.06$ for the model system in Eq. (4) represented in real amplitudes. Observe the clearly expressed separation of the distribution from zero angles that is the indicator of the hyperbolicity.

Figure 5

Wave forms generated by the system in Eq. (7) for $A=3$, $T=10$, $\varepsilon =0.06$, ${\omega }_0=2\pi$.

Figure 6

(a) Attractor in the Poincaré section in two-dimensional projection on the plane of the auxiliary variables introduced in the text and characterizing amplitudes of the first two oscillators and (b) diagram for the angular variables at successive instants $t=nT$ for the system in Eq. (7) at $A=3$, $T=10$, $\varepsilon =0.06$, ${\omega }_0=2\pi$.

Figure 7

Lyapunov exponents of the Poincaré map for the system in Eq. (7) plotted versus parameters: (a) the dependence on the amplitude of the slow modulation $A$ at $T=10$, $\varepsilon =0.06$, ${\omega }_0=2\pi$; (b) the dependence on the coupling parameter $\varepsilon$ at $A=3$, $T=10$, ${\omega }_0=2\pi$; (c) the dependence on the period of the slow modulation $T$ at $A=3$, $\varepsilon =0.06$, ${\omega }_0=2\pi$.

Figure 8

Typical pattern of time dependencies for coordinate variables of four oscillators obtained from numerical solution of Eqs. (9) at $T=10$, $A=3$, $\varepsilon =0.06$, ${\omega }_1=10\pi$, ${\omega }_2=16\pi$.

Figure 9

(a) Attractor in the Poincaré section in two-dimensional projection on the plane of two real variables relating to the first two oscillators of the system in Eq. (9) (see text). (b) Diagram for the angular variables for the system in Eq. (9). Parameters are $T=10$, $\varepsilon =0.06$, $A=3$, ${\omega }_1=10\pi$, ${\omega }_2=16\pi$.

Figure 10

Lyapunov exponents of the Poincaré map for the system in Eq. (9) plotted versus parameters: (a) the dependence on the amplitude of the slow modulation $A$ at $T=10$, $\varepsilon =0.06$, $\varepsilon =0.06$; (b) the dependence on the coupling parameter $\varepsilon$ at $A=3$, $T=10$; (c) the dependence on the period of the slow modulation $T$ at $A=3$, $\varepsilon =0.06$. The values of the other parameters are ${\omega }_1=10\pi$, ${\omega }_2=16\pi$.