^{1}, Sergey P. Kuznetsov

^{1,2}and Igor R. Sataev

^{1}

### Abstract

Formation or destruction of hyperbolic chaotic attractor under parameter variation is considered with an example represented by Smale–Williams solenoid in stroboscopic Poincaré map of two alternately excited non-autonomous van der Pol oscillators. The transition occupies a narrow but finite parameter interval and progresses in such way that periodic orbits constituting a “skeleton” of the attractor undergo saddle-node bifurcation events involving partner orbits from the attractor and from a non-attracting invariant set, which forms together with its stable manifold a basin boundary of the attractor.

In classic theory of oscillations, the dynamical phenomena possessing the structural stability or roughness are those subjected to priority study and the most important for practice. The next step is the examination of the first degree of non-roughness (that corresponds to bifurcations of codimension-1), then of situations separating those of the first degree of non-roughness (the codimension-2 bifurcations), etc. This program successful for 2D flow systems met difficulties for higher dimensions linked particularly with occurrence of chaos. As known from mathematical theory, structurally stable chaotic attractors are those belonging to a uniformly hyperbolic class (“axiom-A-attractors”); their mathematical examples were advanced during 1960–1980. That time, it was expected that they might be relevant to physical situations (like fluid turbulence), but latter it became clear that attractors, which normally occur in applications, do not fall into this special class. The obvious contradiction with the principle of significance of the rough systems recently has been overcome to some extent after introducing physically realizable systems, in which hyperbolic attractors actually occur. It opens an opportunity to return to the mentioned research program, and put a question about formation or destruction of the structurally stable chaotic attractors under variation of parameters (analogous to bifurcations of codimension-1). Also, this is a promising direction of search for real-world systems with the hyperbolic chaotic attractors. In this paper, we consider a scenario of the onset or destruction of the hyperbolic chaotic attractor of Smale–Williams type associated with collision of two invariant sets (an attracting and a non-attracting one) in a physically realizable system composed of two alternately activated non-autonomous self-oscillators.

The work was supported by RFBR Grant No 12-02-00342. O.I. acknowledges support from RFBR Grant No 12-02-31342.

I. INTRODUCTION

II. THE BASIC MODEL

III. NUMERICAL RESULTS AND DISCUSSION

IV. CONCLUSION

### Key Topics

- Attractors
- 56.0
- Bifurcations
- 20.0
- Oscillators
- 12.0
- Orbital dynamics
- 11.0
- Bernoulli's principle
- 8.0

## Figures

The Lyapunov exponents of the system (1) versus parameter *h*: for the chaotic attractor (black curves) and for the trivial attractor, the fixed point at the origin (gray line). Parameter values are , *T* = 6, , *a* = 6.5.

The Lyapunov exponents of the system (1) versus parameter *h*: for the chaotic attractor (black curves) and for the trivial attractor, the fixed point at the origin (gray line). Parameter values are , *T* = 6, , *a* = 6.5.

Constructing the amplitude stroboscopic map at , *a* = 6.5, *T* = 6, for different values of parameter *h*: (a) the original diagram accounting all accumulated computational data, and (b) the idealized plot excluding the widening by the phase averaging.

Constructing the amplitude stroboscopic map at , *a* = 6.5, *T* = 6, for different values of parameter *h*: (a) the original diagram accounting all accumulated computational data, and (b) the idealized plot excluding the widening by the phase averaging.

Diagrams illustrating dynamics of phase on successive stages of excitation of the first oscillator on the attractor *A* (black) and on the non-attracting invariant set *S* (gray).

Diagrams illustrating dynamics of phase on successive stages of excitation of the first oscillator on the attractor *A* (black) and on the non-attracting invariant set *S* (gray).

(a) Periodic solutions of the equations (1) with period plotted in the extended phase parameter space at , *a* = 6.5, *T* = 6, . (b) Superimposed graphs of radial (amplitude) component of the same set of solutions versus parameter *h*; this is actually a longitudinal slice of (a).

(a) Periodic solutions of the equations (1) with period plotted in the extended phase parameter space at , *a* = 6.5, *T* = 6, . (b) Superimposed graphs of radial (amplitude) component of the same set of solutions versus parameter *h*; this is actually a longitudinal slice of (a).

Attractor *A* and non-attracting invariant set *S* in the stroboscopic Poincaré section shown on the plane of variables of the first oscillator (a) and in three-dimensional plot (b) accounting a variable of the second oscillator at , *a* = 6.5, *T* = 6, , and *h* = −1.15. The gray color on the panel (a) indicates roughly the basin of attraction of the stable fixed point in the origin.

Attractor *A* and non-attracting invariant set *S* in the stroboscopic Poincaré section shown on the plane of variables of the first oscillator (a) and in three-dimensional plot (b) accounting a variable of the second oscillator at , *a* = 6.5, *T* = 6, , and *h* = −1.15. The gray color on the panel (a) indicates roughly the basin of attraction of the stable fixed point in the origin.

A plot of radial component of the model map (A1) (a) and disposition of attractors *A*, *O* and of a non-attractive invariant set *S* on the plane of complex variable *z* (b).

A plot of radial component of the model map (A1) (a) and disposition of attractors *A*, *O* and of a non-attractive invariant set *S* on the plane of complex variable *z* (b).

Mutual disposition of attractors *A* and *O* and of a non-attractive invariant set (*S*) of the map (A3) on the plane of complex variable *z* for different values of parameter *R* at .

Mutual disposition of attractors *A* and *O* and of a non-attractive invariant set (*S*) of the map (A3) on the plane of complex variable *z* for different values of parameter *R* at .

## Tables

Bifurcation values of parameter *h* for the unstable periodic orbits of the system (1) at , *a* = 6.5, *T* = 6, .

Bifurcation values of parameter *h* for the unstable periodic orbits of the system (1) at , *a* = 6.5, *T* = 6, .

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