# A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system

## Abstract

A nonautonomous nonlinear system is constructed and implemented as an experimental device. As represented by a 4D stroboscopic Poincare map, the system exhibits a Smale-Williams-type strange attractor. The system consists of two coupled van der Pol oscillators whose frequencies differ by a factor of two. The corresponding Hopf bifurcation parameters slowly vary as periodic functions of time in antiphase with one another; i.e., excitation is alternately transferred between the oscillators. The mechanisms underlying the system's chaotic dynamics and onset of chaos are qualitatively explained. A governing system of differential equations is formulated. The existence of a chaotic attractor is confirmed by numerical results. Hyperbolicity is verified numerically by performing a statistical analysis of the distribution of the angle between the stable and unstable subspaces of manifolds of the chaotic invariant set. Experimental results are in qualitative agreement with numerical predictions.

- Authors:

- Russian Academy of Sciences, Saratov Branch, Institute of Radio Engineering and Electronics (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21067731

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Experimental and Theoretical Physics; Journal Volume: 102; Journal Issue: 2; Other Information: DOI: 10.1134/S1063776106020166; Copyright (c) 2006 Nauka/Interperiodica; Article Copyright (c) 2006 Pleiades Publishing, Inc; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATTRACTORS; BIFURCATION; CHAOS THEORY; DIFFERENTIAL EQUATIONS; EXCITATION; NONLINEAR PROBLEMS; OSCILLATORS; PERIODICITY

### Citation Formats

```
Kuznetsov, S. P., E-mail: spkuz@rambler.ru, and Seleznev, E. P.
```*A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system*. United States: N. p., 2006.
Web. doi:10.1134/S1063776106020166.

```
Kuznetsov, S. P., E-mail: spkuz@rambler.ru, & Seleznev, E. P.
```*A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system*. United States. doi:10.1134/S1063776106020166.

```
Kuznetsov, S. P., E-mail: spkuz@rambler.ru, and Seleznev, E. P. Wed .
"A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system". United States.
doi:10.1134/S1063776106020166.
```

```
@article{osti_21067731,
```

title = {A strange attractor of the Smale-Williams type in the chaotic dynamics of a physical system},

author = {Kuznetsov, S. P., E-mail: spkuz@rambler.ru and Seleznev, E. P.},

abstractNote = {A nonautonomous nonlinear system is constructed and implemented as an experimental device. As represented by a 4D stroboscopic Poincare map, the system exhibits a Smale-Williams-type strange attractor. The system consists of two coupled van der Pol oscillators whose frequencies differ by a factor of two. The corresponding Hopf bifurcation parameters slowly vary as periodic functions of time in antiphase with one another; i.e., excitation is alternately transferred between the oscillators. The mechanisms underlying the system's chaotic dynamics and onset of chaos are qualitatively explained. A governing system of differential equations is formulated. The existence of a chaotic attractor is confirmed by numerical results. Hyperbolicity is verified numerically by performing a statistical analysis of the distribution of the angle between the stable and unstable subspaces of manifolds of the chaotic invariant set. Experimental results are in qualitative agreement with numerical predictions.},

doi = {10.1134/S1063776106020166},

journal = {Journal of Experimental and Theoretical Physics},

number = 2,

volume = 102,

place = {United States},

year = {Wed Feb 15 00:00:00 EST 2006},

month = {Wed Feb 15 00:00:00 EST 2006}

}