Situations of particle motion on a plane under periodic kicks associated with attractors of Smale – Williams type in the stroboscopic map were proposed and studied numerically in Refs. [1, 2].

Consider a particle of unit mass on the plane (x, y) in a stationary potential field

possessing rotational symmetry about the origin, with minimum on the unit circle. We assume that an additional force field with potential

is switched on and off periodically with time interval T, producing short-time kicks of magnitude and direction depending on the instantaneous position of the particle. Also we add the friction force proportional to the velocity; the friction coefficient is assumed unit for simplicity. The equations read

To explain the system functioning, consider a ring of particles resting initially on the unit circle with coordinates x=cos Ф и у=sin Ф. After a kick each particle will get the momentum components P_x=-x+x²-y² and P_y=-y-2xy. Е Without taking into account the field U, the particles would stop due to the friction at the coordinates easily evaluated as

Substituting x=cos Ф and у=sin Ф we obtain x'=cos Ф' and у'=sin Ф', where Ф'=2Ф. t means that the particles settle down again on the unit circle, but a single bypass of the original ring corresponds to a twofold bypass of the newly formed ring in the opposite direction. Thus, for the angular coordinate we have the expanding circle map, or the Bernoulli map. The dynamics of the ensemble of particles is illustrated by the animation. Been given initial state just before the n-th kick, x_n={x, v_x, y, v_y}_t=nT-0, one can determine the state before the next, n+1-th kick from solution of the equations on a time interval T: x_n+1=f(x_n) that corresponds to the four-dimensional Poincaré map of our system. Figures illustrate some results of numerical simulations: a typical trajectory of the particle on the plane (x, y) and attractor in the stroboscopic section in projection on the plane (x, y) are shown. It is the Smale – Williams solenoid because of the outlined topological property of the ensemble of particles after the evolution between the kicks, namely, the emergence of the loop bypassing the origin twice. The third picture shows the iterative diagram for the angular variable determined immediately before each successive kick. One can see that the angular coordinate behaves in accordance with the expanding circle map: one bypass of the circle for the pre-image implies two detours for the image in the opposite direction.

The next plot shows Lyapunov exponents of the Poincaré map versus the parameter at fixed period of kicks T. Observe that the largest Lyapunov exponent remains approximately constant and close to ln 2 that corresponds to the uniformly expanding circle map describing roughly the dynamics of the angular variable. Other exponents are negative and responsible for compression of the phase volume and approach of orbits to the attractor.

The page is elaborated under support of RSF grant No 15-12-20035 in the Udmurt State University (Izhevsk).

REFERENCES

[1] S.P. Kuznetsov, L.V. Turukina. Attractors of Smale–Williams type in periodically kicked model systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 18, 2010, No 5, 80-92. 

[2] S.P. Kuznetsov. Some Mechanical Systems Manifesting Robust Chaos. Nonlinear Dynamics and Mobile Robotics, 1, 2013, No 1, 3–22.