Search

This paper proposes a novel algorithm to accurately calculate the coordinates of homoclinic points observed in discrete-time dynamical systems. The proposed method is based on the particle swarm optimization method. Compared with the current methods, the proposed methodology has the advantages of not requiring the careful selection of the initial conditions and not requiring information related to the derivative of the objective function. It is shown that the proposed method can successfully obtain the homoclinic points of a system, even if the system parameters are close to those that describe the homoclinic bifurcation sets of the system; this is achieved via the construction of an efficient objective function that depends on the Euclidean distance between the points in each manifold. We apply the developed method to two- and three-dimensional discrete-time dynamical systems and demonstrate the validity of the algorithm via the numerical work. The reliability of the proposed algorithm was achieved by evaluating a metric based on the success rate of the method.

The bouncing ball system is a simple mechanical collision system that has been extensively studied for several decades. It is a fundamental problem in impact dynamics. We studied a traditional bouncing ball system numerically and experimentally and discovered novel bifurcation structures where the maximum height of the bouncing ball with respect to the stationary state increases stepwise nonsmoothly when we increase the frequency of the oscillating table continuously even though the bouncing ball is in chaotic states [Karube et al., Chaos 30, 103111 (2020)]. We attempt to reveal the trick causing the stepwise changes of the maximum heights of the bouncing ball. We focus on the time interval for the ball to take off and land on the oscillating table at which the ball takes the maximum height. Let this time interval be denoted by t-interval. In addition, let the oscillation frequency of the table be denoted by f. The stepwise increases in the maximum heights of the bouncing ball in the stationary states occur when the multiplication of the t-interval and f coincides with integer values. This is the mechanism causing the nonsmooth maximum heights. Furthermore, results that are qualitatively consistent with the numerical ones are verified in the actual bouncing ball system using table tennis ball equipment.