Existence of unpredictable solutions and chaos

Abstract

Recently we introduced the concept of Poincaré chaos. In the present paper, by means of the Bebutov dynamical system, an unpredictable solution is considered as a generator of the chaos in a quasilinear system. The results can be easily extended to different types of differential equations. An example of an unpredictable function is provided. A proper irregular behavior in coupled Duffing equations is observed through simulations.

Poincaré chaos and unpredictable functions

“…These are all strong arguments for the insertion of chaos research to the theory of differential equations. In addition to the role of the present paper for the theory of differential equations, the concept of unpredictable points and A C C E P T E D M A N U S C R I P T orbits introduced in our studies [2,3] and the additional results of the present study will bring the chaos research to the scope of the classical theory of dynamical systems. Moreover, not less importantly, these concepts extend the boundaries of the theory of dynamical systems significantly, since we are dealing with a new type of motions, which are behind and next to Poisson stable trajectories.…”

“…This new definition makes homoclinic chaos and the late descriptions of the phenomenon be closer to each other such that a unified background of chaos can be obtained in the future. Besides, in article [3], an unpredictable function was defined as an unpredictable point of the Bebutov dynamical system. In the present paper, we have obtained samples of unpredictable functions and sequences, which are in the basis of Poincaré chaos.…”

“…For example, if one considers non-autonomous or non-smooth systems. Then we can apply unpredictable functions [3], a new type of oscillations which immediately follow almost periodic solutions of differential equations in the row of bounded solutions. They can be investigated for any type of equations, since by our results they can be treated by methods of qualitative theory of differential equations.…”

“…Thus, the Poincaré chaos concept has been eventually shaped in our paper [2]. We have also determined the unpredictable function on the real axis as an unpredictable point of the Bebutov dynamics in paper [3] to involve widely differential and discrete equations to the chaos investigation. Nonetheless, we need a more precise description of what one understands as unpredictable function.…”

Poincaré chaos for a hyperbolic quasilinear system

“…A special type of Poisson stable trajectory called an unpredictable trajectory, which leads to Poincaré chaos in the quasi-minimal set, was introduced in the paper [1]. Moreover, the papers [2]- [4] were concerned with the unpredictable solutions of various types of quasilinear differential equations in which the matrix of coefficients admits eigenvalues all with negative linear part. In this study, we deal with the unpredictable solutions of quasilinear differential equations with hyperbolic linear part such that the matrix of coefficients admits eigenvalues both with negative and positive real parts.…”