Computational aspects of large-length cycle search algorithms for nonlinear discrete systems

Abstract

Even the simplest nonlinear discrete systems dynamics is very complex. It includes both periodic movements and quasi-periodic or recurrent ones. In such systems, almost always present are the chaotic attractors, whose nature is currently well studied, at least for a wide class of model equations. In many cases, chaotic attractors can be modeled using periodic motions characterized with large periods. Such attractors’ and minimal invariant sets’ search represents an important task of applied mathematics, with respect to that the solutions are used in physical, chemical, economic sciences, in coding theory, signal transmission theory and so on. However, mathematical results based on computer calculations require a careful verification, since these calculations themselves are carried out approximately, and the chaotic systems are very sensitive to calculation errors. One of the approaches to solving the cycles search and verification problem is based on the application of these cycles’ stabilization methods. These methods can be divided into two groups: delayed control, that uses knowledge on system’s previous states, and predictive control, which uses the future values of system state in the absence of control. This study purpose is to demonstrate the effectiveness of the cycles search averaged predictive control method on some dynamical systems widely referred to in technical reference sources. Another important goal we aimed onto is to formulate the necessary conditions at which the orbit found actually represents a cycle. The article exposes the elaboration of predictive control methods: the averaged predictive control is used, at that the cycles search algorithms based on such control properties are offered. Noted are various features of algorithms’ functioning that depend on the original discrete system properties. Proposed are the cyclic points’ verification methods in the form of three necessary conditions of point’s cyclicity: checking the smallness of the residual, checking the periodicity and checking the cycle local asymptotic stability. Well-known two-dimensional discrete systems such as Lozi, Henon, Ikeda, Elhadj-Sprott, Multihorseshoe, Prey-Predator have been chosen to demonstrate the algorithm and numerical simulation. These systems’ essential features include the presence of large lengths cycles with a dominant multiplier, i.e. when two-dimensional case one multiplier has larger modulus, and another’s modulus is less than one. With this class of systems, the proposed algorithm operates particularly efficiently. The developed method can also be used to study the discrete dynamical systems’ topological properties dependence on changes in parameters, as well as to study the presence of bifurcations and their types.