In literature many chaotic systems, based on third-order jerk equations with different nonlinear functions, are available. A jerk system is taken to be a part of dynamical systems that can exhibit regular and chaotic behavior. By extension, a hyperjerk system can be described as a dynamical system with nth-order ordinary differential equations where n is 4 or up to. Hyperjerk systems have been ...

Dynamic neural networks are often used for nonlinear system identification. This paper presents a novel series-parallel dynamic neural network structure which is suitable for nonlinear system identification. A theoretical proof is given showing that this type of dynamic neural network is able to approximate finite trajectories of nonlinear dynamical systems. Also, this neural network is trained...

The dynamics of a system composed of two nonlinearly coupled, drastically different nonlinear and eventually oscillatory elements is studied. The rich variety of attractors of the system is studied with the help of phase space analysis and Poincare maps. PACS. 05.45.-a Nonlinear dynamics and nonlinear dynamical systems

Dynamic neural networks are often used for nonlinear system identification. This paper presents a novel series-parallel dynamic neural network structure which is suitable for nonlinear system identification. A theoretical proof is given showing that this type of dynamic neural network is able to approximate finite trajectories of nonlinear dynamical systems. Also, this neural network is trained...

The behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict. Rather than characterizing dynamical behavior at the level of trajectories, we consider following the evolution of sets. There are often collections of sets that behave in a very predictable way, in spite of the fact that individual trajectories are entirely unpredictable. Such special...

Invariant manifolds facilitate the understanding of nonlinear stochastic dynamics. When an invariant manifold is represented approximately by a graph for example, the whole stochastic dynamical system may be reduced or restricted to this manifold. This reduced system may provide valuable dynamical information for the original system. The authors have derived an invariant manifold reduction or r...

In this paper a grey-box model of a nonlinear dynamical system is constructed. This involves using a Gaussian process to emulate model error the error that arises as a result of flaws in one’s physical-law based model of the system. The work shows how such an approach can be extended towards dynamical systems. Specifically, it is applied to experimental data, obtained from a dynamical system wh...

Takens Dynamical Systems and Turbulence, Lecture Notes in Mathematics, edited by D. A. Rand and L. S. Young Springer-Verlag, New York, 1981 , Vol. 898, pp. 366–381 has shown that a dynamical system may be reconstructed from scalar data taken along some trajectory of the system. A reconstruction is considered successful if it produces a system diffeomorphic to the original. However, if the origi...

Many high-dimensional time-varying signals can be modeled as a sequence of noisy nonlinear observations of a low-dimensional dynamical process. Given high-dimensional observations and a distribution describing the dynamical process, we present a computationally inexpensive approximate algorithm for estimating the inverse of this mapping. Once this mapping is learned, we can invert it to constru...

The main aim of this study is to construct an explicit form of an analytic solution of a multidimensional difference equation. To this end, we present a nonlinear Cournot duopoly model with unimodal reaction functions, show that its dynamical system is double logistic when the production externalities involved get stronger, and then give a constructive proof of the existence of an analytic solu...