Stability of Non-Hyperbolic Equilibrium Point for Polynomial System of Differential Equations

In this paper, a polynomial system of plane differential equations is observed. The stability of the non-hyperbolic equilibrium point was analyzed using normal forms. The nonlinear part of the differential equation system is simplified to the maximum. Two nonlinear transformations were used to simplify first the quadratic and then the cubic part of the system of equations.

Dynamical Analysis of a Novel 4-D Hyper-Chaotic System With One Non-Hyperbolic Equilibrium Point and Application in Secure Communication

In this article, a novel hyper-chaotic system has been introduced and its dynamical properties (i.e., phase plots, time series, lyapunov exponents, bifurcation diagrams, equilibrium points, Poincare sections, etc.) have been studied. Also, the novel chaotic systems have been synchronized using novel synchronization technique multi-switching compound difference synchronization and its application have been shown in the field of secure communication. Numerical simulations have been undertaken to validate the efficacy of the synchronization in secure communication.

Subcritical Hopf Equilibrium Points in Boundary of the Stability Region

A complete characterization of the boundary of the stability region of a class of nonlinear autonomous dynamical systems is developed admitting the existence of Subcritical Hopf nonhyperbolic equilibrium points on the boundary of the stability region. The characterization of the stability region developed in this paper is an extension of the characterization already developed in the literature, which considers only hyperbolic equilibrium point. Under the transversality condition, it is shown the boundary of the stability region is comprised of the stable manifolds of all equilibrium points on the boundary of the stability region, including the stable manifolds of the subcritical Hopf equilibrium points of type k, with 0<=k<=(n-2), which belong to the boundary of the stability region.

Successive approximation: A survey on stable manifold of fractional differential systems

This paper outlines a reliable strategy to approximate the local stable manifold near a hyperbolic equilibrium point for nonlinear systems of differential equations of fractional order. Furthermore, the local behavior of these systems near a hyperbolic equilibrium point is investigated based on the fractional Hartman-Grobman theorem. The fractional derivative is described in the Caputo sense. The solution existence, uniqueness, stability and convergence of the proposed scheme is discussed. Finally, the validity and applicability of our approach is examined with the use of a solvable model method.

Seismic Response Analysis on a Chaotic System

This paper is concerned with the dynamical behavior of a chaotic system which is a model for seismic response of structures. The local bifurcation of the non-hyperbolic equilibrium point of the chaotic system is investigated by using center manifold method. The transcritical bifurcation is analyzed in detail. Based on numerical simulations, spectrums of maximal Lyapunov exponent and the bifurcation diagrams are presented for the dynamic analysis. The method proposed can be used as a reference of nonlinear seismic response analysis.

THE CONLEY INDEX AND GLOBAL BIFURCATIONS I: CONCEPTS AND THEORY

The Morse index is defined for a hyperbolic equilibrium point of a vector field as the dimension of its unstable manifold. This paper presents the theory of the Conley index, which is a farreaching generalization of the Morse index. It also outlines how this new index can be used to solve global bifurcation problems. In fact, we shall argue that the Conley index methods constitute the most appropriate and natural set of tools for studying global aspects of flows. The reason for this is that we have known since the time of Poincaré that global questions have always involved topological considerations. (A typical elementary question involving topology is the following: in what subsets of Euclidean space is it possible to define a "potential function" if we are given its "gradient" — in the form of a smooth vector field, say?) The presentation we give is elementary and is addressed to those with no more than a passing acquaintance with the problems and methods of topology. The aim will be to motivate the concepts introduced, to give a number of examples and to only present the theory that is needed for applications. A second part will present illustrative applications in greater detail.