Saddle-node bifurcation

2004

In this paper, we consider a two-dimensional map (a duopoly game) in which the fixed point is destabilized via a subcritical Neimark-Hopf (N-H) bifurcation. Our aim is to investigate, via numerical examples, some global bifurcations associated with the appearance of repelling closed invariant curves involved in the Neimark-Hopf bifurcations. We shall see that the mechanism is not unique, and that it may be related to homoclinic connections of a saddle cycle, that is to a closed invariant curve formed by the merging of a branch of the stable set of the saddle with a branch of the unstable set of the same saddle. This will be shown by analyzing the bifurcations arising inside a periodicity tongue, i.e., a region of the parameter space in which an attracting cycle exists.

The tool developed in this work is a software package which objective is the graphical detection, in phase portraits, of 1-codimension local bifurcation of non-linear discrete-time and continuous-time dynamical systems. Working with continuous-time systems, this tool is able to detect fold (also called tangent or saddle-node) and Hopf (or Andronov-Hopf) bifurcation. With discrete-time systems, it is able to detect fold (or tangent), flip (or period-doubling), and Neimark-Sacker (or secondary Hopf) bifurcation. This tool is presented as a MatLab Toolbox that uses the principles of continuation for simulating the behavior of the system.

Journal of Vibration Testing and System Dynamics, 2024

In this paper, nonlinear dynamics of the product-quadratic systems with self-quadratic and crossing-quadratic vector fields is presented, which is the study continuation of product-quadratic systems with two product-quadratic vector fields. With a self-quadratic field, the stability and bifurcations of product quadratic systems are discussed. The saddle-sink bifurcation for saddle and sink is discussed, and the saddle-source bifurcation for saddle and source is presented. The saddle-saddle bifurcations of the first kind are presented. The appearing bifurcation conditions for sink-source and saddle-saddle are discussed. The inflection sink (or source) bifurcations are presented for the switching bifurcations for hyperbolic flow and saddles with hyperbolic-secant flow and sink (or source). With a crossingquadratic vector field, the dynamics and bifurcations for such a quadratic system are discussed. The saddle-center appearing bifurcations are presented through the parabola-saddle bifurcations. The center-center bifurcation and the saddle-saddle bifurcation of the second kind are discussed through the hyperbolic sink-source and circular sink-source bifurcations. The up-parabola-saddle bifurcations are for the switching of the center and hyperbolic flow with saddle and hyperbolic-secant flows. The down-parabola-saddle bifurcations are for the switching of the center and hyperbolic-secant flow with saddle and hyperbolic flows. The parabola-saddle on the infinite-equilibrium is called the switching bifurcations of saddle and center.

1998

Journal of Sound and Vibration, 1996

Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228), 2001

Physics Letters A, 2013

Bifurcation cascades in conservative systems are shown to exhibit a generalized diagram, which contains all relevant informations regarding the location of periodic orbits (resonances), their width (island size), irrational tori and the infinite higher-order resonances, showing the intricate way they are born. Contraction rates for islands sizes, along period-doubling bifurcations, are estimated to be α I ∼ 3.9. Results are demonstrated for the standard map and for the continuous Hénon-Heiles potential. The methods used here are very suitable to find periodic orbits in conservative systems, and to characterize the regular, mixed or chaotic dynamics as the nonlinear parameter is varied.

Journal of Kufa for Mathematics and Computer, 2015

In this paper we are interested in studying of bifurcation solutions of bifurcation equation (nonlinear system of algebraic equations with four or six parameters). Also, we found a new geometrical description of the Discriminate set(bifurcation set) with bifurcation spreading of the number of regular solutions in every region. In addution, we calculate the topological indices of the solutions of the problem.

Journal of Vibration Testing and System Dynamics, 2023

This paper presents a theory for nonlinear dynamics of dynamical systems with two variable-crossing univariate vector fields. Dynamical systems with a variable-crossing univariate linear and quadratic vector fields are discussed, and the corresponding bifurcation and global dynamics are presented. The saddle-center bifurcations are presented through parabola-saddle bifurcations. Dynamical systems with two crossing-variable univariate quadratic vector fields are discussed, and the switching and appearing bifurcations for saddles and centers are discussed through the first integral manifolds, and the homoclinic networks will be first presented. Double-inflection bifurcations are for appearance of the saddle-center network, and the homoclinic networks with centers are constructed. The saddle-center networks with limit cycles are presented from the first integral manifolds.

This paper is devoted to a bifurcation problems, based on some models, described by algebraic (cubic) equations with real coefficients. Bifurcation analysis, parametric representation of solutions and their asymptotic analysis and expressions are described in the frame of analytical approaches. The results, presented in the paper, can be used to help locate the bifurcation points of the solution curves. The results also allow the development of very efficient procedures for sensitivity analysis of dependences of solutions on the problem parameters.