Bifurcations of Invariant Torus and Knotted Periodic Orbits in Generalized Hopf-Langford Type Equations

2021 ◽  
Author(s):  
Yanggeng Fu ◽  
Jibin Li
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Invariant Torus ◽  
Bifurcation Theory ◽  
Three Dimensional ◽  
Heteroclinic Orbits ◽  
Stable Limit ◽  
Planar System ◽  
Perturbed System ◽  
Stable Limit Cycles

Abstract In this paper, we study the bifurcations of invariant torus and knotted periodic orbits for generalized Hopf-Langford type equations. By using bifurcation theory of dynamical systems, we obtain the exact explicit form of the heteroclinic orbits and knot periodic orbits. Moreover, under small perturbation, we prove that the perturbed planar system has two symmetric stable limit cycles created by Poincare bifurcations. Therefore, the corresponding three-dimensional perturbed system has an attractive invariant rotation torus.

The Nonlinear Behavior of Heavy-Duty Truck Combinations With Respect to Straightline Stability

1989 ◽  
Vol 111 (4) ◽  
pp. 577-582 ◽  
Author(s):  
A. Stribersky ◽  
P. S. Fancher
Keyword(s):  
Nonlinear Stability ◽  
Bifurcation Theory ◽  
Nonlinear Behavior ◽  
Stable Limit ◽  
Heavy Duty ◽  
Stability Boundaries ◽  
Stability Behavior ◽  
Heavy Duty Truck ◽  
Straight Line ◽  
Stable Limit Cycles

A comparison of the nonlinear stability behavior of the steady state straight line motion of truck combinations with and without a second trailer is shown. These investigations have been done by applying bifurcation theory. Stability boundaries in the parameter space and the corresponding bifurcation solutions are given. Depending on the loading conditions, unstable and also stable limit cycles have been found. Particular emphasis is given to the influence of the frictional coupling between tire and road on the nonlinear stability behavior of these vehicles.

scholarly journals Chaos in sociobiology

1995 ◽  
Vol 51 (3) ◽  
pp. 439-451 ◽  
Author(s):  
J.R. Christie ◽  
K. Gopalsamy ◽  
Jibin Li
Keyword(s):  
Periodic Orbits ◽  
Evolutionary Biology ◽  
Mating Behaviour ◽  
Continuous Family ◽  
Heteroclinic Cycle ◽  
Planar System ◽  
Perturbed System ◽  
Battle Of The Sexes ◽  
Game Theoretic ◽  
Time Periodic

It is shown that the dynamical game theoretic mating behaviour of males and females can be modelled by a planar system of autonomous ordinary differential equations. This system occurs in modelling “the battle of the sexes” in evolutionary biology. The existence of a heteroclinic cycle and a continuous family of periodic orbits of the system is established; then the dynamical characteristics of a time-periodic perturbation of the system are investigated. By using the well-known Melnikov's method, a sufficient condition is obtained for the perturbed system to have a transverse heteroclinic cycle and hence to possess chaotic behaviour in the sense of Smale. Finally, subharmonic Melnikov theory is used to obtain a criterion for the existence of subharmonic periodic orbits of the perturbed system.

STUDY ON THE NONLINEAR VIBRATION OF SWCNTs WITH INITIAL AXIAL STRESS BASED ON THE BIFURCATION THEORY OF DYNAMICAL SYSTEMS

2010 ◽  
Vol 24 (18) ◽  
pp. 1979-1986
Author(s):  
YAN YAN ◽  
WENQUAN WANG ◽  
LIXIANG ZHANG
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Initial Conditions ◽  
Axial Stress ◽  
Sufficient Conditions ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors ◽  
Families Of Periodic Orbits

The nonlinear dynamical behaviors of single-walled carbon nanotubes (SWCNTs) with the initial axial stress are investigated based on Donnell's cylindrical shell model. By using the bifurcation theory of dynamical systems on a vibration equation of SWCNTs, the existence of centers, saddles, families of periodic orbits, homoclinic orbits and/or heteroclinic orbits is shown. Under different initial conditions, various sufficient conditions to guarantee the existence of the above solutions are given and the dynamical behaviors of SWCNTs are well-known.

KNOTTED PERIODIC ORBITS AND CHAOTIC BEHAVIOR OF A CLASS OF THREE-DIMENSIONAL FLOWS

2011 ◽  
Vol 21 (09) ◽  
pp. 2505-2523 ◽  
Author(s):  
JIBIN LI ◽  
FENGJUAN CHEN
Keyword(s):  
Periodic Orbits ◽  
Invariant Torus ◽  
Vector Fields ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Symmetric Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Perturbed Systems

This paper considers a class of three-dimensional systems constructed by rotating some planar symmetric polynomial vector fields. It shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on a family of invariant torus. For two three-dimensional systems, exact explicit parametric representations of the knotted periodic orbits are given. For their perturbed systems, the chaotic behavior is discussed by using two different methods.

ON PERIODIC ORBITS AND HOMOCLINIC BIFURCATIONS IN CHUA’S CIRCUIT WITH A SMOOTH NONLINEARITY

1993 ◽  
Vol 03 (02) ◽  
pp. 363-384 ◽  
Author(s):  
ALEXANDER I. KHIBNIK ◽  
DIRK ROOSE ◽  
LEON O. CHUA
Keyword(s):  
Periodic Orbits ◽  
Bifurcation Theory ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Homoclinic Bifurcation ◽  
Chua’S Circuit ◽  
Chua's Circuit ◽  
Homoclinic Bifurcations ◽  
Cubic Polynomial

We present the bifurcation analysis of Chua’s circuit equations with a smooth nonlinearity, described by a cubic polynomial. Our study focuses on phenomena that can be observed directly in the numerical simulation of the model, and on phenomena which are revealed by a more elaborate analysis based on continuation techniques and bifurcation theory. We emphasize how a combination of these approaches actually works in practice. We compare the dynamics of Chua’s circuit equations with piecewise-linear and with smooth nonlinearity. The dynamics of these two variants are similar, but we also present some differences. We conjecture that this similarity is due to the central role of homoclinicity in this model. We describe different ways in which the type of a homoclinic bifurcation influences the behavior of branches of periodic orbits. We present an overview of codimension 1 bifurcation diagrams for principal periodic orbits near homoclinicity for three-dimensional systems, both in the generic case and in the case of odd symmetry. Most of these diagrams actually occurs in the model. We found several homoclinic bifurcations of codimension 2, related to the so called resonant conditions. We study one of these bifurcations, a double neutral saddle loop.

scholarly journals CHAOTIC BEHAVIOR OF THREE COMPETING SPECIES OF MAY–LEONARD MODEL UNDER SMALL PERIODIC PERTURBATIONS

2001 ◽  
Vol 11 (02) ◽  
pp. 435-447 ◽  
Author(s):  
VALENTIN S. AFRAIMOVICH ◽  
SZE-BI HSU ◽  
HUEY-ER LIN
Keyword(s):  
Invariant Torus ◽  
Chaotic Behavior ◽  
System Modeling ◽  
Heteroclinic Orbits ◽  
Volterra System ◽  
Regular Behavior ◽  
Perturbed System ◽  
Periodic Perturbations ◽  
Small Periodic ◽  
Theoretical Results

The influence of periodic perturbations to a Lotka–Volterra system, modeling a competition between three species, is studied, provided that in the unperturbed case the system has a unique attractor — a contour of heteroclinic orbits joining unstable equilibria. It is shown that the perturbed system may manifest regular behavior corresponding to the existence of a smooth invariant torus, and, as well, may have chaotic regimes depending on some parameters. Theoretical results are confirmed by numerical simulations.

STABILITY, SYMMETRY-BREAKING BIFURCATIONS AND CHAOS IN DISCRETE DELAYED MODELS

2008 ◽  
Vol 18 (05) ◽  
pp. 1477-1501 ◽  
Author(s):  
MINGSHU PENG ◽  
YUAN YUAN
Keyword(s):  
Limit Cycles ◽  
Bifurcation Theory ◽  
Invariant Tori ◽  
Pitchfork Bifurcation ◽  
Chaotic Attractors ◽  
Stable Limit ◽  
Stable Equilibria ◽  
Multiple Stable Equilibria ◽  
Stable Limit Cycles ◽  
Stable Invariant

In this paper, we use the standard bifurcation theory to study rich dynamics of time-delayed coupling discrete oscillators. Equivariant bifurcations including equivariant Neimark–Sacker bifurcation, equivariant pitchfork bifurcation and equivariant periodic doubling bifurcation are analyzed in detail. In the application, we consider a ring of identical discrete delayed Ikeda oscillators. Multiple oscillation patterns, such as multiple stable equilibria, stable limit cycles, stable invariant tori and multiple chaotic attractors, are shown.

scholarly journals Three-dimensional geometry and topology effects in viscous streaming

2022 ◽  
Vol 933 ◽  
Author(s):  
Fan Kiat Chan ◽  
Yashraj Bhosale ◽  
Tejaswin Parthasarathy ◽  
Mattia Gazzola
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Direct Numerical Simulations ◽  
Flow Topology ◽  
And Topology ◽  
Streaming Flows ◽  
The Impact

Recent studies on viscous streaming flows in two dimensions have elucidated the impact of body curvature variations on resulting flow topology and dynamics, with opportunities for microfluidic applications. Following that, we present here a three-dimensional characterization of streaming flows as functions of changes in body geometry and topology, starting from the well-known case of a sphere to progressively arrive at toroidal shapes. We leverage direct numerical simulations and dynamical systems theory to systematically analyse the reorganization of streaming flows into a dynamically rich set of regimes, the origins of which are explained using bifurcation theory.

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