NDLI: Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system

Content Provider	Springer Nature Link
Author	Gambi, G. Choudhury, S. Roy Chen, T.
Copyright Year	2012
Abstract	In order to understand the onset of hyperchaotic behavior recently observed in many systems, we study bifurcations in the modified Chen system leading from simple dynamics into chaotic regimes. In particular, we demonstrate that the existence of only one fixed point of the system in all regions of parameter space implies that this simple point attractor may only be destabilized via a Hopf or double Hopf bifurcation as system parameters are varied. Saddle-node, transcritical and pitchfork bifurcations are precluded.The normal form immediately following double Hopf bifurcations is constructed analytically by the method of multiple scales. Analysis of this generalized double Hopf normal form along standard lines reveals possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. However, considering these more carefully, we find that only certain combinations or sequences of these dynamical regimes are possible, while others derived and considered in earlier work are in fact mathematically impossible.We also discuss the post-bifurcation dynamics in the context of two intermittent routes to chaos (routes following either (i) subcritical or (ii) supercritical Hopf or double Hopf bifurcations). In particular, the route following supercritical bifurcations is somewhat subtle. Such behavior following repeated Hopf bifurcations is well-known and widely observed, including in the classical Ruelle–Takens and quasiperiodic routes to chaos. However, to the best of our knowledge, it has not been considered in the context of the double-Hopf normal form, although it has been numerically observed and tracked in the post-double Hopf regime.Numerical simulations are employed to corroborate these various predictions from the normal form. They reveal the existence of stable periodic and toroidal attractors in the post-supercritical-Hopf cases, and either attractors at infinity or bounded chaotic dynamics following subcritical Hopf bifurcations.Future work will map out the remainder of the routes into the chaotic regimes, including further bifurcations of the post-supercritical-Hopf two- and three-tori via either torus doubling or breakdown.
Starting Page	1439
Ending Page	1455
Page Count	17
File Format	PDF
ISSN	0924090X
Journal	Nonlinear Dynamics
Volume Number	69
Issue Number	3
e-ISSN	1573269X
Language	English
Publisher	Springer Netherlands
Publisher Date	2012-02-24
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Double-Hopf bifurcations Normal forms Modified post-bifurcation dynamics Automotive Engineering Mechanics Mechanical Engineering Vibration, Dynamical Systems, Control
Content Type	Text
Resource Type	Article
Subject	Ocean Engineering Applied Mathematics Control and Systems Engineering Mechanical Engineering Electrical and Electronic Engineering Aerospace Engineering

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Modified Post-Bifurcation Dynamics and Routes to Chaos from Double-Hopf Bifurcations in a Hyperchaotic System (2011)

Dynamical analysis in a 4D hyperchaotic system

The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

Dynamics of a hyperchaotic Lorenz-type system

Hopf bifurcation and chaos in fractional-order modified hybrid optical system

Zero–Hopf bifurcation in a hyperchaotic Lorenz system

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system

Hopf and zero-Hopf bifurcations in the Hindmarsh–Rose system

Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system

Similar Documents

Modified Post-Bifurcation Dynamics and Routes to Chaos from Double-Hopf Bifurcations in a Hyperchaotic System (2011)

Dynamical analysis in a 4D hyperchaotic system

The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

Dynamics of a hyperchaotic Lorenz-type system

Hopf bifurcation and chaos in fractional-order modified hybrid optical system

Zero–Hopf bifurcation in a hyperchaotic Lorenz system

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system

Hopf and zero-Hopf bifurcations in the Hindmarsh–Rose system

Modified post-bifurcation dynamics and routes to chaos from double-Hopf bifurcations in a hyperchaotic system