FOLLOWING A SADDLE-NODE OF PERIODIC ORBITS' BIFURCATION CURVE IN CHUA'S CIRCUIT

2009 ◽  
Vol 19 (02) ◽  
pp. 487-495 ◽  
Author(s):  
E. FREIRE ◽  
E. PONCE ◽  
J. ROS
Keyword(s):  
Periodic Orbits ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Numerical Continuation ◽  
Bifurcation Curve ◽  
Saddle Node Bifurcation ◽  
Leading Role ◽  
Chua’S Oscillator ◽  
Saddle Node ◽  
Parametric Region

Starting from previous analytical results assuring the existence of a saddle-node bifurcation curve of periodic orbits for continuous piecewise linear systems, numerical continuation is done to get some primary bifurcation curves for the piecewise linear Chua's oscillator in certain dimensionless parameter plane. The primary period doubling, homoclinic and saddle-node of periodic orbits' bifurcation curves are computed. A Belyakov point is detected in organizing the connection of these curves. In the parametric region between period-doubling, focus-center-limit cycle and homoclinic bifurcation curves, chaotic attractors coexist with stable nontrivial equilibria. The primary saddle-node bifurcation curve plays a leading role in this coexistence phenomenon.

Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?

2015 ◽  
Vol 25 (13) ◽  
pp. 1550185 ◽  
Author(s):  
Tetsushi Ueta ◽  
Daisuke Ito ◽  
Kazuyuki Aihara
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Stable Limit Cycle ◽  
Period Doubling ◽  
Transient Behavior ◽  
Control Input ◽  
Stable Limit ◽  
Unstable Periodic Orbits ◽  
Saddle Node Bifurcation ◽  
Saddle Node

We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark–Sacker bifurcations whose periodic orbits continue to exist through the bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate the suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic orbit can restore a stable limit cycle which disappeared after the saddle-node bifurcation.

scholarly journals Complex Behaviors of Epidemic Model with Nonlinear Rewiring Rate

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Ding Fang ◽  
Yongxin Zhang ◽  
Wendi Wang
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Epidemic Model ◽  
Multiple Equilibria ◽  
Homoclinic Bifurcation ◽  
Propagation Model ◽  
Bifurcation Curve ◽  
Saddle Node Bifurcation ◽  
Saddle Node

An SIS propagation model with the nonlinear rewiring rate on an adaptive network is considered. It is found by bifurcation analysis that the model has the complex behaviors which include the transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Especially, a bifurcation curve with “S” shape emerges due to the nonlinear rewiring rate, which leads to multiple equilibria and twice saddle-node bifurcations. Numerical simulations show that the model admits a homoclinic bifurcation and a saddle-node bifurcation of the limit cycle.

Routes to Chaos in Squeeze-Film Dampers

2004 ◽  
Author(s):  
Jawaid I. Inayat-Hussain ◽  
Njuki W. Mureithi
Keyword(s):  
Period Doubling ◽  
Squeeze Film ◽  
Saddle Node Bifurcation ◽  
Chaotic Vibrations ◽  
Saddle Node ◽  
Highly Nonlinear ◽  
Squeeze Film Dampers ◽  
Route To Chaos ◽  
Type 3 ◽  
Period Doubling Bifurcations

This work reports on a numerical study undertaken to investigate the imbalance response of a rigid rotor supported by squeeze-film dampers. Two types of damper configurations were considered, namely, dampers without centering springs, and eccentrically operated dampers with centering springs. For a rotor fitted with squeeze-film dampers without centering springs, the study revealed the existence of three regimes of chaotic motion. The route to chaos in the first regime was attributed to a sequence of period-doubling bifurcations of the period-1 (synchronous) rotor response. A period-3 (one-third subharmonic) rotor whirl orbit, which was born from a saddle-node bifurcation, was found to co-exist with the chaotic attractor. The period-3 orbit was also observed to undergo a sequence of period-doubling bifurcations resulting in chaotic vibrations of the rotor. The route to chaos in the third regime of chaotic rotor response, which occurred immediately after the disappearance of the period-3 orbit due to a saddle-node bifurcation, was attributed to a possible boundary crisis. The transitions to chaotic vibrations in the rotor supported by eccentric squeeze-film dampers with centering springs were via the period-doubling cascade and type 3 intermittency routes. The type 3 intermittency transition to chaos was due to an inverse period-doubling bifurcation of the period-2 (one-half subharmonic) rotor response. The unbalance response of the squeeze-film-damper supported rotor presented in this work leads to unique non-synchronous and chaotic vibration signatures. The latter provide some useful insights into the design and development of fault diagnostic tools for rotating machinery that operate in highly nonlinear regimes.

Saddle–node bifurcation of invariant cones in 3D piecewise linear systems

2012 ◽  
Vol 241 (5) ◽  
pp. 623-635 ◽  
Author(s):  
Victoriano Carmona ◽  
Soledad Fernández-García ◽  
Emilio Freire
Keyword(s):  
Linear Systems ◽  
Piecewise Linear ◽  
Saddle Node Bifurcation ◽  
Piecewise Linear Systems ◽  
Saddle Node ◽  
Invariant Cones

BIFURCATION IN A PIECEWISE LINEAR CIRCUIT WITH SWITCHING BOUNDARIES

2012 ◽  
Vol 22 (02) ◽  
pp. 1250034 ◽  
Author(s):  
ZHENGDI ZHANG ◽  
QINSHENG BI
Keyword(s):  
Periodic Orbits ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Generalized Jacobian ◽  
Linear Circuit ◽  
The Stability

By introducing time-dependent power source, a periodically excited piecewise linear circuit with double-scroll is established. In the absence of the excitation, all possible equilibrium points as well as the stability conditions are presented. Analyzing the corresponding characteristic equations with perturbation method, Hopf bifurcation conditions associated with the equilibria are derived, which can be demonstrated by the numerical simulations. The Hopf bifurcations of the two symmetric equilibrium points may cause two symmetric periodic orbits, which lead to single-scroll chaotic attractors via sequences of period-doubling bifurcations with the variation of the parameters. The two chaotic attractors expand to interact with each other to form an enlarged chaotic attractor with double-scroll. The behaviors on the switching boundaries are investigated by the generalized Jacobian matrix. When periodic excitation is applied to work on the circuit, three periodic orbits with the frequency of the excitation may exist, which can be called generalized equilibrium points (GEPs) with the same characteristic polynomials as those of the corresponding equilibrium points for the autonomous case. It is shown that when the trajectories do not pass across the switching boundaries, the solutions are the same as the GEPs. However, when the trajectories pass across the switching boundaries, complicated behaviors will take place. Three forms of chaotic attractors via different bifurcations can be observed and the influence of the switching boundaries on the phase portraits is discussed to explore the mechanism of the dynamical evolution.

scholarly journals Algebraic Limit Cycles in Piecewise Linear Differential Systems

2018 ◽  
Vol 28 (03) ◽  
pp. 1850039 ◽  
Author(s):  
Claudio A. Buzzi ◽  
Armengol Gasull ◽  
Joan Torregrosa
Keyword(s):  
Limit Cycles ◽  
Piecewise Linear ◽  
Saddle Node Bifurcation ◽  
Differential Systems ◽  
Saddle Node ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

This paper is devoted to study the algebraic limit cycles of planar piecewise linear differential systems. In particular, we present examples exhibiting two explicit hyperbolic algebraic limit cycles, as well as some one-parameter families with a saddle-node bifurcation of algebraic limit cycles. We also show that all degrees for algebraic limit cycles are allowed.

scholarly journals DELAY-INDUCED MULTISTABILITY NEAR A GLOBAL BIFURCATION

2008 ◽  
Vol 18 (06) ◽  
pp. 1759-1765 ◽  
Author(s):  
J. HIZANIDIS ◽  
R. AUST ◽  
E. SCHÖLL
Keyword(s):  
Fixed Point ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Generic Model ◽  
Stable Node ◽  
Saddle Node Bifurcation ◽  
Infinite Dimensional ◽  
Saddle Node ◽  
Homoclinic Bifurcations ◽  
Time Delayed Feedback

We study the effect of a time-delayed feedback within a generic model for a saddle-node bifurcation on a limit cycle. Without delay the only attractor below this global bifurcation is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit cycles are found in accordance with Shilnikov's theorems.

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