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.

Delay-Induced Multistability in a Generic Model for Excitable Dynamics

2007 ◽  
Author(s):  
Johanne Hizanidis ◽  
Roland Aust ◽  
Eckehard Scho¨ll
Keyword(s):  
Global Bifurcation ◽  
Period Doubling ◽  
Generic Model ◽  
Stable Node ◽  
Neuron Models ◽  
System Variable ◽  
Infinite Dimensional ◽  
Saddle Node ◽  
Excitable Systems ◽  
The Difference

Motivated by real-world excitable systems such as neuron models and lasers, we consider a paradigmatic model for excitability with a global bifurcation, namely a saddle-node bifurcation on a limit cycle. We study the effect of a time-delayed feedback force in the form of the difference between a system variable at a certain time and at a delayed time. In the absence of delay the only attractor in the system in the excitability regime, below the 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 as well as saddle-node bifurcations of limit cycles are found in accordance with Shilnikov’s theorems.

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.

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.

Classification of Bifurcations of Quasi-Periodic Solutions Using Lyapunov Bundles

2014 ◽  
Vol 24 (12) ◽  
pp. 1430034 ◽  
Author(s):  
Kyohei Kamiyama ◽  
Motomasa Komuro ◽  
Tetsuro Endo ◽  
Kazuyuki Aihara
Keyword(s):  
Fixed Point ◽  
Lyapunov Exponent ◽  
Periodic Orbit ◽  
Period Doubling ◽  
Type A ◽  
Double Covering ◽  
Poincaré Section ◽  
Poincare Section ◽  
Mathematical Concepts ◽  
Saddle Node

In continuous-time dynamical systems, a periodic orbit becomes a fixed point on a certain Poincaré section. The eigenvalues of the Jacobian matrix at this fixed point determine the local stability of the periodic orbit. Analogously, a quasi-periodic orbit (2-torus) becomes an invariant closed curve (ICC) on a Poincaré section. From the Lyapunov exponents of an ICC, we can determine the time average of the exponential divergence rate of the orbit, which corresponds to the eigenvalues of a fixed point. We denote the Lyapunov exponent with the smallest nonzero absolute value as the Dominant Lyapunov Exponent (DLE). A local bifurcation manifests as a crossing or touch of the DLE locus with zero. However, the type of bifurcation cannot be determined from the DLE. To overcome this problem, we define the Dominant Lyapunov Bundle (DLB), which corresponds to the dominant eigenvectors of a fixed point. We prove that the DLB of a 1-torus in a map can be classified into four types: A+(annulus and orientation preserving), A-(annulus and orientation reversing), M (Möbius band), and F (focus). The DLB of a 2-torus in a flow can be classified into three types: A+× A+, A-× M (equivalently M × A-and M × M), and F × F. From the results, we conjecture the possible local bifurcations in both cases. For the 1-torus in a map, we conjecture that type A+and A-DLBs correspond to a saddle-node and period-doubling bifurcations, respectively, whereas a type M DLB denotes a double-covering bifurcation, and type F relates to a Neimark–Sacker bifurcation. Similarly, for the 2-torus in a flow, we conjecture that type A+× A+DLBs correspond to saddle-node bifurcations, type A-× M DLBs to double-covering bifurcations, and type F × F DLBs to the Neimark–Sacker bifurcations. After introducing the mathematical concepts, we provide a DLB-calculating algorithm and illustrate all of the above bifurcations by examples.

scholarly journals Global bifurcation analysis of Topp system

2019 ◽  
pp. 244-250
Author(s):  
Valery A. Gaiko ◽  
Henk W. Broer ◽  
Alef E. Sterk
Keyword(s):  
Limit Cycles ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Homoclinic Orbits ◽  
Chaotic Attractors ◽  
Dimensional Model ◽  
Saddle Node Bifurcation ◽  
3 Dimensional ◽  
Suitable Parameter ◽  
Parameter Values

In this paper, we study the 3-dimensional Topp model for the dynamics of diabetes. First, we reduce the model to a planar quartic system. In particular, studying global bifurcations, we prove that such a system can have at most two limit cycles. Next, we study the dynamics of the full 3-dimensional model. We show that for suitable parameter values an equilibrium bifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests that near this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arise through period doubling cascades of limit cycles.

RESONANCE STRUCTURE IN A WEAKLY DETUNED LASER WITH INJECTED SIGNAL

2001 ◽  
Vol 11 (10) ◽  
pp. 2587-2605 ◽  
Author(s):  
CATALINA MAYOL ◽  
MARIO A. NATIELLO ◽  
MARTÍN G. ZIMMERMANN
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Orbit ◽  
Global Bifurcation ◽  
Homoclinic Orbits ◽  
Resonance Structure ◽  
Saddle Node Bifurcation ◽  
Arnold Tongue ◽  
Saddle Node ◽  
Organizing Center ◽  
Qualitative Dynamics

We describe the qualitative dynamics and bifurcation set for a laser with injected signal for small cavity detunings. The main organizing center is the Hopf-saddle-node bifurcation from where a secondary Hopf bifurcation of a periodic orbit originates. We show that the laser's stable cw solution existing for low injections, also suffers a secondary Hopf bifurcation. The resonance structure of both tori interact, and homoclinic orbits to the "off" state are found inside each Arnold tongue. The accumulation of all the above resonances towards the Hopf-saddle-node singularity points to the occurrence of a highly degenerate global bifurcation at the codimension-2 point.

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 REAL CONTINUATION FROM THE COMPLEX QUADRATIC FAMILY: FIXED-POINT BIFURCATION SETS

2000 ◽  
Vol 10 (02) ◽  
pp. 391-414 ◽  
Author(s):  
BRUCE B. PECKHAM ◽  
JAMES MONTALDI
Keyword(s):  
Fixed Point ◽  
Numerical Study ◽  
Period Doubling ◽  
Real Parameter ◽  
Complex Conjugate ◽  
Phase Variable ◽  
Auxiliary Parameter ◽  
Primary Parameter ◽  
Saddle Node ◽  
Quadratic Family

This paper is primarily a numerical study of the fixed-point bifurcation loci — saddle-node, period-doubling and Hopf bifurcations — present in the family: [Formula: see text] where z is a complex dynamic (phase) variable, [Formula: see text] its complex conjugate, and C and A are complex parameters. We treat the parameter C as a primary parameter and A as a secondary parameter, asking how the bifurcation loci projected to the C plane change as the auxiliary parameter A is varied. For A=0, the resulting two-real-parameter family is a familiar one-complex-parameter quadratic family, and the local fixed-point bifurcation locus is the main cardioid of the Mandlebrot set. For A ≠ 0, the resulting two-real-parameter families are not complex analytic, but are still analytic (quadratic) when viewed as a map of ℛ2. Saddle-node and period-doubling loci evolve from points on the main cardioid for A=0 into closed curves for A ≠ 0. As A is varied further from 0 in the complex plane, the three sets interact in a variety of interesting ways. More generally, we discuss bifurcations of families of maps with some parameters designated as primary and the rest as auxiliary. The auxiliary parameter space is then divided into equivalence classes with respect to a specified set of bifurcation loci. This equivalence is defined by the existence of a diffeomorphism of corresponding primary parameter spaces which preserves the specified set of specified bifurcation loci. In our study there is a huge amount of complexity added by specifying the three fixed-point bifurcation loci together, rather than one at a time. We also provide a preliminary classification of the types of codimension-one bifurcations one should expect in general studies of families of two-parameter families of maps of the plane. Comments on numerical continuation techniques are provided as well.

A Novel Threshold Across which the Negative Stimulation Evokes Action Potential Near a Saddle-Node Bifurcation in a Neuronal Model with Ih Current

2019 ◽  
Vol 29 (14) ◽  
pp. 1950198 ◽  
Author(s):  
Linan Guan ◽  
Bing Jia ◽  
Huaguang Gu
Keyword(s):  
Action Potential ◽  
Phase Trajectory ◽  
Neuronal Model ◽  
Stable Node ◽  
Nonlinear Phenomenon ◽  
Negative Pulse ◽  
Dynamical Mechanism ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Pulse Stimulation

The negative or hyperpolarization pulse stimulation induces action potential, i.e. the post-inhibitory rebound spike, which has been widely observed in various single neurons with hyperpolarization-activated cation current ([Formula: see text]) in neuroscience and is suggested to be evoked from a focus near the Hopf bifurcation according to the traditional viewpoint of nonlinear dynamics. In the present paper, a novel viewpoint that post-inhibitory rebound spike can be evoked from a stable node near the saddle-node bifurcation on invariant circle (SNIC) is proposed, which can be well interpreted with hyperpolarization activation characteristic of [Formula: see text] current, bifurcation analysis, and threshold. Especially, the boundary between the subthreshold and suprathreshold initial values which respectively evoke subthreshold potential and action potential is acquired to be a threshold surface containing the saddle. [Formula: see text] current after the negative pulse stimulation for small conductance [Formula: see text] of [Formula: see text] is low enough to evoke just a subthreshold potential while for large [Formula: see text] is high enough to evoke a post-inhibitory rebound spike. For small [Formula: see text], the pulse induces the decrease of membrane potential [Formula: see text] and then the phase trajectory always stays within the subthreshold initial value region locating lower to the threshold surface with a nearly fixed [Formula: see text] value. For large [Formula: see text], the threshold surface changes and is composed of two parts: one part with a nearly fixed [Formula: see text] value and the other with a nearly fixed value of [Formula: see text] variable to describe [Formula: see text] inactivation probability. Although the negative pulse stimulation induces the decrease of [Formula: see text], [Formula: see text] increases to a level high enough and then the phase trajectory runs across the part with a nearly fixed [Formula: see text] value to form a post-inhibitory rebound spike. The appearance of the novel [Formula: see text] threshold is the internal dynamical mechanism for the generation of post-inhibitory rebound spike, and the external cause is that the negative pulse stimulation induces the phase trajectory to run across the [Formula: see text] threshold surface. The results present a novel nonlinear phenomenon and the corresponding dynamical mechanism related to post-inhibitory rebound spike induced by [Formula: see text] current near the SNIC bifurcation point.

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