scholarly journals On born of in-phase limit cycle in ensemble of excitatory coupled FitzHugh-Nagumo elements

2018 ◽  
Vol 20 (3) ◽  
pp. 273-281
Author(s):  
A. G. Korotkov ◽  
T. A. Levanova
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Limit Cycle ◽  
Phenomenological Model ◽  
Coupling Strength ◽  
Active Element ◽  
Smooth Approximation ◽  
Impulse Function

We proposed and studied numerically efficient phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements that are coupled by symmetric synaptic excitatory coupling. This coupling is defined by function that depends on phase of active element and that is smooth approximation of rectangular impulse function. Above-mentioned coupling depends on three parameters that define the beginning of element activation, the duration of the activation and the coupling strength. We show analytically that in the phase space of the model there exists stable in-phase limit cycle that corresponds to regular oscillations with equal phases and frequencies of elements. It is proved that this limit cycle is a result of supercritical Andronov-Hopf bifurcation. The chart of activity regimes is depicted on the plane of parameters that define beginning and duration of activation. The boundaries of bifurcations that lead to birth of this cycle are found.

scholarly journals On born of in-phase limit cycle in ensemble of excitatory coupled FitzHugh-Nagumo elements

2018 ◽  
Vol 20 (3) ◽  
pp. 273-281
Author(s):  
Alexander G. Korotkov ◽  
Tatiana A. Levanova
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Limit Cycle ◽  
Phenomenological Model ◽  
Coupling Strength ◽  
Active Element ◽  
Smooth Approximation ◽  
Impulse Function

We proposed and studied numerically efficient phenomenological model of ensemble of two FitzHugh-Nagumo neuron-like elements that are coupled by symmetric synaptic excitatory coupling. This coupling is defined by function that depends on phase of active element and that is smooth approximation of rectangular impulse function. Above-mentioned coupling depends on three parameters that define the beginning of element activation, the duration of the activation and the coupling strength. We show analytically that in the phase space of the model there exists stable in-phase limit cycle that corresponds to regular oscillations with equal phases and frequencies of elements. It is proved that this limit cycle is a result of supercritical Andronov-Hopf bifurcation. The chart of activity regimes is depicted on the plane of parameters that define beginning and duration of activation. The boundaries of bifurcations that lead to birth of this cycle are found.

Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely-spaced modes

2021 ◽  
Author(s):  
Lukas Woiwode ◽  
Alexander F. Vakakis ◽  
Malte Krack
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Dry Friction ◽  
The Self ◽  
Perturbation Approach ◽  
Friction Damping ◽  
Periodic Oscillations ◽  
Analytical Results ◽  
Fold Bifurcation ◽  
Phase Motion

Abstract It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase of the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of Complexification-Averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes) and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors' knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations.

Influence of Hopf Bifurcation Dynamics on Conduction Failure of Action Potentials Along Nerve C-Fiber

2019 ◽  
Vol 29 (07) ◽  
pp. 1950093 ◽  
Author(s):  
Xinjing Zhang ◽  
Huaguang Gu
Keyword(s):  
Hopf Bifurcation ◽  
Coupling Strength ◽  
The Other ◽  
Dimensional Parameter ◽  
External Stimulation ◽  
C Fiber ◽  
Current Threshold ◽  
Stimulation Period ◽  
Conduction Failure ◽  
A Chain

Contrary to faithful conduction of every action potential or spike along the axon, some spikes induced by the external stimulation with a high frequency at one end of the unmyelinated nerve fiber (C-fiber) disappear during the conduction process to the other end, which leads to conduction failure. Many physiological functions such as information coding or pathological pain are involved. In the present paper, the dynamic mechanism of the conduction failure is well interpreted by two characteristics of the focus near Hopf bifurcation of the Hodgkin–Huxley (HH) model. One is that the current threshold to evoke a spike from the after-potential corresponding to the focus exhibits damping oscillations, and the other is that the damping oscillations exhibit an internal period. A chain network model composed of HH neurons and stimulated by the external periodic stimulation is used to stimulate C-fiber. In the two-dimensional parameter space of the stimulation period and coupling strength, the conduction failure appears for the coupling strength lower than that of the faithful conduction, which is due to some maximal values of the coupling current for low coupling strength not being strong enough to evoke spikes, and the coupling strength threshold between the faithful conduction and conduction failure exhibiting damping oscillations with respect to the stimulation period, due to the damping oscillations of the current threshold. The damping oscillations of the coupling strength exhibit close correlations to those of the current threshold. The coupling strength for the conduction failure exhibits maximal values as the stimulation period is approximated to 1-, 2-, 3- or 4-times of the internal period and the maximal values decrease with increasing stimulation period. In addition, the correspondence between the simulation results and the previous experimental observations is discussed. The results present deep insights into the dynamics of the conduction failure with Hopf bifurcation and are helpful to investigate the influence of other modulation factors on the conduction failure.

scholarly journals Rendering a subcritical Hopf bifurcation supercritical

1996 ◽  
Vol 317 ◽  
pp. 91-109 ◽  
Author(s):  
Po Ki Yuen ◽  
Haim H. Bau
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Thermal Convection ◽  
Control Strategy ◽  
Chaotic Motion ◽  
Feedback Controller ◽  
Nonlinear Feedback ◽  
Naturally Occurring ◽  
Subcritical Hopf Bifurcation ◽  
Stable Periodic

It is demonstrated experimentally and theoretically that through the use of a nonlinear feedback controller, one can render a subcritical Hopf bifurcation supercritical and thus dramatically modify the nature of the flow in a thermal convection loop heated from below and cooled from above. In particular, we show that the controller can replace the naturally occurring chaotic motion with a stable, periodic limit cycle. The control strategy consists of sensing the deviation of fluid temperatures from desired values at a number of locations inside the loop and then altering the wall heating to counteract such deviations.

Limit Cycles in a Class of Quartic Kolmogorov Model with Three Positive Equilibrium Points

2015 ◽  
Vol 25 (06) ◽  
pp. 1550080 ◽  
Author(s):  
Chaoxiong Du ◽  
Yirong Liu ◽  
Qi Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Equilibrium Points ◽  
Positive Equilibrium ◽  
Bifurcation Problem ◽  
Kolmogorov Model ◽  
Limit Cycle Bifurcation ◽  
Perturbed Model

Limit cycle bifurcation problem of Kolmogorov model is interesting and significant both in theory and applications. In this paper, we will focus on investigating limit cycles for a class of quartic Kolmogorov model with three positive equilibrium points. Perturbed model can bifurcate three small limit cycles near (1, 2) or (2, 1) under a certain condition and can bifurcate one limit cycle near (1, 1). In addition, we have given some examples of simultaneous Hopf bifurcation and the structure of limit cycles bifurcated from three positive equilibrium points. The limit cycle bifurcation problem for Kolmogorov model with several positive equilibrium points are less seen in published references. Our result is good and interesting.

scholarly journals Hopf bifurcations in plasma layers between rigid planes in thermal magnetohydrodynamics, via a simple formula

2020 ◽  
Vol 31 (4) ◽  
pp. 985-997
Author(s):  
Salvatore Rionero
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Simple Formula ◽  
Nobel Laureate ◽  
Critical Value ◽  
Transverse Magnetic ◽  
Hopf Bifurcations ◽  
Linearized Problem ◽  
Globally Attractive ◽  
Chandrasekhar Number

Abstract The phenomenon produced by the Hopf bifurcations is of notable importance. In fact, a Hopf bifurcation—guaranteeing the existence of an unsteady periodic solution of the linearized problem at stake—is also an optimum limit cycle candidate of the nonlinear associated problem and, if non linearly globally attractive, is an absorbing set and an effective limit cycle. The present paper deals with the onset of Hopf bifurcations in thermal magnetohydrodynamics (MHD). Precisely, it is devoted to characterization—via a simple formula—of the Hopf bifurcations threshold in horizontal plasma layers between rigid planes, heated from below and embedded in a constant transverse magnetic field. This problem, remarked clearly and notably by the Nobel Laureate Chandrasekhar (Nature 175:417–419, 1955), constitutes a difficulty met by him and—for plasma layers between rigid planes electricity perfectly conducting—is, as far as we know, still not removed. Let $$m_0$$ m 0 be the thermal conduction rest state and let $$P_r, P_m, R, Q$$ P r , P m , R , Q , be the Prandtl, the Prandtl magnetic, the Rayleigh and the Chandrasekhar number, respectively. Recognized (according to Chandrasekhar) that the instability of $$m_0$$ m 0 via Hopf bifurcation can occur only in a plasma with $$P_m>P_r$$ P m > P r , in this paper it is shown that the Hopf bifurcations occur if and only if $$\begin{aligned} Q>Q_c=\displaystyle \frac{4\pi ^2[1+P_r(\mu /2\pi )^4]}{P_m-P_r}, \end{aligned}$$ Q > Q c = 4 π 2 [ 1 + P r ( μ / 2 π ) 4 ] P m - P r , with $$ \mu =7.8532$$ μ = 7.8532 . Moreover, the critical value of R at which the Hopf bifurcation occurs is characterized via the smallest zero of the second invariant of the spectrum equation governing the most destabilizing perturbation. The critical value of Q, in the free-rigid and rigid-free cases is shown to be $$\displaystyle \frac{1}{4}$$ 1 4 of the previous value.

The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

2014 ◽  
Vol 11 (4) ◽  
pp. 045002 ◽  
Author(s):  
Aurore Woller ◽  
Didier Gonze ◽  
Thomas Erneux
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Goodwin Model

A Perturbation Procedure for Limit Cycle and Heteroclinic Connection Analysis of Certain Self-Excited Oscillator

2012 ◽  
Vol 204-208 ◽  
pp. 4529-4532
Author(s):  
Yang Yang Chen ◽  
Wei Zhao ◽  
Le Wei Yan
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Method ◽  
Limit Cycle ◽  
Present Method ◽  
Elliptic Functions ◽  
Hyperbolic Functions ◽  
Perturbation Procedure ◽  
Heteroclinic Connection ◽  
Excited Oscillator

A perturbation procedure, in which the elliptic perturbation method and the hyperbolic perturbation method are applied, is presented for predicting heteroclinic connection of limit cycle or self-excited ocsillator. The limit cycle can be analytically　constructed first by the elliptic perturbation method after Hopf bifurcation, in which the amplitude of limit cycle can be　controlled by the modulus of elliptic functions. The heteroclinic trajectories, which are formed by the heteroclinic connection of limit cycle, can also be constructed by similar perturbation　procedure but adopting the hyperbolic functions instead of　elliptic functions. And the criterion of heteroclinic connection is given in the perturbation procedure. A typical self-excited oscillator is studied in detail to assess the present method.

scholarly journals QUANTUM GRAVITY PHENOMENOLOGY, LORENTZ INVARIANCE AND DISCRETENESS

2004 ◽  
Vol 19 (24) ◽  
pp. 1829-1840 ◽  
Author(s):  
FAY DOWKER ◽  
JOE HENSON ◽  
RAFAEL D. SORKIN
Keyword(s):  
Phase Space ◽  
Cosmic Rays ◽  
Quantum Gravity ◽  
Phenomenological Model ◽  
Lorentz Invariance ◽  
Minkowski Spacetime ◽  
High Energy ◽  
High Energy Cosmic Rays ◽  
Quantum Gravity Phenomenology ◽  
Massive Particles

Contrary to what is often stated, a fundamental spacetime discreteness need not contradict Lorentz invariance. A causal set's discreteness is in fact locally Lorentz invariant, and we recall the reasons why. For illustration, we introduce a phenomenological model of massive particles propagating in a Minkowski spacetime which arises from an underlying causal set. The particles undergo a Lorentz invariant diffusion in phase space, and we speculate on whether this could have any bearing on the origin of high energy cosmic rays.

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