Bifurcation Of Homoclinic Orbits Research Articles

Existence conditions for bifurcations of homoclinic orbits in a railway wheelset model

This paper investigates the bifurcations of homoclinic orbits to hyperbolic saddle points in a simplified railway wheelset model with cubic and quintuple nonlinear terms. Using Melnikov’s method, the sufficient conditions for the existence of the supercritical and the subcritical pitchfork bifurcations of homoclinic orbits are proven. To determine the integrability of the variational equations around homoclinic orbits in the meaning of differential Galois theory, the corresponding Fuchsian second-order differential equation for the normal variational equation and the Riemann P function are obtained. It is shown that the coefficients of the linear terms and the cubic coupling terms play a very significant role on influencing the existence of homoclinic orbits. While, the cubic coupling terms have little effect on the size of the left-hand and right-hand potential wells of homoclinic orbits. These results are beneficial to explore the key mechanism of hunting stability of a simplified railway wheelset model.

Homoclinic Bifurcations in a Class of Three-Dimensional Symmetric Piecewise Affine Systems

Many physical and engineering systems have certain symmetric properties. Homoclinic orbits play an important role in studying the global dynamics of dynamical systems. This paper focuses on the existence and bifurcations of homoclinic orbits to a saddle in a class of three-dimensional one-parameter three-zone symmetric piecewise affine systems. Based on the analysis of the Poincaré maps, the systems have two types of limit cycles and do not have chaotic invariant sets near the homoclinic orbits. In addition, the paper provides a constant [Formula: see text] to study the homoclinic bifurcations to limit cycles for the case [Formula: see text]. Two examples with simulations of the homoclinic orbits and the limit cycles are given to illustrate the effectiveness of the results.

Temperature elevations can induce switches to homoclinic action potentials that alter neural encoding and synchronization

Almost seventy years after the discovery of the mechanisms of action potential generation, some aspects of their computational consequences are still not fully understood. Based on mathematical modeling, we here explore a type of action potential dynamics – arising from a saddle-node homoclinic orbit bifurcation - that so far has received little attention. We show that this type of dynamics is to be expected by specific changes in common physiological parameters, like an elevation of temperature. Moreover, we demonstrate that it favours synchronization patterns in networks – a feature that becomes particularly prominent when system parameters change such that homoclinic spiking is induced. Supported by in-vitro hallmarks for homoclinic spikes in the rodent brain, we hypothesize that the prevalence of homoclinic spikes in the brain may be underestimated and provide a missing link between the impact of biophysical parameters on abrupt transitions between asynchronous and synchronous states of electrical activity in the brain.

The nonlinear mechanisms underlying the various stochastic dynamics evoked from different bursting patterns in a neuronal model

The stochastic dynamics such as coherence resonance (CR) and anti-CR show that noise plays constructive roles in the nervous system, which are always evoked from the resting-state or the nearby coexisting behaviors instead of bursting. In the present paper, various stochastic dynamics are evoked from different bursting patterns and the corresponding nonlinear mechanisms are presented. Anti-CR, transition from CR to anti-CR, and non-CR characterized by the coefficient of variation (CV) of interspike intervals are induced by noise from the “Homoclinic/Homoclinic” bursting, “Fold/Big Homoclinic” bursting with quiescent state, and “Fold/Big Homoclinic” bursting with subthreshold oscillations, respectively, which are determined by different stochastic transition regularities between the burst/spike and quiescent state/subthreshold oscillations within the stochastic bursting. The more transitions from spike to subthreshold oscillations to form long duration of subthreshold oscillations evoked by noise near a homoclinic orbit bifurcation, noise-induced more transition from quiescent state to spikes prior to a fold bifurcation, and frequent transitions between spikes and quiescent state are the dominant dynamical mechanisms for the anti-CR, CR, and non-CR, respectively. More detailed causes for the three different stochastic dynamics can be acquired through building relationships between the transitions and the different nonlinear dynamics of the fast subsystem such as the bifurcations, coexisting behaviors, attraction domains, and the distance between the trajectory of bursting and the bifurcation point or border between attraction domains. The comprehensive viewpoint and nonlinear mechanism for the various stochastic dynamics of the different bursting patterns extend the occurrence conditions for the CR or anti-CR and presents modulation measures and potential functions of bursting neurons receiving synaptic noise.

Bifurcations underlying different excitability transitions modulated by excitatory and inhibitory memristor and chemical autapses

Memristor is an emerging modulation factor to mimic neuronal synapse, and the switch or transition between neuronal excitability classes is an important topic in nonlinear dynamics and neurophysiology. In the present investigation on the Morris-Lecar model, the memristive autapse is identified to induce excitability transitions different from those of the chemical one. The inhibitory and excitatory chemical autapses induce excitability switch from class I to II and from class II to I, respectively. However, the inhibitory and excitatory memristive autapses induce the transition from class II to I and from class I to II, respectively. Furthermore, comprehensive bifurcation mechanisms underlying the transitions are acquired. The class I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC), and class II excitability to the saddle-node (SN) bifurcation and Hopf bifurcation. Type I spiking corresponds to the SNIC or big homoclinic (BHom) orbit bifurcations, and type II spiking to the Fold Limit Cycle (FLC) bifurcation or supercritical Hopf bifurcation. Among various codimension-2 bifurcations, the saddle-node homoclinic orbit bifurcation, which is related to the SNIC, BHom, and SN bifurcations, is responsible for the excitability transition. A nameless degenerate bifurcation associated with the BHom and FLC bifurcations, is responsible for the spiking transition. In addition, the nullcline shapes present the geometric mechanism for the transitions. The obtained results present different roles of memristive and chemical autapses on modulating neuronal electrical activities and the underlying dynamical mechanisms, which are helpful for the design and development of novel memristive synapses with excitatory or inhibitory effects.

Analytic and Algebraic Conditions for Bifurcations of Homoclinic Orbits II: Reversible Systems

Following Part I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under some conditions. We next extend the Melnikov method of Part I to reversible systems and obtain some theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones.

Activity-mediated accumulation of potassium induces a switch in firing pattern and neuronal excitability type.

During normal neuronal activity, ionic concentration gradients across a neuron's membrane are often assumed to be stable. Prolonged spiking activity, however, can reduce transmembrane gradients and affect voltage dynamics. Based on mathematical modeling, we investigated the impact of neuronal activity on ionic concentrations and, consequently, the dynamics of action potential generation. We find that intense spiking activity on the order of a second suffices to induce changes in ionic reversal potentials and to consistently induce a switch from a regular to an intermittent firing mode. This transition is caused by a qualitative alteration in the system's voltage dynamics, mathematically corresponding to a co-dimension-two bifurcation from a saddle-node on invariant cycle (SNIC) to a homoclinic orbit bifurcation (HOM). Our electrophysiological recordings in mouse cortical pyramidal neurons confirm the changes in action potential dynamics predicted by the models: (i) activity-dependent increases in intracellular sodium concentration directly reduce action potential amplitudes, an effect typically attributed solely to sodium channel inactivation; (ii) extracellular potassium accumulation switches action potential generation from tonic firing to intermittently interrupted output. Thus, individual neurons may respond very differently to the same input stimuli, depending on their recent patterns of activity and/or the current brain-state.

Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron

<p style='text-indent:20px;'>Post-inhibitory rebound (PIR) spike induced by the negative stimulation, which plays important roles and presents counterintuitive nonlinear phenomenon in the nervous system, is mainly related to the Hopf bifurcation and hyperpolarization-active caution (<inline-formula><tex-math id="M1">$ I_h $</tex-math></inline-formula>) current. In the present paper, the emerging condition for the PIR spike is extended to the bifurcation of the big homoclinic (BHom) orbit in a model without <inline-formula><tex-math id="M2">$ I_h $</tex-math></inline-formula> current. The threshold curve for a spike evoked from a mono-stable or coexisting steady state surrounds the steady state from left, to below, and to right, because the BHom orbit is big enough to surround the steady state. The right part of the threshold curve coincides with the stable manifold of the saddle and acts the threshold for the spike induced by the positive stimulation, resembling that of the saddle-node bifurcation on an invariant cycle, and the left part acts the threshold for the PIR spike, resembling that of the Hopf bifurcation. The bifurcation curve and a codimension-2 bifurcation point related to the BHom orbit are acquired in the two-parameter plane. The results present a comprehensive viewpoint to the dynamics near the BHom orbit bifurcation, which presents a novel threshold curve and extends the conditions for the PIR spike.</p>

Bifurcation mechanism of not increase but decrease of spike number within a neural burst induced by excitatory effect

Nonlinear dynamics is identified to play very important roles in identifying the complex phenomenon, dynamical mechanism, and physiological functions of neural electronic activities. In the present paper, a novel viewpoint that the excitatory stimulus cannot enhance but reduce the number of the spikes within a burst, the novel viewpoint which is different from the traditional viewpoint, is proposed and is explained with the nonlinear dynamics. When the impulse current or the autaptic current with suitable strength is used in the suitable phase within the quiescent state of the bursting pattern of the Rulkov model, a novel firing pattern with reduced number of spikes within a burst is evoked. The earlier the application phase of the current within the quiescent state, the higher the threshold of the current strength to evoke the novel firing pattern is and the less the number of the spikes within a burst of the novel firing pattern. Moreover, such a novel phenomenon can be explained by the intrinsic nonlinear dynamics of the bursting combined with the characteristics of the current. The nonlinear behaviors of the fast subsystem of the Rulkov model are acquired by the fast and slow variable dissection method, respectively. For the fast subsystem, there exist a stable node with lower membrane potential, a stable limit cycle with higher membrane potential, a saddle serving as the border between the stable node and limit cycle, a saddle-node bifurcation, and a homoclinic orbit bifurcation. When external simulation is not received, the bursting pattern of the Rulkov model exhibits behavior alternating between the spikes corresponding to the limit cycle of the fast subsystem and quiescent state of the fast subsystem, which is located within the parameter region between the saddle-node bifurcation point and the homoclinic orbit bifurcation point of the fast subsystem. The spikes begin with the saddle-node bifurcation and end with the homoclinic orbit bifurcation. As the bifurcation parameter turns close to the homoclinic orbit bifurcation, the disturbation or stimulus that can induce the transition from the quiescent state to the spikes becomes strong. Therefore, as the application phase of the current within the quiescent state becomes earlier, the strength threshold of the current that can induce the transition from the quiescent state to the spikes becomes stronger, and the initial phase of the spikes becomes closer to the homoclinic orbit bifurcation, which leads the parameter region of the spikes to become shorter and then leads the number of spikes within a burst to turn less. It is the dynamical mechanism of the decrease of the spike number induced by the excitatory currents. The results enrich the nonlinear phenomenon and dynamical mechanism, present a novel viewpoint for the excitatory effect, and provide a new approach to modulating the neural bursting patterns.

A direct algorithm with square wave perturbation for calculating phase response curve

Neuron is a typical dynamic system, therefore, it is quite natural to study the firing behaviors of neurons by using the dynamical system theory. Two kinds of firing patterns, i.e., the periodic spiking and the periodic bursting, are the limit cycle oscillators from the point of view of nonlinear dynamics. The simplest way to describe the limit cycle is to use the phase of the oscillator. A complex state space model can be mapped into a one-dimensional phase model by phase transformation, which is helpful for obtaining the analytical solution of the oscillator system. The response characteristics of the oscillator system in the motion state of the limit cycle to the external stimuli can be characterized by the phase response curve. A phase response curve illustrates the transient change in the cycle period of an oscillation induced by a perturbation as a function of the phase at which it is received. Now it is widely believed that the phase response curve provides a new way to study the behavior of the neuron. Existing studies have shown that the phase response curve of the periodic spiking can be divided into two types, which are closely related to the bifurcation mechanism of neurons from rest to repetitive firing. However, there are few studies on the relationship between the phase response curve and the bifurcation type of the periodic bursting. Clearly, the first prerequisite to understand this relationship is to calculate the phase response curve of the periodic bursting. The existing algorithms for computing the phase response curve are often unsuccessful in the periodic bursting. In this paper, we present a method of calculating the phase response curve, namely the direct algorithm with square wave perturbation. The phase response curves of periodic spiking and periodic bursting can be obtained by making use of the direct algorithm, which is verified in the four neuron models of the Hodgkin-Huxley, FitzHugh-Nagumo, Morris-Lecar and Hindmarsh-Rose. This algorithm overcomes the limitations to other algorithms in the application. The calculation results show that the phase response curve of the periodic spiking is determined by the bifurcation type. We find a suprathreshold periodic oscillation starting from a Hopf bifurcation and terminating at a saddle homoclinic orbit bifurcation as a function of the applied current strength in the Morris-Lecar model, and its phase response curve belongs to Type II. A large amount of calculation indicates that the relative size of the phase response and its positive or negative value depend only on the time of imposing perturbation, and the phase response curve of periodic bursting is more complicated than that of periodic spiking.

Mixed-mode oscillations in a nonlinear time delay oscillator with time varying parameters

In this study, the mechanism for the action of time-invariant delay on a non-autonomous system with slow parametric excitation is investigated. The complex mix-mode oscillations (MMOs) are presented when the parametric excitation item slowly passes through critical bifurcation values of this nonlinear time delay oscillator. We use bifurcation theory to clarify certain generation mechanism related to three complex spiking formations, i.e., ``symmetric sup-pitchfork bifurcation'', ``symmetric sup-pitchfork/sup-Hopf bifurcation'', and ``symmetric sup-pitchfork/sup-Hopf/homoclinic orbit bifurcation''. Such bifurcation behaviors result in various hysteresis loops between the spiking attractor and the quasi-stationary process, which are responsible for the generation of MMOs. We further identify that the occurrence and evolution of such complex MMOs depend on the magnitude of the delay. Specifically, with the increase of time delay, the two limit cycles bifurcated from Hopf bifurcations may merge into an enlarged cycle, which is caused by a saddle homoclinic orbit bifurcation. We can conclude that time delay plays a vital role in the generation of MMOs. Our findings enrich the routes to spiking process and deepen the understanding of MMOs in time delay systems.

Kuramoto Phase Model with Inertia: Bifurcations Leading to the Loss of Synchrony and to the Emergence of Chaos

We consider a finite-dimensional model of phase oscillators with inertia in the case of star configuration of coupling. The system of equations is reduced to a nonlinearly coupled system of pendulum equations. We prove that the transition from synchronous to asynchronous oscillations occurs via bifurcation of saddle-node equilibrium. In this connection the asynchronous regime can be partially synchronous rotations. We find that the reverse transition from asynchronous to synchronous regime occurs via bifurcation of homoclinic orbit both of the saddle equilibrium point and of the saddle periodic orbit. In the case of homoclinic loop of the saddle point the synchrony appears only from asynchronous mode without partially synchronized rotations. In the case of the homoclinic curve of the saddle periodic orbit the system undergoes a chaotic rotation regime which results in a random return to synchrony. We establish that return transitions are hysteretic in the case of large inertia.

Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikovʼs method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg–Landau partial differential equations, and demonstrate the theoretical results by numerical ones.

Climbing-fiber induced state transitions in cerebellar Purkinje cells are controlled by synaptic conductance changes

Purkinje cells have been previously modeled as a systemundergoing a saddle-node bifurcation of fixed pointsfrom rest to firing and a saddle homoclinic orbit bifur-cation from firing to rest. In vitro, this dynamical struc-ture is a result of the neuron’s intrinsic membraneproperties and is associated with bistability within a lim-ited range of low firing frequencies, where a unidirec-tional climbing fiber (CF) input is able to toggle the cellbetween a firing (“up”)andrest(“down”)state.Weidentified several factors that contribute to bistabilityand the ability for a unidirectional input (CF) to togglecell output, including a slow K+ current activated duringspike discharge or followingsynaptic depolarizations.However, input conditions that determine the probabil-ity for a Purkinje cell to express bistabilityin vivo havenot been determined. A key difference in vivo is the pre-sence of tonic background input to the dendrites fromparallel fiber (PF) and stellate cell inputs. We tested thehypothesis that dendritic inputs control the dynamics ofPurkinje cell firing, and can thus regulate the ability forCFs to induce toggling of Purkinje cell output.Presentation of mixed excitatory and inhibitory dendri-tic current noise (I-noise) or conductance noise (g-noise)to a two-compartment 5-equation model of the Purkinjeneuron had differing effects on spike output. Mixed I-noise increased the probability of observing CF-evokedtransitions to a down state whether the model was in thelow frequency-bistable regime or not. The size and timecourse of the currents associated with different statetransitions suggested that properly timed PF and/or stel-late cell inputs could affect the ability for CFs to invokePurkinje cell transitions. However, conductance noiseprevented any CF-evoked transitions and the model washighly sensitive to the E:I ratio. Spike trains with physio-logical mean frequencies and high coefficient of variation(CV) were also found. Spike triggered averages duringg-noise revealed that the spikes were being driven bysynaptic inputs and not intrinsic dynamics, indicating ashift in computational properties between high and lowconductance states.ConclusionsHere we show that bistabilityinaPurkinjeneuroncanbe controlled by the amount of synaptic input itreceives. Of the two types of noise we used, I-noisecould cause spontaneous state transitions, but g-noisecould not, suggesting that CF-associated toggling wouldnot occur in high conductance states. These resultscould explain the discrepancy between in vivo and in vitrorecordings regarding CF-induced state transitions.

Orbits homoclinic to resonances in a harmonically excited and undamped circular plate

Orbits homoclinic to resonances in mode interactions of an imperfect circular plate with 1:1 internal resonance are investigated. The case of primary resonance is considered. The damping force is not included in the analysis. The energy-phase criterion is used to give a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonances. A saddle-node bifurcation of homoclinic orbits occurs. The existence of homoclinic orbits in the unperturbed system may lead to chaos in the sense of Smale horseshoes under perturbation.

Bifurcation of degenerate homoclinic orbits to saddle-center in reversible systems

The authors study the bifurcation of homoclinic orbits from a degenerate homoclinic orbit in reversible system. The unperturbed system is assumed to have saddle-center type equilibrium whose stable and unstable manifolds intersect in two-dimensional manifolds. A perturbation technique for the detection of symmetric and nonsymmetric homoclinic orbits near the primary homoclinic orbits is developed. Some known results are extended.

Homoclinic snaking near a heteroclinic cycle in reversible systems

Snaking curves of homoclinic orbits have been found numerically in a number of ODE models from water wave theory and structural mechanics. Along such a curve infinitely many fold bifurcation of homoclinic orbits occur. Thereby the corresponding solutions spread out and develop more and more bumps (oscillations) about their own centre. A common feature of the examples is that the systems under consideration are reversible. In this paper it is shown that such a homoclinic snaking can be caused by a heteroclinic cycle between two equilibria, one of which is a bi-focus. Using Lin’s method a snaking of 1-homoclinic orbits is proved to occur in an unfolding of such a cycle. Further dynamical consequences are discussed. As an application a system of Boussinesq equations is considered, where numerically a homoclinic snaking curve is detected and it is shown that the homoclinic orbits accumulate along a heteroclinic cycle between a real saddle and a bi-focus equilibrium.

Master Equation Simulations of Bistable and Excitable Dynamics in a Model of a Thermochemical System

The effect of fluctuations on the dynamics of a model of a bistable thermochemical system is studied by means of the master equation. The system has three stationary states and exhibits two types of bistability: the coexistence of two stable focuses and the coexistence of a stable focus with a stable limit cycle separated by a saddle point. Stochastic effects are important when the system is close to the bifurcation, in which the stable limit cycle disappears through a homoclinic orbit. In this case the distribution of the first passage time from the stable limit cycle to the stable focus has a multipeak form. The dependence of this distribution on the number of particles is presented. Near the homoclinic orbit bifurcation, the system also exhibits excitability due to a particular shape of the basin of attraction of the stable focus.

Homoclinic bifurcation of strongly nonlinear oscillators by frequency-incremental method

With the criterion to predict bifurcation of homoclinic orbits, this paper considers strongly nonlinear oscillators described by x ̈ +g(x)=λf(μ,x, x ̇ ) x ̇ , in which λ is a parameter of arbitrary magnitude. The approximation of a limit cycle near homoclinicity is computed by frequency-incremental method, and the bifurcation of homoclinic orbits is predicted when the limit cycle collides with equilibrium points. The result of this method is consistent with that of Mel’nikov’s method when λ is small.

Symmetry-Breaking Homoclinic Bifurcations in Diffusively Coupled Equations

In this study we examine a symmetry-breaking bifurcation of homoclinic orbits in diffusively coupled ordinary differential equations. We prove that asymmetric homoclinic orbits can bifurcate from a symmetric homoclinic orbit when the equilibria to which the latter is homoclinic undergoes a pitchfork bifurcation. A condition which defines the direction of the bifurcation in a parameter space is given. All hypotheses of the main theorem are verified for a diffusively coupled logistic system and the twistedness of the bifurcating homoclinic orbits is computed for a range of coupling strengths.