Homoclinic Bifurcation Research Articles

Complex dynamics of an epidemic model with saturated media coverage and recovery.

During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically and numerically investigate the dynamics of the proposed model. Given great difficulties caused by high nonlinearity in theoretical analysis, we separately considered subsystems with only nonlinear recovery or with only saturated media impact. For the model with only nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. Numerical simulations reveal the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, saddle-node bifurcation, homoclinic bifurcation and unstable limit cycle. So the limitation of medical resources induces rich dynamics and causes much difficulties in eliminating the infectious diseases. We then investigated the dynamics of the system with only saturated media impact and concluded that saturated media impact hardly induces the complicated dynamics. Further, we parameterized the proposed model on the basis of the COVID-19 case data in mainland China and data related to news items, and estimated the basic reproduction number to be 2.86. Sensitivity analyses were carried out to quantify the relative importance of parameters in determining the cumulative number of infected individuals at the end of the first month of the outbreak. Combining with numerical analyses, we suggested that providing adequate medical resources and improving media response to infection or individuals’ response to mass media may reduce the cumulative number of the infected individuals, which mitigates the transmission dynamics during the early stage of the COVID-19 pandemic.

A unified physiological framework of transitions between seizures, sustained ictal activity and depolarization block at the single neuron level

The majority of seizures recorded in humans and experimental animal models can be described by a generic phenomenological mathematical model, the Epileptor. In this model, seizure-like events (SLEs) are driven by a slow variable and occur via saddle node (SN) and homoclinic bifurcations at seizure onset and offset, respectively. Here we investigated SLEs at the single cell level using a biophysically relevant neuron model including a slow/fast system of four equations. The two equations for the slow subsystem describe ion concentration variations and the two equations of the fast subsystem delineate the electrophysiological activities of the neuron. Using extracellular K+ as a slow variable, we report that SLEs with SN/homoclinic bifurcations can readily occur at the single cell level when extracellular K+ reaches a critical value. In patients and experimental models, seizures can also evolve into sustained ictal activity (SIA) and depolarization block (DB), activities which are also parts of the dynamic repertoire of the Epileptor. Increasing extracellular concentration of K+ in the model to values found during experimental status epilepticus and DB, we show that SIA and DB can also occur at the single cell level. Thus, seizures, SIA, and DB, which have been first identified as network events, can exist in a unified framework of a biophysical model at the single neuron level and exhibit similar dynamics as observed in the Epileptor.Author Summary: Epilepsy is a neurological disorder characterized by the occurrence of seizures. Seizures have been characterized in patients in experimental models at both macroscopic and microscopic scales using electrophysiological recordings. Experimental works allowed the establishment of a detailed taxonomy of seizures, which can be described by mathematical models. We can distinguish two main types of models. Phenomenological (generic) models have few parameters and variables and permit detailed dynamical studies often capturing a majority of activities observed in experimental conditions. But they also have abstract parameters, making biological interpretation difficult. Biophysical models, on the other hand, use a large number of variables and parameters due to the complexity of the biological systems they represent. Because of the multiplicity of solutions, it is difficult to extract general dynamical rules. In the present work, we integrate both approaches and reduce a detailed biophysical model to sufficiently low-dimensional equations, and thus maintaining the advantages of a generic model. We propose, at the single cell level, a unified framework of different pathological activities that are seizures, depolarization block, and sustained ictal activity.

FLIP BIFURCATION WITH RANDOM EXCITATION

<abstract> In this paper, flip bifurcation with random excitation is studied by employing the methods of normal forms, Picard iterations and orthogonal polynomial approximation. For the codimension one case, a Neimark-Sacker bifurcation, a 1:2 resonance and a fold-flip bifurcation are detected. It is found that the system undergoes heteroclinic bifurcation and homoclinic bifurcation near 1:2 resonance point, a hopf bifurcation and a cusp bifurcation near fold-flip bifurcation point. For the codimension two case, the system undergoes only a flip bifurcation when random excitation is imposed on the nonlinear term. In addition, numerical simulations are given to show the disparity between the codimension one and two cases. </abstract>

A competition model in the chemostat with allelopathy and substrate inhibition

&lt;p style='text-indent:20px;'&gt;A model of two microbial species in a chemostat competing for a single resource is considered, where one of the competitors that produces a toxin, which is lethal to the other competitor (allelopathic inhibition), is itself inhibited by the substrate. Using general growth rate functions of the species, necessary and sufficient conditions of existence and local stability of all equilibria of the four-dimensional system are determined according to the operating parameters represented by the dilution rate and the input concentration of the substrate. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. If a non monotonic growth rate is considered (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. We describe its operating diagram, which is the bifurcation diagram giving the behavior of the system with respect to the operating parameters. By means of this bifurcation diagram, we show that the general model presents a set of fifteen possible behaviors: washout, competitive exclusion of one species, coexistence, multi-stability, occurrence of stable limit cycles through a super-critical Hopf bifurcations, homoclinic bifurcations and flip bifurcation. This diagram is very useful to understand the model from both the mathematical and biological points of view.&lt;/p&gt;

Isochron foliations and global bifurcations: a case study

Abstract The basins of attraction of periodic orbits or focus equilibria of a given vector field are foliated by forward-time isochrons, defined as all initial conditions that synchronize under the flow with a given phase. Similarly, backward-time isochrons of repelling periodic orbits or focus equilibria foliate their respective basins of repulsion. We present a case study of a planar system that features a sequence of bifurcations, including a saddle-node bifurcation of periodic orbits, a homoclinic bifurcation and Hopf bifurcations, that change the nature and existence of periodic orbits. We explain how the basins and isochron foliations change throughout the sequence of bifurcations. In particular, we identify structurally stable tangencies between the foliations by forward-time and backward-time isochrons, which are curves in the plane, in regions of phase space where they exist simultaneously. Such tangencies are generically quadratic and associated with sharp turns of isochrons and phase sensitivity of the system. In contrast to the earlier reported cubic isochron foliation tangency (CIFT) mechanism, which generates a pair of tangency orbits, we find isochron foliation tangencies that occur along single specific orbits in the respective basin of attraction or repulsion. Moreover, the foliation tangencies we report arise from actual bifurcations of the system, while a CIFT is not associated with a topological change of the underlying phase portrait. The properties and interactions of isochron foliations are determined and illustrated by computing a representative number of forward-time and backward-time isochrons as arclength-parametrized curves with a boundary value problem set-up. Our algorithm for computing isochrons has been further refined and implemented in the Matlab package CoCo; it is made available as Matlab code in the supplementary material of this paper, together with a guide that walks the user through the computation of two specific isochron foliations.

Hopf bifurcation control of the ML neuron model with Hc bifurcation type

&lt;abstract&gt;&lt;p&gt;It is shown that many neurological diseases are caused by the changes of firing patterns induced by bifurcations. Therefore, the bifurcation control may provide a potential therapeutic method of these neurodegenerative diseases. In this paper, we investigate the Hopf bifurcation control of the Morris-Lecar (ML) model with Homoclinic (Hc) bifurcation type by introducing a dynamic state-feedback control. The results indicate that the linear term can change the ML model from Hc bifurcation type to SNIC bifurcation type without changing the firing patterns. The cooperation of linear and cubic term can transform the ML model from the Hc bifurcation type to the Hopf bifurcation type, resulting in the transformation of firing patterns from type I to type II. Besides, we utilize the Poincare Birkhoff (PB) normal form method to derive the analytical expression of the bifurcation stability index for the controlled ML model with Hc bifurcation type, and the results show that the cubic term can regulate the criticality of the Hopf bifurcation. Numerical simulation results are consistent with the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Complex dynamics and Bogdanov-Takens bifurcations in a retarded van der Pol-Duffing oscillator with positional delayed feedback.

In this article, we will investigate a retarded van der Pol-Duffing oscillator with multiple delays. At first, we will find conditions for which Bogdanov-Takens (B-T) bifurcation occurs around the trivial equilibrium of the proposed system. The center manifold theory has been used to extract second order normal form of the B-T bifurcation. After that, we derived third order normal form. We also provide a few bifurcation diagrams, including those for the Hopf, double limit cycle, homoclinic, saddle-node, and Bogdanov-Takens bifurcation. In order to meet the theoretical requirements, extensive numerical simulations have been presented in the conclusion.

Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $

&lt;p style='text-indent:20px;'&gt;The Bose-Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose-Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that describes two specific coupled oscillators, where both the amplitude and the phase are important. We perform a bifurcation analysis of this system to identify regions of different behavior as the pump power &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ f $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and the detuning &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ \delta $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; of the driving signal are varied, for the case of fixed positive coupling. The bifurcation diagram in the &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ (f, \delta) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-plane is organized by two points of codimension-two bifurcations——a &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \mathbb{Z}_2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-equivariant homoclinic flip bifurcation and a Bykov T-point——and provides a roadmap for the observable dynamics, including different types of chaotic behavior. To illustrate the overall structure and different accumulation processes of bifurcation curves and associated regions, our bifurcation analysis is complemented by the computation of kneading invariants and of maximum Lyapunov exponents in the &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ (f, \delta) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-plane. The bifurcation diagram displays a menagerie of dynamical behavior and offers insights into the theory of global bifurcations in a four-dimensional phase space, including novel bifurcation phenomena such as degenerate singular heteroclinic cycles.&lt;/p&gt;

A predator-prey model with cooperative hunting in the predator and group defense in the prey

&lt;p style='text-indent:20px;'&gt;In this paper we propose a predator-prey model with a non-differentiable functional response in which the prey exhibits group defense and the predator exhibits cooperative hunting. There is a separatrix curve dividing the phase portrait. The species with initial population above the separatrix result in extinction of prey in finite time, and the species with initial population below it can coexist, oscillate sustainably or leave the prey surviving only. Detailed bifurcation analysis is carried out to explore the effect of cooperative hunting in the predator and aggregation in the prey on the existence and stability of the coexistence state as well as the dynamics of system. The model undergoes transcritical bifurcation, Hopf bifurcation, homoclinic (heteroclinic) bifurcation, saddle-node bifurcation, and Bogdanov-Takens bifurcation, and through numerical simulations it is found that it possesses rich dynamics including bubble loop of limit cycles, and open ended branch of periodic orbits disappearing through a homoclinic cycle or a loop of heteroclinic orbits. Also, a continuous transition of different types of Hopf branches are investigated which forms a global picture of Hopf bifurcation in the model.&lt;/p&gt;

Continuation methods for principal foliations of embedded surfaces

&lt;p style='text-indent:20px;'&gt;Continuation methods are a well established tool for following equilibria and periodic orbits in dynamical systems as a parameter is varied. Properly formulated, they locate and classify bifurcations of these key components of phase portraits. Principal foliations of surfaces embedded in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathbb{R}^3 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; resemble phase portraits of two dimensional vector fields, but they are not orientable. In the spirit of dynamical systems theory, Gutierrez and Sotomayor investigated qualitative geometric features that characterize structurally stable principal foliations and their bifurcations in one parameter families. This paper computes return maps and applies continuation methods to obtain new insight into bifurcations of principal foliations.&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;Umbilics are the singularities of principal foliations and lines of curvature connecting umbilics are analogous to homoclinic and heteroclinic bifurcations of vector fields. Here, a continuation method tracks a periodic line of curvature in a family of surfaces that deforms an ellipsoid. One of the bifurcations of these periodic lines of curvature are connections between lemon umbilics. Differences between these bifurcations and analogous saddle connections in two dimensional vector fields are emphasized. A second case study tracks umbilics in a one parameter family of surfaces with continuation methods and locates their bifurcations using Taylor expansions in "Monge coordinates." Return maps that are generalized interval exchange maps of a circle are constructed for generic surfaces with no monstar umbilics.&lt;/p&gt;

Post inhibitory rebound spike related to nearly vertical nullcline for small homoclinic and saddle-node bifurcations

&lt;abstract&gt;&lt;p&gt;A spike induced by inhibitory stimulation instead of excitatory stimulation, called post-inhibitory rebound (PIR) spike, has been found in multiple neurons with important physiological functions, which presents counterintuitive behavior mainly related to focus near Hopf bifurcation. In the present paper, the condition for the PIR spike is extended to small homoclinic orbit (SHom) and saddle-node (SN) bifurcations, and the underlying mechanism is acquired in a neuron model. Firstly, PIR spike is evoked from a stable node near the SHom or SN bifurcation by a strong inhibitory stimulation. Then, the dynamics of threshold curve for a spike, vector fields, and nullcline of recovery variable are used to well explain the cause for the PIR spike. The shape of threshold curve for the node resembles that of focus. The nullcline plays an important role in forming PIR spike, which is analytically identified at last. Besides, a sufficient condition is acquired from the integration to a differential equation, and the range of parameters for the PIR spike is presented. The extended bifurcation types and the underlying mechanisms for the PIR spike such as the nullcline present comprehensive and deep understandings for the PIR spike, which also provides potential strategy to modulate the PIR phenomenon and even related physiological functions of neurons.&lt;/p&gt;&lt;/abstract&gt;

Sub-Synchronous Oscillations and Transient Stability of Islanded Microgrid

Sustained sub-synchronous oscillations are observed in an islanded microgrid consisting of grid-forming converters under a large transient disturbance. Such oscillations are studied in terms of various types of hidden attractors, namely, hidden periodic orbit, hidden quasi-periodic orbit, and hidden chaotic attractor, which are shown to coexist with the stable equilibrium point, share different basins of attraction, and be transformed mutually for different parameter values. Moreover, the basins of attraction have fractal boundaries, making the nearby trajectories unpredictable. The transient stability of the islanded microgrid can be improved by eliminating the hidden attractors through adjusting control parameters and triggering a homoclinic bifurcation. Furthermore, transient disturbances can be mitigated by protecting critical converters and accelerating the restarting process. Besides, the emergence of sub-synchronous oscillations can be eliminated through tripping off one or more specific converters. Full-circuit cycle-by-cycle simulations are performed for islanded microgrids comprising i) 5 grid-forming converters, ii) 5 grid-forming converters and a synchronous generator, and iii) 30 grid-forming converters to verify the analytical findings.

Basins of attraction in a modified ratio-dependent predator-prey model with prey refugee

&lt;abstract&gt;&lt;p&gt;In this paper, we analyze a modified ratio-dependent predator-prey model with a strong Allee effect and linear prey refugee. The model exhibits rich dynamics with the existence of separatrices in the phase plane in-between basins of attraction associated with oscillation, coexistence, and extinction of the interacting populations. We prove that if the initial values are positive, all solutions are bounded and stay in the interior of the first quadrant. We show that the system undergoes several bifurcations such as transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations. Consequently, a homoclinic bifurcation curve exists generating an unstable periodic orbit. Moreover, we find that the Bogdanov-Takens bifurcation acts as an organizing center for the scenario of surviving or extinction of both interacting species. Topologically different phase portraits with all possible trajectories and equilibria are depicted illustrating the behavior of the system.&lt;/p&gt;&lt;/abstract&gt;

Homoclinic bifurcation analysis for logistic map

Homoclinic bifurcation analysis for logistic map

Chaos explosion and topological horseshoe in three-dimensional impacting hybrid systems with a single impact surface

For a class of three-dimensional impacting hybrid systems comprising a linear system of ordinary differential equations and a reset map, and having a single impact surface, this paper studies the phenomena of impacting homoclinic bifurcation leading to periodic orbits, horseshoes and chaos explosions. More precisely, it is proved that the homoclinic bifurcation can result in the impacting periodic sinks or impacting periodic saddle orbits when the impact surface is a plane and the reset map satisfies some basic conditions, but it is not easy to find horseshoes in this case. Furthermore, when the single impact surface is not a plane and the reset map has a certain rotational property, it is proved that a topological horseshoe will appear suddenly when bifurcation parameter passes through some threshold, thus, a kind of chaos explosion takes place. These main results are illustrated by several examples, in which it can be seen that the newborn chaotic invariant sets might be chaotic attractors from the perspective of numerical simulation.

Operon dynamics with state dependent transcription and/or translation delays.

Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.

Global dynamics and performance of vibration reduction for a new vibro-impact bistable nonlinear energy sink

A new vibro-impact bistable nonlinear energy sink (VI-BNES) with relatively light weight attached to a damped linear oscillator (LO) is proposed to form a two-degree-of-freedom nonlinear mechanical system with bilateral rigid constraints. The state of the system after instantaneous impact is described by the simplified shock theory and the momentum conservation. When the LO subjected to impulse excitations with different levels is considered as the driver of the VI-BNES, a two-dimensional non-smooth reduced impact system with a harmonic perturbation and an irrational nonlinear restoring force is obtained to study the threshold for the homoclinic bifurcations and chaotic responses of the VI-BNES. Bistability of this mechanical model makes the reduced unperturbed system have a pair of symmetric homoclinic orbits as the pseudo-separatrix for in-well and cross-well oscillations. The Melnikov method developed by us for non-smooth planar systems with bilateral rigid constraints is firstly employed to detect the threshold of the amplitude of excitations of the LO for the onset of chaotic oscillations. Furthermore, cubic polynomial approximation of the restoring force on the premise of maintaining the equilibria of the system is carried out such that the analytical Melnikov analysis for the global dynamics becomes possible. The performance of the VI-BNES is also studied and compared with other NESs in detail through numerical analysis and experiments to show the high efficiency of vibration reduction of the VI-BNES due to the dual beneficial role of the bistability and the impacts with the bilateral rigid constraints.

Role of phase-dependent influence function in the Winfree model of coupled oscillators.

We consider a globally coupled Winfree model comprised of a phase-dependent influence function and sensitive function, and unravel the impact of offset and integer parameters, characterizing the shape of the influence function, on the phase diagram of the Winfree model. The decreasing value of the offset parameter decreases the degree of positive phase shift among the oscillators by promoting the negative phase shift, which indeed favors the onset of multistability among the synchronous oscillatory state and asynchronous stable steady states in a large region of the phase diagram. Further, large integer parameters lead to brief pulses of the influence function, which again enhances the effect of the offset parameter. There is an explosive transition to a synchronous oscillatory state from an asynchronous steady state via a Hopf bifurcation. Dynamical transitions and multistability emerge through saddle-node, pitchfork, and homoclinic bifurcations in the phase diagram. We deduce two ordinary differential equations corresponding to the two macroscopic variables from the population of globally coupled Winfree oscillators using the Ott-Antonsen ansatz. We also deduce various bifurcation curves analytically from the reduced low-dimensional macroscopic variables for the exactly solvable case. The analytical curves exactly match the simulation boundaries in the phase diagram.

Self-Sustained Libration Regime in Nonlinear Microelectromechanical Devices

We present a mode of operation for Duffing-type nonlinear microelectromechanical system devices whereby a self-sustained multifrequency output is generated. This self-sustained librator regime creates a limit cycle around a dynamical fixed point, i.e., around fixed points within the rotating frame, whereas a traditional oscillator generates a limit cycle around a static fixed point. The librator limit cycles thus created do not change the global topology of the rotating frame phase space, but are constrained by it. Because of the Duffing nonlinearity, different types of limit cycles could be generated within the same phase space, with each type possessing distinct dynamical features. Transitioning between these limit cycles requires crossing homoclinic bifurcations, which is done without generating chaos as the phase space dynamics are constrained in two dimensions. This work provides an alternative experimental tool for the study of microelectromechanical and nanoelectromechanical system dynamics and their application.

Singularly perturbed dynamics of the tippedisk.

The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disc, being reminiscent of (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown how the presence of friction leads to slow–fast dynamics and the creation of a two-dimensional slow manifold. Furthermore, it is revealed that a bifurcation scenario involving a homoclinic bifurcation and a Hopf bifurcation leads to an explanation of the inversion phenomenon. In particular, a closed-form condition for the critical spinning speed for the inversion phenomenon is derived. Hence, the tippedisk forms an excellent mathematical-mechanical problem for the analysis of global bifurcations in singularly perturbed dynamics.