Degenerate Bifurcation Research Articles

Multiple and complex codimension-2 bifurcations underlying post-inhibitory rebound spike and excitability transition

Neuron exhibits nonlinear dynamics such as excitability transition and post-inhibitory rebound (PIR) spike related to bifurcations, which are associated with information processing, locomotor modulation, or brain disease. PIR spike is evoked by inhibitory stimulation instead of excitatory stimulation, which presents a challenge to the threshold concept. In this paper, 7 codimension-2 or degenerate bifurcations related to 10 codimension-1 bifurcations are acquired in a neuronal model, which presents the bifurcations underlying the excitability transition and PIR spike. Type I excitability corresponds to saddle-node bifurcation on an invariant cycle (SNIC) bifurcation, and type II excitability to saddle-node (SN) bifurcation or sub-critical Hopf (SubH) bifurcation or sup-critical Hopf (SupH) bifurcation. The excitability transition from type I to II corresponds to the codimension-2 bifurcation, Saddle-Node Homoclinic orbit (SNHO) bifurcation, via which SNIC bifurcation terminates and meanwhile big homoclinic orbit (BHom) bifurcation and SN bifurcation emerge. A degenerate bifurcation via which BHom bifurcation terminates and limit point of cycle (LPC) bifurcation emerges is responsible for spiking transition from type I to II, and the roles of other codimension-2 bifurcations (Cusp, Bogdanov-Takens, and Bautin) are discussed. In addition, different from the widely accepted viewpoint that PIR spike is mainly evoked near Hopf bifurcation rather than SNIC bifurcation, PIR spike is identified to be induced near SNIC or BHom or LPC bifurcations, and threshold curves resemble that of Hopf bifurcation. The complex bifurcations present comprehensive and deep understandings of excitability transition and PIR spike, which are helpful for the modulation to neural firing activities and physiological functions.

Degenerate Transcritical Bifurcation Point can be an Attractor: A Case Study in a Slow–Fast Modified Leslie–Gower Model

Degenerate Transcritical Bifurcation Point can be an Attractor: A Case Study in a Slow–Fast Modified Leslie–Gower Model

Double-zero degeneracy and heteroclinic cycles in a perturbation of the Lorenz system

In this paper we consider a 3D three-parameter unfolding close to the normal form of the triple-zero bifurcation exhibited by the Lorenz system. First we study analytically the double-zero degeneracy (a double-zero eigenvalue with geometric multiplicity two) and two Hopf bifurcations. We focus on the more complex case in which the double-zero degeneracy organizes several codimension-one singularities, namely transcritical, pitchfork, Hopf and heteroclinic bifurcations. The analysis of the normal form of a Hopf-transcritical bifurcation allows to obtain the expressions for the corresponding bifurcation curves. A degenerate double-zero bifurcation is also considered. The theoretical information obtained is very helpful to start a numerical study of the 3D system. Thus, the presence of degenerate heteroclinic and homoclinic orbits, T-point heteroclinic loops and chaotic attractors is detected. We find numerical evidence that, at least, four curves of codimension-two global bifurcations are related to the triple-zero degeneracy in the system analyzed.

Degenerate Bogdanov-Takens bifurcation of codimension 4 in Holling-Tanner model with harvesting

In this paper, we revisit the Holling-Tanner model with constant-yield prey harvesting. It is shown that the highest codimension of a nilpotent cusp is 4, and the model can undergo degenerate Bogdanov-Takens bifurcation of codimension 4. Moreover, when the model has a center-type equilibrium, we show that it is a weak focus with order at least 3 and at most 4, and the model can exhibit Hopf bifurcation of codimension 3. Some algebraic methods including resultant elimination and pseudo-division are used to solve the semi-algebraic varieties of normal form coefficients or focal values. Our results indicate that constant-yield prey harvesting can cause not only richer dynamics and bifurcations, but also the coextinction of both populations with some positive initial densities. Finally, numerical simulations, including the coexistence of limit cycle and homoclinic cycle, and three limit cycles, are presented to illustrate the theoretical results.

An analysis of two degenerate double-Hopf bifurcations

<abstract><p>The generic double-Hopf bifurcation is presented in detail in literature in textbooks like references. In this paper we complete the study of the double-Hopf bifurcation with two degenerate (or nongeneric) cases. In each case one of the generic conditions is not satisfied. The normal form and the corresponding bifurcation diagrams in each case are obtained. New possibilities of behavior which do not appear in the generic case were found.</p></abstract>

Border Collision Bifurcations and Chaos in a Ring of Three Unidirectionally Coupled Nonmonotonic Piecewise Constant Neurons

The bifurcations and chaos in a ring of three nonmonotonic piecewise constant neurons with unidirectional inhibitory coupling are examined. Periodic solutions in the system undergo border collision bifurcations, which are characteristic of piecewise smooth systems. Conditions for the bifurcations of periodic solutions are derived analytically by using a piecewise constant function as the output function of neurons. Multiple periodic solutions are generated through border collision bifurcations and saddle-node bifurcations. Chaotic attractors are generated through border collision bifurcations and degenerate period-doubling bifurcations. The existence of multiple periodic solutions and chaotic attractors is proven to be common in rings of unidirectionally coupled nonmonotonic neurons.

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.

Multistability Regions and Related Basins of Multiattractors in Piecewise Linear Discontinuous Map from Financial Markets

By adding trend followers, we extend the model given by Tramontana et al. from one-dimensional ([Formula: see text]D) piecewise linear discontinuous (PWLD) map to a new 2D PWLD map. Using this map in financial markets, we describe the bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: border collision bifurcations; Poincaré equator collision bifurcations; degenerate flip bifurcations in both supercritical and subcritical cases. We investigate the multistability regions in the parameter plane and related basins of multiattractors to uncover the reason for the unpredictability of the internal law of price fluctuations in financial market.

Global analysis in Bazykin's model with Holling II functional response and predator competition

In this paper, we study the well-known Bazykin's model with Holling II functional response and predator competition. A detailed bifurcation analysis, depending on all four parameters, reveals a rich bifurcation structure, including supercritical and subcritical Bogdanov-Takens bifurcation, degenerate Hopf bifurcation of codimension at most 2, and a focus type degenerate Bogdanov-Takens bifurcation of codimension 3, originating from a nilpotent focus of codimension 3 which acts as the organizing center for the bifurcation set. Moreover, some sufficient conditions to guarantee the global asymptotical stability of the semi-trivial equilibrium or the unique positive equilibrium are also given. Our analysis indicates that we can classify the long-time dynamics of the model with a threshold value c0 for the natural mortality rate c of predators, in detail, the following are true. (i) When c≥c0, the prey will persist and predators will eventually go extinct for all positive initial populations. (ii) When c<c0, the prey and predators will coexist, for all positive initial populations, in the form of multiple positive equilibria or multiple periodic orbits. Our results can be seen as a complement to the work by Bazykin et al. [2–5], Hainzl [22,23], Kuznetsov [30].

On the significance of borders: the emergence of endogenous dynamics

We propose a prototype model of market dynamics in which all functional relationships are linear. We take into account three borders, defined by linear functions, that are intrinsic to the economic reasoning: non-negativity of prices; downward rigidity of capacity (depreciation); and a capacity constraint for the production decision. Given the linear specification, the borders are the only source for the emerging of cyclical and more complex dynamics. In particular, we discuss centre bifurcations, border collision bifurcations and degenerate flip bifurcations—dynamic phenomena the occurrence of which are intimately related to the existence of borders.

Degenerate grazing-sliding bifurcations in planar Filippov systems

In this paper we contribute to a class of degenerate grazing-sliding bifurcations in planar Filippov systems, assuming that the unperturbed system has a generalized homoclinic loop to a fold-fold point. Depending on the type of this fold-fold point, we exhibit some codimension-2 bifurcation diagrams as well as the asymptotics of all bifurcation curves. Our work not only reports several complete unfoldings in the investigation of degenerate grazing-sliding bifurcations, but also answers the open problem about the maximum number of limit cycles bifurcating from the generalized homoclinic loop for the case of fold-fold presented in Liang et al. (2013) [30].

Influence of gravity on the frozen wave instability in immiscible liquids

Gravity is crucial to the frozen wave instability because it both sets the threshold of applied forcing and limits the growth of unstable perturbations. Through numerical simulations of immiscible liquids in finite containers we examine the transition from a supercritical pitchfork bifurcation in ordinary gravity, which is accompanied by stable finite-amplitude waves, to a degenerate bifurcation in weightlessness, which is followed by large columnar patterns. The vibroequilibria effect plays a symmetry-breaking role in the bifurcation, selecting frozen waves with the heavier fluid displaced upward along the lateral walls.

Discontinuous Bifurcation of a Soft-Impact System

In this paper, we investigate discontinuous bifurcations of a soft-impact system, which is nonsmooth at the switching boundary consisting of two parts. We find that there are no periodic orbits located only in the impact zone, and when grazing bifurcation on one part of the switching boundary occurs, the tangency point changes may occur for different bifurcation parameters, which is also verified by numerical simulation. In particular, we discover degenerate inner and external corner bifurcations, which can produce chaotic behavior, for example, period-doubling cascades and a degenerate inner corner bifurcation that induce chaotic responses. In this way, our investigation reveals the presence of narrow bands of chaotic motion induced by the afore mentioned dynamical phenomena.

Hidden attractors, singularly degenerate heteroclinic orbits, multistability and physical realization of a new 6D hyperchaotic system

This paper reports a sequential design of linearly controlling a three-dimensional (3D) quadratic system to a simple six-dimensional hyperchaotic system with complex dynamics. By adding three linear dynamical controllers, the resulting 6D system has no equilibrium and a hidden attractor, which has four positive Lyapunov exponents (LEs). This paper focuses on the 6D system, to reveal its unusual dynamics such as infinitely many singularly degenerate heteroclinic cycles and bifurcations from such singular orbits to hidden hyperchaotic attractors. Detailed numerical investigations are carried out, including bifurcation diagram, LE spectrum and phase portrait. Furthermore, the system has multistability corresponding to three types of equilibria, including no equilibrium and infinite non-isolated equilibria. In particular, we find that at least seven different attractors coexist when the system has one equilibrium line. Finally, this 6D hyperchaotic system is verified by 0–1 test and a circuit.

Bifurcation Sequences and Multistability in a Two-Dimensional Piecewise Linear Map

Bifurcation mechanisms in piecewise linear or piecewise smooth maps are quite different with respect to those occurring in smooth maps, due to the role played by the borders. In this work, we describe bifurcation mechanisms associated with the appearance/disappearance of cycles, which may be related to several cases: (A) fold border collision bifurcations, (B) degenerate flip bifurcations, supercritical and subcritical, (C) degenerate transcritical bifurcations and (D) supercritical center bifurcations. Each of these is characterized by a particular dynamic behavior, and may be related to attracting or repelling cycles. We consider different bifurcation routes, showing the interplay between all these kinds of bifurcations, and their role in the phase plane in determining attracting sets and basins of attraction.

Explanation of the onset of bouncing cycles in isotropic rotor dynamics; a grazing bifurcation analysis.

The dynamics associated with bouncing-type partial contact cycles are considered for a 2 degree-of-freedom unbalanced rotor in the rigid-stator limit. Specifically, analytical explanation is provided for a previously proposed criterion for the onset upon increasing the rotor speed Ω of single-bounce-per-period periodic motion, namely internal resonance between forward and backward whirling modes. Focusing on the cases of 2 : 1 and 3 : 2 resonances, detailed numerical results for small rotor damping reveal that stable bouncing periodic orbits, which coexist with non-contacting motion, arise just beyond the resonance speed Ω p:q . The theory of discontinuity maps is used to analyse the problem as a codimension-two degenerate grazing bifurcation in the limit of zero rotor damping and Ω = Ω p:q . An analytic unfolding of the map explains all the features of the bouncing orbits locally. In particular, for non-zero damping ζ, stable bouncing motion bifurcates in the direction of increasing Ω speed in a smooth fold bifurcation point that is at rotor speed beyond Ω p:q . The results provide the first analytic explanation of partial-contact bouncing orbits and has implications for prediction and avoidance of unwanted machine vibrations in a number of different industrial settings.

Novel two-parameter dynamics of impact oscillators near degenerate grazing points

Following the previous work on the degenerate grazing bifurcations of impact oscillators, this paper aims to explore novel two-parameter dynamics near the degenerate grazing points using GPU parallel computing technology. By using the technology, a further understanding of the near-grazing dynamics can be developed for impact oscillators. Three main indicators, i.e., the largest Lyapunov exponent, number of excitation periods and number of impacts, are calculated for each grid of the two-parameter plane chosen. Based on these indicators, the dynamic response in the vicinity of degenerate grazing points can be characterized and more dynamic behaviors than the published results can be discovered. Phenomena of coexisting attractors and chaotic transitions including crisis are also discussed. The single and two degree-of-freedom impact oscillators are selected as illustrative examples to demonstrate the results.

Bifurcation analysis in a host-generalist parasitoid model with Holling II functional response

In this paper we study a host-generalist parasitoid model with Holling II functional response where the generalist parasitoids are introduced to control the invasion of the hosts. It is shown that the model can undergo a sequence of bifurcations including cusp, focus and elliptic types degenerate Bogdanov-Takens bifurcations of codimension three, and a degenerate Hopf bifurcation of codimension at most two as the parameters vary, and the model exhibits rich dynamics such as the existence of multiple coexistent steady states, multiple coexistent periodic orbits, homoclinic orbits, etc. Moreover, there exists a critical value for the carrying capacity of generalist parasitoids such that: (i) when the carrying capacity of the generalist parasitoids is smaller than the critical value, the invading hosts can always persist despite of the predation by the generalist parasitoids, i.e., the generalist parasitoids cannot control the invasion of hosts; (ii) when the carrying capacity of the generalist parasitoids is larger than the critical value, the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations, i.e., whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations; (iii) in both cases, the generalist parasitoids always persist. Numerical simulations are presented to illustrate the theoretical results.

Almost-periodic bifurcations for one-dimensional degenerate vector fields

ABSTRACT Quasi-periodic high order degenerate bifurcation theories have been well established, but works which are related to almost-periodic bifurcations seem to be very few. In this paper, we consider the almost-periodic time-dependent perturbations of one-dimensional degenerate vector field With the KAM theory and singularity theory, we show that the universal unfolding of the vector field can persist under small almost-periodic perturbation if some appropriate non-resonant conditions are satisfied, which implies strongly non-resonant invariant tori in the integrable part and all bifurcation scenario can survive under any small almost-periodic perturbation.

A critically degenerate elliptic Dirichlet problem, spectral theory and bifurcation

In an open, bounded subset Ω of RN such that 0∈Ω we consider the nonlinear eigenvalue problem −∑i,j=1N∂i{Aij(x)∂ju}+V(x)u+n(x,∇u)+g(x,u)=λuinΩ∫Ωu2+∑i,j=1NAij∂ju∂iudx<∞andu=0on∂Ω, where V∈L∞(Ω) and the nonlinear terms n and g are of higher order near 0 so that the formal linearization about the trivial solution u≡0 is −∑i,j=1N∂i{Aij(x)∂ju}+Vu=λu. The leading term is degenerate elliptic on Ω because it is assumed that there are constants C2≥C1>0 such that C1|x|2|ξ|2≤∑i,j=1NAij(x)ξjξi≤C2|x|2|ξ|2for allξ∈RNand almost allx∈Ω. This is the lowest level of degeneracy at x=0 for which the linearization has a non-empty essential spectrum. Furthermore, elliptic regularity theory does not apply at x=0. Eigenfunctions of the linearization and solutions of the nonlinear problem having finite energy may be singular at the origin. The main results establish conditions for the existence or not of eigenvalues of the linearization, describe the behaviour of eigenfunctions as x→0 and determine values of the parameter λ at which bifurcation from the line of trivial solutions occurs. Standard bifurcation theory does not apply, even when n and g are smooth functions, since the nonlinear terms generate operators which are Gâteaux but not Fréchet differentiable at the trivial solution.