Showing papers on "Infinite-period bifurcation published in 2012"

Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

Abstract: We perform a bifurcation analysis of a discrete predator–prey model with Holling functional response. We summarize stability conditions for the three kinds of fixed points of the map, further called F 1 , F 2 and F 3 and collect complete information on this in a single scheme. In the case of F 2 we also compute the critical normal form coefficient of the flip bifurcation analytically. We further obtain new information about bifurcations of the cycles with periods 2, 3, 4, 5, 8 and 16 of the system by numerical computation of the corresponding curves of fixed points and codim-1 bifurcations, using the software package MatContM . Numerical computation of the critical normal form coefficients of the codim-2 bifurcations enables us to determine numerically the bifurcation scenario around these points as well as possible branch switching to curves of codim-1 points. Using parameter-dependent normal forms, we compute codim-1 bifurcation curves that emanate at codim-2 bifurcation points in order to compute the stability boundaries of cycles with periods 4, 5, 8 and 16.

Bifurcation control of a parametric pendulum

Abstract: In this paper, we apply chaos control methods to modify bifurcations in a parametric pendulum-shaker system. Specifically, the extended time-delayed feedback control method is employed to maintain stable rotational solutions of the system avoiding period doubling bifurcation and bifurcation to chaos. First, the classical chaos control is realized, where some unstable periodic orbits embedded in chaotic attractor are stabilized. Then period doubling bifurcation is prevented in order to extend the frequency range where a period-1 rotating orbit is observed. Finally, bifurcation to chaos is avoided and a stable rotating solution is obtained. In all cases, the continuous method is used for successive control. The bifurcation control method proposed here allows the system to maintain the desired rotational solutions over an extended range of excitation frequency and amplitude.

The shape of phase-resetting curves in oscillators with a saddle node on an invariant circle bifurcation

Abstract: We introduce a simple two-dimensional model that extends the Poincare oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle (SNIC) for certain parameter values. Arbitrarily close to this bifurcation, the phase-resetting curve (PRC) continuously depends on parameters, where its shape can be not only primarily positive or primarily negative but also nearly sinusoidal. This example system shows that one must be careful inferring anything about the bifurcation structure of the oscillator from the shape of its PRC.

Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle

Abstract: Homoclinic bifurcation is a difficult and important topic of bifurcation theory. As we know, a general theory for a homoclinic loop passing through a hyperbolic saddle was established by [Roussarie, 1986]. Then the method of stability-changing to find limit cycles near a double homoclinic loop passing through a hyperbolic saddle was given in [Han & Chen, 2000], and further developed by [Han et al., 2003; Han & Zhu, 2007]. For a homoclinic loop passing through a nilpotent saddle there are essentially two different cases, which we distinguish by cuspidal type and smooth type, respectively. For the cuspidal type a general theory was recently established in [Zang et al., 2008]. In this paper, we consider limit cycle bifurcation near a double homoclinic loop passing through a nilpotent saddle by studying the analytical property of the first order Melnikov functions for general near-Hamiltonian systems and obtain the conditions for the perturbed system to have 8, 10 or 12 limit cycles in a neighborhood of the loop with seven different distributions. In particular, for the homoclinic loop of smooth type, a general theory is obtained as a consequence. We finally consider some polynomial systems and find a lower bound of the maximal number of limit cycles as an application of our main results.

The “resultant bifurcation diagram” method and its application to bifurcation behaviors of a symmetric railway bogie system

Abstract: The concept of symmetric bifurcation for a symmetric wheel-rail system is defined. After that, the time response of the system can be achieved by the numerical integration method, and an unfixed and dynamic Poincare section and its symmetric section for the symmetric wheel-rail system are established. Then the ‘resultant bifurcation diagram’ method is constructed. The method is used to study the symmetric/asymmetric bifurcation behaviors and chaotic motions of a two-axle railway bogie running on an ideal straight and perfect track, and a variety of characteristics and dynamic processes can be obtained in the results. It is indicated that, for the possible sub-critical Hopf bifurcation in the railway bogie system, the stable stationary solutions and the stable periodic solutions coexist. When the speed is in the speed range of Hopf bifurcation point and saddle-node bifurcation point, the coexistence of multiple solutions can cause the oscillating amplitude change for different kinds of disturbance. Furthermore, it is found that there are symmetric motions for lower speeds, and then the system passes to the asymmetric ones for wide ranges of the speed, and returns again to the symmetric motions with narrow speed ranges. The rule of symmetry breaking in the system is through a blue sky catastrophe in the beginning.

New Bifurcation Critical Criterion of Flip-Neimark-Sacker Bifurcations for Two-Parameterized Family of -Dimensional Discrete Systems

Abstract: A new bifurcation critical criterion of flip-Neimark-Sacker bifurcation is proposed for detecting or anticontrolling this type of codimension-two bifurcation of discrete systems in a general sense. The criterion is built on the properties of coefficients of characteristic equations instead of the properties of eigenvalues of Jacobian matrix of nonlinear system, which is formulated using a set of simple equalities and inequalities consisting of the coefficients of characteristic polynomial equation. The inequality conditions enable us to easily pick off the fake parameter domain whereas the equality conditions are used to accurately locate the critical bifurcation point. In particular, after the bifurcation parameter piont is determined, the inequality conditions can be used to figure out the feasible region of other system parameters. Thus, the criterion is suitable for two-parameterized family of -dimensional discrete systems. As compared with the classical critical criterion (or definition) of flip-Neimark-Sacker bifurcation stated in terms of the properties of eigenvalues, the proposed criterion is preferable in anticontrolling or detecting the existence of flip-Neimark-Sacker bifurcation in high-dimension nonlinear systems, due to its explicit parameter mechanism of the bifurcation.

“Quorum sensing” generated multistability and chaos in a synthetic genetic oscillator

Abstract: We model the dynamics of the synthetic genetic oscillator Repressilator equipped with quorum sensing In addition to a circuit of 3 genes repressing each other in a unidirectional manner, the model includes a phase-repulsive type of the coupling module implemented as the production of a small diffusive molecule—autoinducer (AI) We show that the autoinducer (which stimulates the transcription of a target gene) is responsible for the disappearance of the limit cycle (LC) through the infinite period bifurcation and the formation of a stable steady state (SSS) for sufficiently large values of the transcription rate We found conditions for hysteresis between the limit cycle and the stable steady state The parameters’ region of the hysteresis is determined by the mRNA to protein lifetime ratio and by the level of transcription-stimulating activity of the AI In addition to hysteresis, increasing AI-dependent stimulation of transcription may lead to the complex dynamic behavior which is characterized by the appearance of several branches on the bifurcation continuation, containing different regular limit cycles, as well as a chaotic regime The multistability which is manifested as the coexistence between the stable steady state, limit cycles, and chaos seems to be a novel type of the dynamics for the ring oscillator with the added quorum sensing positive feedback

The singular limit of a Hopf bifurcation

Abstract: Hopf bifurcation in systems with multiple time scales takes several forms, depending upon whether the bifurcation occurs in fast directions, slow directions or a mixture of these two. Hopf bifurcation in fast directions is influenced by the singular limit of the fast time scale, that is, when the ratio $\epsilon$ of the slowest and fastest time scales goes to zero. The bifurcations of the full slow-fast system persist in the layer equations obtained from this singular limit. However, the Hopf bifurcation of the layer equations does not necessarily have the same criticality as the corresponding Hopf bifurcation of the full slow-fast system, even in the limit $\epsilon \to 0$ when the two bifurcations occur at the same point. We investigate this situation by presenting a simple slow-fast system that is amenable to a complete analysis of its bifurcation diagram. In this model, the family of periodic orbits that emanates from the Hopf bifurcation accumulates onto the corresponding family of the layer equations in the limit as $\epsilon \to 0$; furthermore, the stability of the orbits is dictated by that of the layer equation. We prove that a torus bifurcation occurs $O(\epsilon)$ near the Hopf bifurcation of the full system when the criticality of the two Hopf bifurcations is different.

Detailed study of bifurcations in an epidemic model on a dynamic network

Abstract: The bifurcations in a four-variable ODE model of an SIS type epidemic on an adaptive network are studied. The model describes the propagation of the epidemic on a network where links (or edges) of different type (i.e. SI;II and SS ) can be activated or deleted according to a simple rule consisting of random link activation and deletion. In the case when II links cannot be neither deleted nor created it is proved that the system can have at most three steady states with the trivial, disease-free steady state being one of them. It is shown that a stable endemic steady state can appear through a transcritical bifurcation, or a stable and an unstable endemic steady state arise as a result of saddle-node bifurcation. Moreover, at the endemic steady state a Hopf bifurcation may occur giving rise to stable oscillation. The bifurcation curves in the parameter space are determined analytically using the parametric representation method. For certain parameter regimes or bifurcation types, analytical results based on the ODE model show good agreement when compared to results based on individual- based network simulations. When agreement between the two modelling approaches holds, the ODE-based model provides a faster and more reliable tool that can be used to explore full spectrum of model behaviour.

The dynamics and bifurcation control of a singular biological economic model

Abstract: The objective of this paper is to study systematically the dynamics and control strategy of a singular biological economic model that is described by a differential-algebraic equation. It is shown that when the economic profit passes through zero, this model exhibits the transcritical bifurcation, the Hopf bifurcation, and the limit cycle. In particular, the system undergoes the singularity induced bifurcation at the positive equilibrium, which can result in impulse. Then, state feedback controllers closer to the actual control strategies are designed to eliminate the unexpected singularity induced bifurcation and stabilize the positive equilibrium under the positive profit. Finally, numerical simulations verify the results and illustrate the effectiveness of the controllers. Also, the model with positive economic profit is shown numerically to have different dynamics.

On 3-Parameter Families of Piecewise Smooth Vector Fields in the Plane

Abstract: This paper is concerned with the local bifurcation analysis around typical singularities of piecewise smooth planar dynamical systems. Three-parameter families of a class of nonsmooth vector fields are studied, and the bifurcation diagrams are exhibited. Our main results describe a particular unfolding of the so-called fold-cusp singularity by means of the variation of 3 parameters.

Bifurcations of Indifference Points in Discrete Time Optimal Control Problems

Abstract: This thesis develops new methods to analyse non-convex discrete time optimal control problems. A distinctive feature of such problems is that indifference states may occur: these are initial states at which several optimising trajectories originate. In the thesis, the genesis of such points through indifference-attractor bifurcations is studied as system parameters are varied. This necessitates an analysis of heteroclinic bifurcation scenarios of the state-costate dynamics. In particular, it is found that infinitely many indifference points exist at certain parameter values, or, equivalently, that the associated value function is not differentiable at infinitely many points in state space. The results make it possible to analyse the bifurcation structure of the discrete-time lake pollution management problem.

Stability and sustained oscillations in a ventricular cardiomyocyte model.

Abstract: The Luo-Rudy I model, describing the electrophysiology of a ventricular cardiomyocyte, is associated with an 8-dimensional discontinuous dynamical system with logarithmic and exponential non-linearities depending on 15 parameters. The associated stationary problem was reduced to a nonlinear system in only two unknowns, the transmembrane potential V and the intracellular calcium concentration [Ca] i . By numerical approaches appropriate to bifurcation problems, sections in the static bifurcation diagram were determined. For a variable steady depolarizing or hyperpolarizing current (I st), the corresponding projection of the static bifurcation diagram in the (I st, V) plane is complex, featuring three branches of stationary solutions joined by two limit points. On the upper branch oscillations can occur, being either damped at a stable focus or diverted to the lower branch of stable stationary solutions when reaching the unstable manifold of a homoclinic saddle, thus resulting in early after-depolarizations (EADs). The middle branch of solutions is a series of unstable saddle points, while the lower one a series of stable nodes. For variable slow inward and K+ current maximal conductances (g si and g K), in a range between 0 and 4-fold normal values, the dynamics is even more complex, and in certain instances sustained oscillations tending to a limit cycle appear. All these types of behavior were correctly predicted by linear stability analysis and bifurcation theory methods, leading to identification of Hopf bifurcation points, limit points of cycles and period doubling bifurcations. In particular settings, e.g. one-fifth-of-normal g si, EADs and sustained high amplitude oscillations due to an unstable resting state may occur simultaneously.

Analytical approximation of heteroclinic bifurcation in a 1:4 resonance

Abstract: Bifurcation of heteroclinic cycle near 1:4 resonance in a self-excited parametrically forced oscillator with quadratic nonlinearity is investigated analytically in this paper. This bifurcation mechanism leads to the disappearance of a slow flow limit cycle giving rise to frequency-locking near the resonance. The analytical approach used to approximate the bifurcation is based on a collision criterion between the slow flow limit cycle and saddles involved in the bifurcation. The amplitudes of the 1:4-subharmonic solution and the slow flow limit cycle are approximated using a double perturbation procedure and the heteroclinic bifurcation is captured applying the collision criterion. For validation, the analytical results are compared to those obtained by numerical simulations.

Nonsmooth and smooth bifurcations in a discontinuous piecewise map

Abstract: In this paper, a piecewise map with singularity of the power (-1/2) is introduced. For this piecewise map, there is an infinite discontinuous gap on the origin. The conditions of nonsmooth border-collision bifurcation and smooth fold or flip bifurcation are analytically derived. For period-1 fixed point, two-parameter-plane can be divided into seven ranges according to different bifurcation structures. For period-n orbits, codimension-2 bifurcation point may lead to different period-increment sequence, and a peculiar feature is found that there are two different period-increment sequences in the same bifurcation diagram.

Bifurcation of Limit Cycles from a Non-Hamiltonian Quadratic Integrable System with Homoclinic Loop

Abstract: In this chapter, we study quadratic perturbations of a non-Hamiltonian quadratic integrable system with a homoclinic loop. We prove that the perturbed system has at most two limit cycles in the finite phase plane, and the bound is exact. The proof relies on an estimation of the number of zeros of related Abelian integrals.

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

Abstract: The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the well-known Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameter three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems. In the piecewise system, the noose bifurcation involves reversible periodic orbits that intersect the separation plane at two or four points. This work is focused on those reversible periodic orbits that intersect the separation plane twice (RP2-orbits). It is established that for every T between 2 π and a critical point, there exists a unique value of the parameter for which the system has an RP2-orbit with period T . Moreover, this critical value, that separates periodic orbits with two or four points of intersection with the separation plane, corresponds to an RP2-orbit that crosses the separation plane tangentially. It is also proved that in a parameter versus period bifurcation diagram, the curve of this family of periodic orbits has a unique maximum point, which corresponds to the saddle-node bifurcation of periodic orbits that appears in the noose bifurcation.

Slow passage through multiple bifurcation points

Abstract: The slow passage problem, the slow variation of a control parameter, is explored in a model problem that posses several co-existing equilibria (fixed points, limit cycles and 2-tori), and these are either created or destroyed or change their stability as control parameters are varied through Hopf, Neimark-Sacker and torus break-up bifurcations. The slow passage through the Hopf bifurcation behaves as determined in previous studies (the delay in the observation of oscillations depends only on how far from critical the ramped parameter is at the start of the ramp--a memory effect), and that through the Neimark-Sacker bifurcation also behaves similarly. We show that the range of the ramped parameter over which a Hopf oscillation can be observed (limited by the subsequent onset of torus oscillations from the Neimark-Sacker bifurcation) is twice that predicted from a static-parameter bifurcation analysis, and this is a memory-less result independent of the initial value of the ramped parameter. These delay and memory effects are independent of the ramp rate, for small enough ramp rates. The slow passage through the torus break-up bifurcation is qualitatively different. It does not depend on the initial value of the ramped parameter, but instead is found to depend, on average, on the square-root of the ramp rate. This is typical of transient behavior. We show that this transient behavior is due to the stable and unstable manifolds of the saddle limit cycles forming a very narrow escape tunnel for trajectories originating near the unstable 2-torus no matter how slow a ramp speed is used. The type of bifurcation sequence in the model problem studied (Hopf, Neimark-Sacker, torus break-up) is typical of those for the transition to spatio-temporal chaos in hydrodynamic problems, and in those physical problems the transition can occur over a very small range of the control parameter, and so the inevitable slow drift of the parameter in an experiment may lead to observations where the slow passage results reported here need to be taken into account.

A novel result in the field of nonlinear stability analysis of boiling water reactors

Abstract: The nonlinear stability analysis of boiling water nuclear reactors (BWRs) is conducted with the aid of so-called advanced, well validated, system codes and an advanced reduced order model to build a detailed mathematical understanding of the BWR behavior in the practical relevant parameter space. In the last years, the existence of Hopf-bifurcation points was confirmed by some researchers. In the framework of this paper, a parameter region was analyzed in which the coexistence of different stability states is realized. As a novel result, we found a parameter region in which stable fixed points, unstable limit cycles and stable limit cycles coexist. This system behavior can be explained by a saddle-node bifurcation of cycles (turning point). The existence of this solution type in a BWR system indicates the possibility of large amplitude limit cycle oscillations in the linear stable region.

Saddle-node Bifurcation of Power Systems Analysis in the Simplest Normal Form

Abstract: That using of the simplest normal form theory, derives the second-order simplest normal form of the power system original differential equation model at the saddle-node bifurcation point, by transforming the higher order solves the problem of resonance coefficient singular, Analysis the system instable phenomenon that is caused by saddle-node bifurcation in use of approximate analytical solution of the power system, In participation factors analysis the impact of the resonance modal to others, reveals the physical mechanism of power system instability caused by saddle-node bifurcation, the saddle-node bifurcation in the voltage collapse at the same time, due to the resonance modal influence, the angle also. There are some values for profound understand the mechanism that system instability be lead by saddle-node bifurcation. Using a power system model verify the above conclusions.

Bifurcation analysis of a discrete-time kaldor model of business cycle

Abstract: This paper reports bifurcation dynamics of a discrete-time Kaldor model of business cycle. By using center manifold theorem and bifurcation theory, it is shown that the model not only undergoes flip bifurcation and Neimark–Sacker bifurcation, but also 1 : 1 resonance of codimension two bifurcation occurs. Some numerical examples are given to support the analytic results.

Coexistence of periods in a bifurcation

Abstract: A particular type of order-to-chaos transition mediated by an infinite set of coexisting neutrally stable limit cycles of different periods is studied in the Varley–Gradwell–Hassell population model. We prove by an algebraic method that this kind of transition can only happen for a particular bifurcation parameter value. Previous results on the structure of the attractor at the transition point are here simplified and extended.

Multiple attractors and bifurcations in hard oscillators driven by constant inputs

Abstract: Hard oscillators are dynamical systems that show the coexistence of qualitatively different attractors, in the form of limit cycles and equilibrium points. In the presence of external inputs their dynamic behavior is significantly different from those of oscillators, called soft, with a limit cycle as unique attractor. This paper studies the dynamics of a simple hard oscillator under the influence of a constant external input. It is shown that, despite the apparent simplicity, when the input strength and the oscillator's natural frequency are varied the system exhibits many different bifurcation phenomena, including global bifurcations as saddle-node on limit cycle and homoclinic bifurcations. The model under investigation can play a role in neuroscience, as it exhibits two different mechanisms of class I neural excitability and one mechanism for class II. It also highlights a mechanism of transition between the two classes.