Showing papers on "Infinite-period bifurcation published in 2006"

On multi-parametric bifurcations in a scalar piecewise-linear map

Abstract: In this work a one-dimensional piecewise-linear map is considered. The areas in the parameter space corresponding to specific periodic orbits are determined. Based on these results it is shown that the structure of the 2D and 3D parameter spaces can be simply described using the concept of multi-parametric bifurcations. It is demonstrated that an infinite number of two-parametric bifurcation lines starts at the origin of the 3D parameter space. Along each of these lines an infinite number of bifurcation planes starts, whereas the origin represents a three-parametric bifurcation.

Global bifurcations in duopoly when the Cournot Point is destabilized through a subcritical Neimark bifurcation

Abstract: An adaptive oligopoly model, where the demand function is isoelastic and the competitors operate under constant marginal costs, is considered. The Cournot equilibrium point then loses stability through a subcritical Neimark bifurcation. The present paper focuses some global bifurcations, which precede the Neimark bifurcation, and produce other attractors which coexist with the still attractive Cournot fixed point.

Stability and dynamics of a controlled van der Pol-Duffing oscillator

Abstract: The trivial equilibrium of a van der Pol–Duffing oscillator under a linear-plus-nonlinear feedback control may change its stability either via a single or via a double Hopf bifurcation if the time delay involved in the feedback reaches certain values. It is found that the trivial equilibrium may lose its stability via a subcritical or supercritical Hopf bifurcation and regain its stability via a reverse subcritical or supercritical Hopf bifurcation as the time delay increases. A stable limit cycle appears after a supercritical Hopf bifurcation occurs and disappears through a reverse supercritical Hopf bifurcation. The interaction of the weakly periodic excitation and the stable bifurcating solution is investigated for the forced system under primary resonance conditions. It is shown that the forced periodic response may lose its stability via a Neimark–Sacker bifurcation. Analytical results are validated by a comparison with those of direct numerical integration.

Co-dimension-Two Grazing Bifurcations in Single-Degree-of-Freedom Impact Oscillators

Abstract: Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.

Remarks on interacting Neimark–Sacker bifurcations

Abstract: We study codimension-2 bifurcations of fixed points of dissipative diffeomorphisms with a pair of complex eigenvalues together with either an eigenvalue − 1 or another such a pair. In the previous studies only cubic normal forms were considered. However, in some cases the unfolding requires higher-order terms and these are investigated here. We (re)derive the normal forms and reduce them to a single amplitude map. This map is similar to the amplitude system for the double-Hopf bifurcation of vector fields. We show how the critical normal form coefficients determine the general bifurcation picture for this amplitude map. Representative nonsymmetric perturbations of the normal forms are studied numerically. Our case studies show a detailed picture near various bifurcation curves, which is richer than known theoretical predictions. For arbitrary maps with these bifurcations we give explicit formulas for critical normal form coefficients on center manifolds. We apply them to an example from robotics where we ...

On period doubling bifurcations of cycles and the harmonic balance method

Abstract: This works attempts to give quasi-analytical expressions for subharmonic solutions appearing in the vicinity of a Hopf bifurcation. Starting with well-known tools as the graphical Hopf method for recovering the periodic branch emerging from classical Hopf bifurcation, precise frequency and amplitude estimations of the limit cycle can be obtained. These results allow to attain approximations for period doubling orbits by means of harmonic balance techniques, whose accuracy is established by comparison of Floquet multipliers with continuation software packages. Setting up a few coefficients, the proposed methodology yields to approximate solutions that result from a second period doubling bifurcation of cycles and to extend the validity limits of the graphical Hopf method.

Delay induced Hopf bifurcation in a simplified network congestion control model

Abstract: This paper studies delay induced Hopf bifurcation in a simplified network congestion control model. It is shown that there exists a critical value of delay for the stability of the network. When the value of feedback delay passes through the critical value, the system will lose its stability and Hopf bifurcation will occur. The stability and direction of the bifurcating periodic solutions are determined by perturbation procedure. A numerical example is also worked out to verify the theoretical analysis.

Stability and bifurcation in a harvested one-predator–two-prey model with delays ☆

Abstract: It is known that one-predator–two-prey system with constant rate harvesting can exhibit very rich dynamics. If such a system contains time delayed component, it can have more interesting behavior. In this paper we study the effects of the time delay on the dynamics of the harvested one-predator–two-prey model. It is shown that time delay can cause a stable equilibrium to become unstable. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhaes. An example is given and numerical simulations are finally performed for justifying the theoretical results.

Bifurcations of global reinjection orbits near a saddle-node Hopf bifurcation

Abstract: The saddle-node Hopf bifurcation (SNH) is a generic codimension-two bifurcation of equilibria of vector fields in dimension at least three. It has been identified as an organizing centre in numerous vector field models arising in applications. We consider here the case that there is a global reinjection mechanism, because the centre manifold of the zero eigenvalue returns to a neighbourhood of the equilibrium. Such a SNH bifurcation with global reinjection occurs naturally in applications, most notably in models of semiconductor lasers.We construct a three-dimensional model vector field that allows us to study the possible dynamics near a SNH bifurcation with global reinjection. This model follows on from our earlier results on a planar (averaged) vector field model, and it allows us to find periodic and homoclinic orbits with global excursions out of and back into a neighbourhood of the SNH point. Specifically, we use numerical continuation techniques to find a two-parameter bifurcation diagram for a well-known and complicated case of a SNH bifurcation that involves the break-up of an invariant sphere. As a particular feature we find a concrete example of a phenomenon that was studied theoretically by Rademacher: a curve of homoclinic orbits that accumulates on a segment in parameter space while the homoclinic orbit itself approaches a saddle-periodic orbit.

Codimension-two bifurcation analysis on firing activities in Chay neuron model

Abstract: Using codimension-two bifurcation analysis in the Chay neuron model, the relationship between the electric activities and the parameters of neurons is revealed. The whole parameter space is divided into two parts, that is, the firing and silence regions of neurons. It is found that the transition sets between firing and silence regions are composed of the Hopf bifurcation curves of equilibrium states and the saddle-node bifurcation curves of limit cycles, with some codimension-two bifurcation points. The transitions from silence to firing in neurons are due to the Hopf bifurcation or the fold limit cycle bifurcation, but the codimension-two singularities lead to complexity in dynamical behaviour of neuronal firing.

Dynamics Analysis and Limit Cycle in a Delayed Model for Tumor Growth with Quiescence

Abstract: The model analyzed in this paper is based on the unstructured model set forth by Gyllenberg and Webb (1989) without delay, which describes an interaction between the proliferating and quiescent cells tumor. In the present paper we consider the model with one delay and a unique positive equilibrium E∗ and the other is trivial. Their dynamics are studied in terms of the local stability of the two equilibrium points and of the description of the Hopf bifurcation at E∗ , that is proven to exists as the delay (taken as a parameter) crosses some critical value. We suggest to examine in laboratory experiments how to employ these results for containing tumor growth.

Chapter 4 Bifurcation Theory of Limit Cycles of Planar Systems

Abstract: Publisher Summary This chapter discusses the bifurcation theory of limit cycles of planar systems with relatively simple dynamics The theory studies the changes of orbital behavior in the phase space, especially the number of limit cycles as one varies the parameters of the system This theory has been considered by many mathematicians starting with Poincare who first introduced the notion of limit cycles A fundamental step towards modem bifurcation theory in differential equations occurred with the definition of structural stability and the classification of structurally stable systems Andronov and Pontryagin introduced the notion of a rough system and presented the necessary and sufficient conditions of roughness for systems on the plane The chapter concentrates on an in-depth study of limit cycles with general methods of both local and global bifurcations in high codimensional case

Multiple limit cycles in the standard model of three species competition for three essential resources.

Abstract: We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the ``inside'' one is an unstable separatrix and the ``outside'' one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix.

Bifurcations of limit cycles in a z2-equivariant planar polynomial vector field of degree 7 *

Abstract: By using the bifurcation theory of planar dynamical systems and the method of detection functions, the bifurcations of limit cycles in a Z2-equivariant planar perturbed Hamiltonian polynomial vector fields of degree 7 are studied. An example of a special Z2-equivariant vector field having 50 limit cycles with a configuration of compound eyes are given.

Hopf bifurcation in a delayed model for tumor-immune system competition with negative immune response

Abstract: The dynamics of the model for tumor-immune system competition with negative immune response and with one delay investigated. We show that the asymptotic behavior depends crucially on the time delay parameter. We are particularly interested in the study of the Hopf bifurcation problem to predict the occurrence of a limit cycle bifurcating from the nontrivial steady state, by using the delay as a parameter of bifurcation. The obtained results provide the oscillations given by the numerical study in M. Galach (2003), which are observed in reality by Kirschner and Panetta (1998).

Hopf-flip bifurcations of vibratory systems with impacts

Abstract: Two vibro-impact systems are considered. The period n single-impact motions and Poincare maps of the vibro-impact systems are derived analytically. Stability and local bifurcations of single-impact periodic motions are analyzed by using the Poincare maps. A center manifold theorem technique is applied to reduce the Poincare map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. It is found that near the point of codim 2 bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Period doubling bifurcation of period one single-impact motion is commonly existent near the point of codim 2 bifurcation. However, no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of Hopf bifurcation associated with period two fixed points. The results from simulation shows that there exists an interest torus doubling bifurcation occurring near the value of Hopf-flip bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transit to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems.

Double-hopf bifurcation in an oscillator with external forcing and time-delayed feedback control

Abstract: In this paper, an oscillator with time delayed velocity feedback controls is studied in detail. Particular attention is given to internal double-Hopf bifurcation with an external exciting force. Linear analysis is used to find the critical conditions under which double-Hopf bifurcation occurs. Then center manifold theory is applied to obtain an ODE system described on a four-dimensional center manifold. Further, the technique of multiple-time scales is employed to find the approximate solutions of periodic and quasi-periodic motions. Finally, numerical simulation results are presented to not only validate the analytical predictions, but also show chaotic motions for some certain parameter values.

Bifurcations in a predator-prey model with diffusion and memory

Abstract: In this paper we formulate a delayed predator-prey system in two patches in which the per capita migration rate of each species is influenced only by its own density, i.e. there is no response to the density of the other and the growth rate of the predator depends on the prey that was available in the past. If the equilibrium point lies in the Allee effect zone and when the diffusion is present only, we show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation, patterns emerge, the spatially homogeneous equilibrium loses its stability and two new spatially non-constant stable equilibria emerge which are asymptotically stable. When the delay is present only, the increase of delay destabilizes the system and causes the occurrence of periodic oscillations, Andronov–Hopf bifurcation. For the full general model (with both diffusion and delay) if the bifurcation parameters are increased through critical values of diffusion and delay the two new spatially nonconstant stationary solutions lose their stability by Hopf bifurcation.

Gluing bifurcations in chua oscillator

Abstract: Gluing bifurcation in a modified Chua's oscillator is reported. Keeping other parameters fixed when a control parameter is varied in the modified oscillator model, two symmetric homoclinic orbits to saddle focus at origin, which are mirror images of each other, are glued together for a particular value of the control parameter. In experiments, two asymmetric limit cycles are homoclinic to the saddle focus origin for different values of the control parameter. However, imperfect gluing bifurcation has been observed, in experiments, when one stable and unstable limit cycles merge to the saddle focus origin via saddle-node bifurcation.

Hopf bifurcation at k-fold resonances in reversible systems

Abstract: We study the bifurcation of families of periodic orbits near a symmetric equilibrium in a reversible system, when for a critical value of the parameters the linearization at the equilibrium has a pair of purely imaginary eigenvalues of geometric multiplicity one but algebraic multiplicity k — we call this a k-fold resonance. Combining the general reduction results of [2] with a particular normal form result for linear reversible operators we can reduce the problem to a scalar polynomial bifurcation equation. The problem has codimension k − 1, and the resulting bifurcation set is a cuspoid of order k. When crossing the codimension one strata of the bifurcation set families of periodic orbits disappear or merge in a way which is similar to what happens at a Krein instability in Hamiltonian systems.

Computation of multiple bifurcation point

Abstract: Purpose – The aim of this paper is to develop a new method for finding multiple bifurcation points in structures.Design/methodology/approach – A brief review of nonlinear analysis is presented. A powerful method (called arc‐length) for tracing nonlinear equilibrium path is described. Techniques for monitoring critical points are discussed to find the rank deficiency of the stiffness matrix. Finally, by using eigenvalue perturbation of tangent stiffness matrix, load parameter associated with multiple bifurcation points is obtained.Findings – Since other methods of finding simple bifurcation points diverge in the neighborhood of critical points, this paper introduces a new method to find multiple bifurcation points. It should be remembered that a simple bifurcation point is a multiple bifurcation point with rank deficiency equal to one. Therefore, the method is applicable to simple critical points as well.Practical implications – Global buckling of the structures should be considered in design. Many structu...

Global bifurcation destroying the experimental torus T 2

Abstract: We show experimentally the scenario of a two-frequency torus ${T}^{2}$ breakdown, in which a global bifurcation occurs due to the collision of a torus with an unstable periodic orbit, creating a heteroclinic saddle connection, followed by an intermittent behavior.