Showing papers on "Infinite-period bifurcation published in 2004"
TL;DR: An optimal velocity model which includes the reflex time of drivers is investigated, which finds branches of oscillating solutions connecting Hopf bifurcation points, which may be super- or subcritical, depending on parameters.
Abstract: We investigate an optimal velocity model which includes the reflex time of drivers After an analytical study of the stability and local bifurcations of the steady-state solution, we apply numerical continuation techniques to investigate the global behavior of the system Specifically, we find branches of oscillating solutions connecting Hopf bifurcation points, which may be super- or subcritical, depending on parameters This analysis reveals several regions of multistability
130 citations
TL;DR: In this article, a delayed predator-prey system with Beddington-DeAngelis functional response was considered and the stability of the interior equilibrium was analyzed by analyzing the associated characteristic transcendental equation.
70 citations
TL;DR: The normal form of this singularity is calculated explicitly and both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf b ifurcation of ODEs.
Abstract: The fold-flip bifurcation occurs if a map has a fixed point with multipliers +1 and -1 simultaneously. In this paper the normal form of this singularity is calculated explicitly. Both local and global bifurcations of the unfolding are analyzed by exploring a close relationship between the derived normal form and the truncated amplitude system for the fold-Hopf bifurcation of ODEs. Two examples are presented, the generalized Henon map and an extension of the Lorenz-84 model. In the latter example the first-, second- and third-order derivatives of the Poincare map are computed using variational equations to find the normal form coefficients.
69 citations
TL;DR: It is found that Hopf bifurcation occurs for the strong kernel of a general two-neuron model with distributed delays and a strong kernel, which means that a family of periodic solutions b ifurcates from the equilibrium when the bIfurcation parameter exceeds a critical value.
66 citations
TL;DR: In this paper, an inertial shaker as a vibratory system with impact is considered and the theoretical solution of periodic n-1 impacting motion can be obtained and the Poincare map is established by means of differential equations, periodicity and matching conditions.
51 citations
TL;DR: A few mathematical problems arising in the classical synchronization theory are discussed, especially those relating to complex dynamics, and the roots of the theory originate in the pioneeringExperimental synchronization theory.
Abstract: A few mathematical problems arising in the classical synchronization theory are discussed; especially those relating to complex dynamics. The roots of the theory originate in the pioneering experim...
47 citations
TL;DR: In this article, the number of limit cycles in a cubic system is investigated and two different distributions are given by using the methods of bifurcation theory and qualitative analysis, respectively.
Abstract: This paper is concerned with the number of limit cycles in a cubic system. Eleven limit cycles are found and two different distributions are given by using the methods of bifurcation theory and qualitative analysis.
36 citations
TL;DR: It is proved that the Neimark-Sacker bifurcation occurs when the bifURcation parameter exceeds a critical value.
31 citations
TL;DR: In this article, the stability and bifurcation in a mutual model with a delay τ, where τ is regarded as a parameter, were studied. And it was shown that there are stability switches and Hopf bifurbation occurs when the delay τ passes through a sequence of critical values.
Abstract: In this paper, we study the stability and bifurcation in a mutual model with a delay τ, where τ is regarded as a parameter. It is found that there are stability switches, and Hopf bifurcation occur when the delay τ passes through a sequence of critical values. A formula for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions in the first bifurcation value is given using the normal form method and center manifold theorem.
30 citations
TL;DR: Using bifurcation methods, the maximal number of limit cycles in global bifURcation is obtained in a family of polynomial systems.
Abstract: In this paper, we study the number of limit cycles in a family of polynomial systems. Using bifurcation methods, we obtain the maximal number of limit cycles in global bifurcation.
30 citations
TL;DR: The number of limit cycles in a cubic system is found and two different distributions are given by using the methods of bifurcation theory and qualitative analysis.
Abstract: This paper concerns the number of limit cycles in a cubic system. Eleven limit cycles are found and two different distributions are given by using the methods of bifurcation theory and qualitative analysis.
TL;DR: In this article, the authors presented a dynamic bifurcation analysis of the non-linear Duffing's equation on a simple elastic structure, which is a two-bar elastic truss with a damper and possesses geometrical nonlinear stiffness.
Abstract: In this paper, we present a dynamic bifurcation analysis of the non-linear Duffing's equation on a simple elastic structure. The structure is a two-bar elastic truss with a damper, and possesses geometrical non-linear stiffness. We consider the dynamic instability of its structure based on Duffing's oscillation, which shows bifurcation behavior of the homoclinic orbit. We could numerically forecast the trajectory near the invariant saddle point of homoclinic bifurcation on this model, and we found that it is possible to solve dynamic bifurcation and strange attractors (chaos) on this non-linear structure. On this truss, we could investigate the dynamic stability of the strange attractor using Lyapunov exponents under the frequency and/or the amplitude parameter of periodic load.
TL;DR: In this article, the authors give sufficient conditions for the existence of one or two limit cycles of singular Lienard systems through the construction of a Poincare-Bendixson domain.
Abstract: We give sufficient conditions for the existence of one or two limit cycles of singular Lienard systems through the construction of a Poincare–Bendixson domain. With the help of the theory of rotated vector fields,we develop a method to compute bifurcation value at Saddle-node bifurcation of limit cycles and homoclinic or symmetric heteroclinic bifurcations. We also present application examples and prove the existence of duck cycles.
TL;DR: It is proved that for the case a 0 the system has at most six limit cycles bifurcated from Hopf bifurstcation or has at least seven limit cyclesbifurCated from the double homoclinic loop.
Abstract: The Hopf bifurcation, saddle connection loop bifurcation and Poincare bifurcation of the generalized Rayleigh–Lienard oscillator Ẍ+aX+2bX3+e(c3+c2X2+c1X4+c4Ẋ2)Ẋ=0 are studied. It is proved that for the case a 0 the system has at most six limit cycles bifurcated from Hopf bifurcation or has at least seven limit cycles bifurcated from the double homoclinic loop. For the case a>0, b<0 the system has at most three limit cycles bifurcated from Hopf bifurcation or has three limit cycles bifurcated from the heteroclinic loop.
TL;DR: It is proved that the Hopf cyclicity is two, and it is also given by the new configurations of the limit cycles bifurcated from the homoclinic loop or heteroclinics loop for quintic system with quintic perturbations by using the methods of b ifurcation theory and qualitative analysis.
TL;DR: In this paper, the global stability of limit cycle oscillations for a particular class of systems and networks was investigated and the results were proven for values of the parameter in the vicinity of a bifurcation value.
TL;DR: In this article, the bifurcation behavior of an equilibrium solution to the Gierer-Meinhardt activator-inhibitor system is analyzed for different ranges of the inhibitor diffusivity.
Abstract: In the limit of small activator diffusivity $\varepsilon$, and in a bounded
domain in $\mathbb{R}^{N}$ with $N=1$ or $N=2$ under homogeneous Neumann
boundary conditions, the bifurcation behavior of an equilibrium
one-spike solution to the Gierer-Meinhardt activator-inhibitor system is
analyzed for different ranges of the inhibitor diffusivity $D$. When
$D=\infty$, it is well-known that a one-spike solution for the resulting
shadow Gierer-Meinhardt system is unstable, and the locations of
unstable equilibria coincide with the points in the domain that are
furthest away from the boundary. For a unit disk domain it is shown
that as $D$ is decreased below a critical bifurcation value $D_{c}$,
with $D_{c}=O(\varepsilon^2 e^{2/\varepsilon})$, the spike at the origin becomes
stable, and unstable spike solutions bifurcate from the origin. The
locations of these bifurcating spikes tend to the boundary of the domain
as $D$ is decreased further. Similar bifurcation behavior is studied in
a one-parameter family of dumbbell-shaped domains. This motivates a
further analysis of the existence of certain near-boundary spikes. Their
location and stability is given in terms of the modified Green's
function. Finally, for the dumbbell-shaped domain, an intricate
bifurcation structure is observed numerically as $D$ is decreased below
some $O(1)$ critical value.
TL;DR: In this article, the authors studied the bifurcation of limit cycles near a polycycle with n hyperbolic saddle points and established necessary and sufficient conditions for the existence of a separatrix connecting any two saddle points.
Abstract: In the study of Hilbert 16th problem the most difficult part is to find the maximal number of limit cycles appearing near a polycycle by perturbations. In this paper we study the bifurcation of limit cycles near a polycycle with n hyperbolic saddle points. We obtain a sufficient condition for the polycycle to generate at least n limit cycles. We also establish a necessary and sufficient condition for the existence of a separatrix connecting any two saddle points.
01 Jan 2004
TL;DR: In this paper, a general Liapunov-Schmidt type of reduction for the study of the bifurcation of periodic points from a symmetric fixed point in families of reversible diffeomorphisms is presented.
Abstract: In this paper we survey a general Liapunov-Schmidt type of reduction for the study of the bifurcation of periodic points from a symmetric fixed point in families of reversible diffeomorphisms. The approach is strongly interwoven with normal form theory for reversible mappings and also addresses the stability problem for the bifurcating periodic points. The paper concludes with an application to subharmonic bifurcation in reversible vectorfields.
TL;DR: In this article, the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback was studied, and the method of Fredholm alternative was applied to determine the bifurcating periodic motions and their stability.
Abstract: This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions.
TL;DR: In this article, the authors study the evolution of the self-sustained current oscillation (SSCO) solutions (limit cycles) in sequential tunnelling of superlattices under dc bias.
Abstract: Within a discrete drift model, we study the evolution of the self-sustained current oscillation (SSCO) solutions (limit cycles) in sequential tunnelling of superlattices under dc bias. We propose two possible modes: one is co-existence of both fixed point and limit cycle solutions, exchanging stabilities at the bifurcation point, termed the transcritical Hopf bifurcation and the other is that, at high doping densities and inside SSCO regime, the breathing motion of the limit cycles, in which the amplitude and frequency of SSCOs oscillate as a function of an applied dc bias, in contrast with a square-root dependence expected in a conventional Hopf bifurcation.
TL;DR: In this paper, the authors investigated the number, location and stability of limit cycles in a class of perturbed polynomial polynomial systems with (2n + 1) or ( 2n + 2)-degree by constructing a detection function and using qualitative analysis.
Abstract: In this paper, we investigate the number, location and
stability of limit cycles in a class of perturbed polynomial
systems with (2n + 1) or
(2n + 2)-degree by
constructing detection function and using qualitative analysis.
We show that there are at most n limit cycles in the perturbed
polynomial system, which is similar to the result of Perko in
[8] by using Melnikov method. For n = 2, we establish the general
conditions depending on polynomial’s coefficients for the
bifurcation, location and stability of limit cycles. The
bifurcation parameter value of limit cycles in [5] is also
improved by us. When n = 3
the sufficient and necessary conditions for the appearance of 3
limit cycles are given. Two numerical examples for the location
and stability of limit cycles are used to demonstrate our
theoretical results.
TL;DR: In this article, the authors study a reaction diffusion problem with delay and make an analysis of the stability of solutions by means of bifurcation theory, taking the delay constant as a parameter.
Abstract: In this work we study a reaction–diffusion problem with delay and we make an analysis of the stability of solutions by means of bifurcation theory. We take the delay constant as a parameter. Special conditions on the vector field assure existence of a spatially nonconstant positive equilibrium Uk , which is stable for small values of the delay. An increase of the delay destabilizes the equilibrium of Uk and leads to super or subcritical Hopf bifurcation. Dedicated to Professor Jose Geraldo Dos Reis. E-mail: mabena@ffclrp.usp.br
TL;DR: In this article, the authors studied a small Prandtl number fluid contained in a two-dimensional vertical cavity subject to a lateral thermal gradient and found a transition from a steady flow to periodic solutions through a supercritical Hopf bifurcation that maintains the centro-symmetry of the basic circulation.
Abstract: Convective flows of a small Prandtl number fluid contained in a two-dimensional vertical cavity subject to a lateral thermal gradient are studied numerically. The chosen geometry and the values of the material parameters are relevant to semiconductor crystal growth experiments in the horizontal configuration of the Bridgman method. For increasing Rayleigh numbers we find a transition from a steady flow to periodic solutions through a supercritical Hopf bifurcation that maintains the centro-symmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf bifurcation, the periodic solution loses stability in a subcritical Neimark–Sacker bifurcation, which gives rise to a branch of quasiperiodic states. In this branch, several intervals of frequency locking have been identified. Inside the resonance horns the stable limit cycles lose and gain stability via some typical scenarios in the bifurcation of periodic solutions. After a complicated bifurcation diagram of the stable limit cycle of the 1:10 resonance horn, a soft transition to chaos is obtained.
TL;DR: In this paper, the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for general planar systems is investigated. But the existence conditions of 4 homoclineic bifurbation curves and small and large limit cycles are not investigated.
Abstract: This paper concerns with the bifurcation of limit cycles from a double homoclinic loop under multiple parameter perturbations for general planar systems. The existence conditions of 4 homoclinic bifurcation curves and small and large limit cycles are especially investigated.
TL;DR: In this article, the steady state and dynamic behavior of two-phase systems in physical equilibrium is investigated and the effect of the forcing amplitude and frequency on the behavior of the non-autonomous system is investigated at a number of chosen positions of the center of forcing.
TL;DR: In this article, the authors studied the number of bifurcation points of the Lipschitz continuous differential equation with at most two periodic solutions for each λ ∈ R.
Abstract: We study the number of bifurcation points of x′=F(t,x,λ), where F is periodic in t, continuous, and locally Lipschitz continuous with respect to x, by assuming that the differential equation has at most two periodic solutions for each λ∈ R . Under some additional assumptions we prove that there are at most two bifurcation points and we find sufficient conditions under which this equation has exactly k bifurcation values, where k=0,1,2.
TL;DR: This paper reports some computation of periodic solutions arising from Hopf bifurcations in order to build up a more accurate procedure for semi-analytical approximations to detect limit cycle bifURcations.
Abstract: This paper reports some computation of periodic solutions arising from Hopf bifurcations in order to build up a more accurate procedure for semi-analytical approximations to detect limit cycle bifurcations. The approximation formulas are derived using nonlinear feedback systems theory and the harmonic balance method. The monodromy matrix is computed for several simple nonlinear flows to detect the first bifurcation of the cycles in the neighborhood of the original Hopf bifurcation.
23 May 2004
TL;DR: A method to control limit cycles in smooth planar systems making use of the theory of nonsmooth bifurcations is presented, and the amplitude and stability properties of the target limit cycle are controlled.
Abstract: In this paper we present a method to control limit cycles in smooth planar systems making use of the theory of nonsmooth bifurcations. By designing an appropriate switching controller, the occurrence of a corner-collision bifurcation is induced in the system and the amplitude and stability properties of the target limit cycle are controlled. The technique is illustrated through a representative example.
TL;DR: In this paper, a complex phenomenon, concerned both with the coexistence of attractors and the crises of chaotic attractors, was observed in the course of constructing domains of steady state solutions of the H?non map in parameter space by numerical methods.
Abstract: A simple branch of solution on a bifurcation diagram, which begins at static bifurcation and ends at boundary crisis (or interior crisis in a periodic window), is generally a period-doubling cascade. A domain of solution in parameter space, enclosed by curves of static bifurcation and that of boundary crisis (or the interior of a periodic window), is the trace of branches of solution. A P-n branch of solution refers to the one starting from a period-n (n?1) solution and the corresponding domain in parameter space is named the P-n domain of solution. Because of the co-existence of attractors, there may be several branches within one interval on a bifurcation diagram and different domains of solution may overlap each other in some areas of the parameter space. A complex phenomenon, concerned both with the co-existence of attractors and the crises of chaotic attractors, was observed in the course of constructing domains of steady state solutions of the H?non map in parameter space by numerical methods. A narrow domain of period-m solutions firstly co-exists with (lies on) a big period-n (m?3n) domain. Then it enters the chaotic area of the big domain and becomes period-m windows. The co-existence of attractors disappears and is called the landing phenomenon. There is an interaction between the two domains in the course of landing: the chaotic area in the big domain is enlarged and there is a crisis step near the landing area.