Showing papers on "Infinite-period bifurcation published in 2000"
TL;DR: BIFurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input.
Abstract: Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying t...
350 citations
TL;DR: In this paper, a system of ODEs which describes the transmission dynamics of childhood diseases is considered, and a three-parameter unfolding of the normal form is studied to capture possible complex dynamics of the original system which is subjected to certain constraints on the state space due to biological considerations.
93 citations
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31 Jul 2000
TL;DR: In this paper, the Liapunov-Schmidt method is used to detect and compute Bifurcation Points and Branch Switching at Simple Bifurlcation Points, and Hopf/Steady State Mode Interactions.
Abstract: 1. Reaction-Diffusion Equations.- 2. Continuation Methods.- 3. Detecting and Computing Bifurcation Points.- 4. Branch Switching at Simple Bifurcation Points.- 5. Bifurcation Problems with Symmetry.- 6. Liapunov-Schmidt Method.- 7. Center Manifold Theory.- 8. A Bifurcation Function for Homoclinic Orbits.- 9. One-Dimensional Reaction-Diffusion Equations.- 10. Reaction-Diffusion Equations on a Square.- 11. Normal Forms for Hopf Bifurcations.- 12. Steady/Steady State Mode Interactions.- 13. Hopf/Steady State Mode Interactions.- 14. Homotopy of Boundary Conditions.- 15. Bifurcations along a Homotopy of BCs.- 16. A Mode Interaction on a Homotopy of BCs.- List of Figures.- List of Tables.
79 citations
TL;DR: In this article, it was shown that in the neighbourhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area-preserving map, there is a ''twistless'' torus.
Abstract: We show that in the neighbourhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area-preserving map, there is generically a bifurcation that creates a `twistless' torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created and eventually collides with the saddle-centre bifurcation that creates the period-three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the non-degeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.
61 citations
57 citations
TL;DR: In this article, experimental results obtained by operating the Bray−Liebhafsky (BL) reaction in the CSTR are presented, and the dynamic behavior of the reaction is examined at several operation points in the concentration phase space, by varying different parameters, including specific flow rate, temperature, and mixed inflow concentrations of the feed species, one at a time.
Abstract: Experimental results obtained by operating the Bray−Liebhafsky (BL) reaction in the CSTR are presented. The dynamic behavior of the BL reaction is examined at several operation points in the concentration phase space, by varying different parameters, the specific flow rate, temperature, and mixed inflow concentrations of the feed species, one at a time. Different types of bifurcation leading to simple periodic orbits, supercritical and subcritical Hopf bifurcations, saddle node infinite period bifurcation (SNIPER), saddle loop infinite period bifurcation, and jug handle bifurcation, are observed. Moreover, complex dynamic behavior, including transition from simple periodic oscillations to complex mixed-mode oscillations and chaos, and bistability are also discovered.
42 citations
TL;DR: In this paper, a class of cubic Hamiltonion systems with the higher-order perturbed term of degree n = 5, 7, 9, 11, 13 is investigated, and it is shown that there exist at least 13 limit cycles with the distribution C 1 9 ⊃2[C 2 3 ⊂2 C 2 2 ] (let C m k denote a nest of limit cycles which encloses m singular points, and the symbol ''⊂'' is used to show the enclosing relations between limit cycles, while the sign ''+'' is divided
Abstract: A class of cubic Hamiltonion system with the higher-order perturbed term of degree n =5, 7, 9, 11, 13 is investigated. We find that there exist at least 13 limit cycles with the distribution C 1 9 ⊃2[ C 2 3 ⊃2 C 2 2 ] (let C m k denote a nest of limit cycles which encloses m singular points, and the symbol `⊂' is used to show the enclosing relations between limit cycles, while the sign `+' is used to divide limit cycles enclosing different critical points. Denote simply C m k + C m k =2 C m k , etc.) in the Hamiltonian system under the perturbed term of degree 7, and give the complete bifurcation diagrams and classification of the phase portraits by using bifurcation theory and qualitative method and numerical simulations. These results in this paper are useful for the study of the weaken Hilbert 16th problem.
32 citations
TL;DR: In this article, a parametrically excited Lienard system is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling, and two coupled equations for the amplitude and the phase of solutions are derived.
Abstract: A parametrically excited Lienard system is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Two coupled equations for the amplitude and the phase of solutions are derived. Their fixed points correspond to limit cycles for the Lienard system and we determine stability of steady-state response as well as response-parametric excitation and response-frequency curves. We use the Poincare–Bendixson theorem, the Dulac's criterion and energy considerations to study existence and characteristics of limit cycles of the two coupled equations. A limit cycle corresponds to a modulated motion in the Lienard system. We show that modulated motion can also be obtained for very low values of the parametric excitation and construct an approximate analytic solution. Moreover, we observe an unusual infinite-period homoclinic bifurcation, because in certain cases due to the symmetry of the two coupled equations two stable limit cycles approach a saddle point and merge to form a greater stable limit cycle. Subsequently, this limit cycle and another unstable limit cycle coalesce and annihilate through a saddle-node bifurcation. Comparison with the solution obtained by the numerical integration confirms the validity of our analysis.
25 citations
TL;DR: In this paper, the concept of critical points is revisited in a more general framework within this context and the concepts of bifurcation points, branching points, and valley ridge inflection points are investigated.
Abstract: Most of the time, the definitions of minima, saddle points or more generally order p (p=0,…,n) critical points, do not mention the possibility of having zero Hessian eigenvalues. This feature reflects some flatness of the potential energy hypersurface in a special eigendirection which is not often taken into account. Thus, the definitions of critical points are revisited in a more general framework within this context. The concepts of bifurcation points, branching points, and valley ridge inflection points are investigated. New definitions based on the mathematical formulation of the reaction path are given and some of their properties are outlined.
23 citations
TL;DR: In this paper, the authors considered a forced Duffing oscillator with a double-well potential, which behaves as an asymmetric soft oscillator in each potential well, and obtained the phase diagram showing rich bifurcation structure in the ω-A plane.
Abstract: We consider a forced Duffing oscillator with a double-well potential, which behaves as an asymmetric soft oscillator in each potential well. Bifurcations associated with resonances of the asymmetric attracting periodic orbits, arising from the two stable equilibrium points of the potential, are investigated in details by varying the two parameters A (the driving amplitude) and ω (the driving frequency). We thus obtain the phase diagram showing rich bifurcation structure in the ω-A plane. For the subharmonic resonances, the corresponding period-doubling bifurcation curves become folded back, within which diverse bifurcation phenomena such as "period bubblings" are observed. For the primary and superharmonic resonances, the corresponding saddle-node bifurcation curves form "horns", leaning to the lower frequencies. With decreasing ω, resonance horns with successively increasing torsion numbers recur in a similar shape. We note that recurrence of self-similar resonance horns is a "universal" feature in the bifurcation structure of many driven nonlinear oscillators.
20 citations
TL;DR: In this paper, the existence and stability of large cycles for systems with nonsmooth nonlinearities, asymptotically homogeneous at infinity, was investigated. But the method combines the parameter functionalization technique and methods of monotone concave operators.
Abstract: An autonomous system which have a principal linear part at infinity and a continuous nonlinear term of sublinear growth is investigated. Results for systems with smooth nonlinearities are presented, where accurate asymptotics of periodic cycles were introduced and cycle stability analyzed by the usual linearization of the system in a neighborhood of the cycle. A new method to study both existence and stability of large cycles for systems with nonsmooth nonlinearities, asymptotically homogeneous at infinity is developed. The method combines the parameter functionalization technique and methods of monotone concave operators.
TL;DR: An asymptotic averaging formalism is applied which permits an analytic development of the bifurcation dynamics, and, within quantifiable limits, prediction of the instability of the period one orbit in terms of the system control parameters.
Abstract: Oscillations described by autonomous three-dimensional differential equations display multiple periodicities and chaos at critical parameter values. Regardless of the subsequent scenario the key instability is often an initial bifurcation from a single period oscillation to either its subharmonic of period two, or a symmetry breaking bifurcation. A generalized third-order nonlinear differential equation is developed which embraces the dynamics vicinal to these bifurcation events. Subsequently, an asymptotic averaging formalism is applied which permits an analytic development of the bifurcation dynamics, and, within quantifiable limits, prediction of the instability of the period one orbit in terms of the system control parameters. Illustrative applications of the general formalism, are made to the Rossler equations, Lorenz equations, three-dimensional replicator equations and Chua's circuit equations. The results provide the basis for discussion of the class of systems which fall within the framework of the formalism.
TL;DR: In this paper, a car-following model of single-lane traffic is studied, where traffic flow is modeled by a system of Newton-type ordinary differential equations, and different solutions (equilibria and limit cycles) correspond to different phases of traffic.
Abstract: A car-following model of single-lane traffic is studied. Traffic flow is modeled by a system of Newton-type ordinary differential equations. Different solutions (equilibria and limit cycles) of this system correspond to different phases of traffic. Limit cycles appear as results of Hopf bifurcations (with density as a parameter) and are found analytically in small neighborhoods of bifurcation points. A study of the development of limit cycles with an aid of numerical methods is performed. The experimental finding of the presence of a two-dimensional region in the density-flux plane is explained by the finding that each of the cycles has its own branch of the fundamental diagram.
TL;DR: It will be shown, that in systems with more than one parameter the scaling constants can depend on the values of the parameters.
Abstract: Two one-dimensional dynamical systems discrete in time are presented, where the variation of one parameter causes a sequence of global bifurcations; at each bifurcation the period increases by a constant value (period-increment scenario, usually denoted as a period-adding scenario). We determine all the bifurcation points and the scaling constants of the period-increment scenario analytically. A re-injection mechanism, leading to the period-increment scenario, is discussed. It will be shown, that in systems with more than one parameter the scaling constants can depend on the values of the parameters.
TL;DR: It is shown that the organizing centers for the long time dynamics are Takens--Bogdanov bifurcation points in a broad range of parameters and the results are cast in a $\phi$-$
u$ phase diagram.
Abstract: We present a complete description of the stationary and dynamical behavior of semiconductor superlattices in the framework of a discrete drift model by means of numerical continuation, singular perturbation analysis, and bifurcation techniques. The control parameters are the applied DC voltage ($\phi$) and the doping ($
u$) in nondimensional units. We show that the organizing centers for the long time dynamics are Takens--Bogdanov bifurcation points in a broad range of parameters and we cast our results in a $\phi$-$
u$ phase diagram. For small values of the doping, the system has only one uniform solution where all the variables are almost equal. For high doping we find multistability corresponding to domain solutions and the stationary solutions may exhibit chaotic spatial behavior. In the intermediate regime of $
u$ the solution can be time-periodic depending on the bias. The oscillatory regions are related to the appearance and disappearance of Hopf bifurcation tongues which can be sub- or supercri...
TL;DR: Lienard systems of the form , with f(x) an even continuous function, are considered and the bifurcation curves of limit cycles are calculated exactly in the weak (e → 0) and in the strongly nonlinear regime.
Abstract: Lienard systems of the form , with f(x) an even continuous function, are considered. The bifurcation curves of limit cycles are calculated exactly in the weak (e → 0) and in the strongly (e → ∞) nonlinear regime in some examples. The number of limit cycles does not increase when e increases from zero to infinity in all the cases analyzed.
TL;DR: In this paper, the authors used singularity theory methods in bifurcation theory to investigate an oscillatory flow at the onset of a so-called saline oscillation, which is generated when a cup containing saline water, with a small orifice in its base, is placed within an outer vessel containing pure water.
TL;DR: This paper focuses on chemical systems and presents an approach based on the theory of normal forms for determining numerical estimates of the limit cycle that branches off at the Hopf bifurcation point, and shows that inclusion of higher-order terms in the normal form expansion of thelimit cycle provides a significant improvement of the limits cycle estimates.
Abstract: In this paper, we numerically investigate local properties of dynamical systems close to a Hopf bifurcation instability. We focus on chemical systems and present an approach based on the theory of normal forms for determining numerical estimates of the limit cycle that branches off at the Hopf bifurcation point. For several numerically ill-conditioned examples taken from chemical kinetics, we compare our results with those obtained by using traditional approaches where an approximation of the limit cycle is restricted to the center subspace spanned by critical eigenvectors, and show that inclusion of higher-order terms in the normal form expansion of the limit cycle provides a significant improvement of the limit cycle estimates. This result also provides an accurate initial estimate for subsequent numerical continuation of the limit cycle.
TL;DR: In this paper, a new approach is proposed to compute Hopf bifurcation points, which could produce small extended systems and therefore could reduce the computational effort and storage, and one numerical example is presented to demonstrate that the method is efficient.
Abstract: A new approach is proposed to compute Hopf bifurcation points. The method could produce small extended systems and therefore could reduce the computational effort and storage. One numerical example is presented to demonstrate that the method is efficient.
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TL;DR: In this article, the existence of bifurcation points of the stationary solution of the Vlasov-Maxwell system with bifurbation direction was proved, and it was shown that the stationary solutions of the system can not be found in the stationary setting.
Abstract: The theorem on the existence of bifurcation points of the stationary solutions for the Vlasov-Maxwell system with bifurcation direction is proved.
TL;DR: It is demonstrated that a three-dimensional model, consisting of a two-dimensional excitable system plus a slow inward current, is sufficient to produce the behaviour observed in the original model, and the models have a stable resting potential for all values of the bifurcation parameter.
TL;DR: In this paper, a bifurcation control of a subcritical Hopf Bifurcation is proposed to convert it to a supercritical one, where the stable periodic solution will have a relatively small amplitude at least close to the Hopf point.
01 Jan 2000
TL;DR: It is shown that, under increasing levels of input, departures from the fixed point occur as a result of a supercritical Hopf bifurcation, giving rise to limit cycle dynamics.
Abstract: We perform the stability and bifurcation analysis of an oscillatory cortical network. An explicit stability constraint for the individual coupling strength between oscillatory units is given, which, if satisfied, ensures the stability of the background fixed-point state. It is shown that, under increasing levels of input, departures from the fixed point occur as a result of a supercritical Hopf bifurcation, giving rise to limit cycle dynamics. Explicit conditions for the onset of the Hopf bifurcation are derived. The full range of dynamical behavior of the network, under typical scenarios of input variations, is illustrated. The result of this work is expected to be applicable to general oscillatory networks coupled through sigmoid type of nonlinear functions.
01 Jan 2000
TL;DR: In this paper, the Hopf bifurcation problem of a predator-prey model with undercrowding effect is considered and the model possessing at least two or three limit cycles under distinct parametic conditions is attained.
Abstract: The Hopf bifurcation problem of a predator-prey model with undercrowding effect is considered. We have attained the model possessing at least two or three limit cycles under distinct parametic conditions by the Hopf bifurcation theorem. And we have the same conclusion by the homoclinlc bifurcation theorem.
TL;DR: In this paper, a study on the plane quadratic polynomial differential systems with two or three parameters was conducted and bifurcation curves were drawn in the cross-section of parameter space, dividing the section into several regions.
Abstract: We conducted a study on the plane quadratic polynomial differential systems with two or three parameters. Bifurcation curves were drawn in the cross-section of parameter space, dividing the section into several regions. Different number of limit cycles can be identified in different regions. Diagrams of variation of amplitude of limit cycles versus parameters are shown and realistic examples of systems having different number of limit cycles are constructed. As an example, a quadratic equation with limit cycles in (1,3) distribution is shown.