Showing papers on "Bifurcation diagram published in 2001"

Global analysis in a predator-prey system with nonmonotonic functional response ∗

Abstract: A predator-prey system with nonmonotonic functional response is considered. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The bifurcation a...

Spectral Theory and Nonlinear Functional Analysis

Abstract: INTRODUCTION General Assumptions and Basic Concepts Some New Results Historical Remarks BIFURCATION FROM SIMPLE EIGENVALUES Simple Eigenvalues and Transversality The Theorem of M.G. Crandall and P.H. Rabinowitz Local Bifurcation Diagrams The Exchange Stability Principle Applications FIRST GENERAL BIFURCATION RESULTS Lyapunov-Schmidt Reductions The theorem of J. Ize The Global Alternative of P.H. Rabinowitz The Theorem of D. Westreich THE ALGEBRAIC MULTIPLICITY Motivating the Concept of Transversality Transversal Eigenvalues Algebraic Eigenvalues Analytic Families Simple Degenerate Eigenvalues FUNDAMENTAL PROPERTIES OF THE MULTIPLICITY The Multiplicity of R.J. Magnus Relations between c and m The Fundamental Theorem The Classical Algebraic Multiplicity Finite Dimensional Characterizations The Parity of the Crossing Number GLOBAL BIFURCATION THEORY Preliminaries Local Bifurcation Global Behavior of the Bounded Components Unilateral Global Bifurcation Unilateral Bifurcation for Positive Operators APPLICATIONS Positive Solutions o Semilinear Elliptic Problems Coexistence States for Elliptic Systems Examples A Further Application REFERENCES INDEX

Chaotic characteristics of a one-dimensional iterative map with infinite collapses

Abstract: A one-dimensional iterative chaotic map with infinite collapses within symmetrical region [-1, O)/spl cup/(O, +1] is proposed. The stability of fixed points and that around the singular point are analyzed. Higher Lyapunov exponents of proposed map show stronger chaotic characteristics than some iterative and continuous chaotic models usually used. There exist inverse bifurcation phenomena and special main periodic windows at certain positions shown in the bifurcation diagram, which can explain the generation mechanism of chaos. The chaotic model with good properties can be generated if choosing the parameter of the map properly. Stronger inner pseudorandom characteristics can also be observed through /spl chi//sup 2/ test on the supposition of even distribution. This chaotic model may have many advantages in practical use.

Dewetting films: bifurcations and concentrations

Abstract: Under the influence of long-range attractive and short-range repulsive forces, thin liquid films rupture and form complex dewetting patterns. This paper studies this phenomenon in one space dimension within the framework of fourth-order degenerate parabolic equations of lubrication type. We derive the global structure of the bifurcation diagram for steady-state solutions. A stability analysis of the solution branches and numerical simulations suggest coarsening occurs. Furthermore, we study the behaviour of solutions in the limit that short-range repulsive forces are neglected. Both asymptotic analysis and numerical experiments show that this limit can concentrate mass in δ-distributions.

Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function

Abstract: A simple neural network model with discrete time delay is investigated. The linear stability of this model is discussed by analyzing the associated characteristic transcendental equation. For the case with inhibitory influence from the past state, it is found that Hopf bifurcation occurs when this influence varies and passes through a sequence of critical values. The stability of bifurcating periodic solutions and the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Chaotic behavior of a single delayed neuron equation with non-monotonously increasing transfer function has been observed in computer simulation. Some waveform diagrams, phase portraits, power spectra and plots of the largest Lyapunov exponent will also be given.

Nonlocal problems in MEMS device control

Abstract: Perhaps the most ubiquitous phenomena associated with electrostatically actuated MEMS devices is the `pull-in' voltage instability. In this instability, when applied voltages are increased beyond a certain critical voltage there is no longer a steady-state configuration of the device where mechanical members remain separate. This instability severely restricts the range of stable operation of many devices. Here, a mathematical model of an idealized electrostatically actuated MEMS device is constructed for the purpose of analyzing various schemes proposed for the control of the pull-in instability. This embedding of a device into a control circuit gives rise to a nonlinear and nonlocal elliptic problem which is analyzed through a variety of asymptotic, analytical, and numerical techniques. The pull-in voltage instability is characterized in terms of the bifurcation diagram for the mathematical model. Variations in various capacitive control schemes are shown to give rise to variations in the bifurcation diagram and hence to effect the pull-in voltage and pull-in distance.

Peakons and their bifurcation in a generalized camassa–holm equation

Abstract: Camassa and Holm [1993] recently derived a new dispersive shallow water equation known as the Camassa–Holm equation. They showed that it also has solitary wave solutions which have a discontinuous first derivative at the wave peak and thus are called "peakons". In this paper, from the mathematical point of view, we study the peakons and their bifurcation of the following generalized Camassa–Holm equation \[ u_t+2ku_x-u_{xxt}+au^mu_x = 2u_xu_{xx} + uu_{xxx} \] with a>0, k∈ℝ, m∈ℕ and the integral constants taken as zero. Using the bifurcation method of the phase plane, we first give the phase portrait bifurcation, then give the integral expressions of peakons through the bifurcation curves and the phase portraits, and finally obtain the peakon bifurcation parameter value and the number of peakons. For m=1, 2, 3, we give the explicit expressions for the peakons. It seems that the bifurcation method of phase planes is good for the study of peakons in nonlinear integrable equations.

Generating chaos via x|x|

Abstract: This paper investigates the role of the function x|x| as a chaos generator in nonautonomous systems. A Duffing-like nonautonomous oscillator is used for illustration. It is rigorously proven via the Melnikov function method that this particular quadratic function induces Smale horseshoes to the Duffing-like system. Moreover, its physical meaning as an energy function is demonstrated, which provides a critical value for the emergence of chaos. Simulations with bifurcation analysis are given for better understanding of the underlying dynamics.

Topology of vortex breakdown bubbles in a cylinder with a rotating bottom and a free surface

Abstract: The flow patterns in the steady, viscous flow in a cylinder with a rotating bottom and a free surface are investigated by a combination of topological and numerical methods. Assuming the flow is axisymmetric, we derive a list of possible bifurcations of streamline structures on varying two parameters, the Reynolds number and the aspect ratio of the cylinder. Using this theory we systematically perform numerical simulations to obtain the bifurcation diagram. The stability limit for steady flow is found and established as a Hopf bifurcation. We compare with the experiments by Spohn, Mory & Hopfinger (1993) and find both similarities and differences.

Anti-control of Hopf bifurcations

Abstract: Bifurcation control generally means to design a controller that is capable of modifying the bifurcation characteristics of a bifurcating nonlinear system, thereby achieving some desirable dynamical behaviors. A typical objective is to delay and/or stabilize an existing bifurcation. In this paper, we consider the problem of anti-controlling bifurcations, that is, a certain bifurcation is created at a desired location with preferred properties by appropriate control. Washout-filter-aided dynamic feedback control laws are developed for the creation of Hopf bifurcations. As Hopf bifurcations give rise to limit cycles, anti-control of Hopf bifurcations suggests a new approach for designing limit cycles with specified oscillatory behaviors into a system via feedback control when such dynamical behaviors are desirable,.

Theory of center-focus for a class of higher-degree critical points and infinite points

Abstract: For the real planar autonomous differential system, the questions of detection between center and focus, successor function, formal series, central integration, integration factor, focal values, values of singular point and bifurcation of limit cycles for a class of higher-degree critical points and infinite points are expounded.

Codimension two bifurcations in a predator-prey system with group defense

Abstract: In this paper we study the qualitative behavior of a predator–prey system with nonmonotonic functional response. The system undergoes a series of bifurcations including the saddle-node bifurcation, the supercritical Hopf bifurcation, and the homoclinic bifurcation. For different parameter values the system could have a limit cycle or a homoclinic loop, or exhibit the so-called "paradox of enrichment" phenomenon. In the generic case, the model has the bifurcation of cusp-type codimension two (i.e. the Bogdanov–Takens bifurcation) but no bifurcations of codimension three.

Nonlinear oscillations in business cycle model with time lags

Abstract: In this paper we analyse the dynamics of the Kaldor–Kalecki business cycle model. This model is based on the classical Kaldor model in which capital stock changes are caused by past investment decisions. This lag is connected with time delay needed for new capital to be installed. The dynamics of the model is reduced to the form of damped oscillator with negative feedback connected with lag parameter and next it is analysed in terms of bifurcation theory. We find conditions for existence and persistence of oscillatory behaviour which is represented by limit cycle on some central manifold in phase space, i.e., single Hopf bifurcation. We demonstrate that the Hopf cycles may be exhibited for nonzero measure set of the parameter space. The conditions for bifurcation of co-dimension two connected with interaction of bifurcations as well as bifurcation diagrams are also given. Finally, we obtain numerical values describing an amplitude and a period of oscillation for different parameter of the system. It is also proved that while the investment function is not nonlinear a quasi-periodic solution (a 1:2 resonant double Hopf point) can appear. The source of such a behaviour is rather a consequence of time lag than nonlinearity of the investment function. Our results confirm the existence of asymmetric (two periodic) cycles in the Kaldor–Kalecki model with time-to-build.

Spatiotemporal dynamics near a supercritical Turing-Hopf bifurcation in a two-dimensional reaction-diffusion system.

Abstract: Pattern formation in semiconductor heterostructures is studied on the basis of a spatially two-dimensional model of reaction-diffusion type. In particular, we investigate the neighborhood of a codimension-two Turing-Hopf instability by analytical methods. Amplitude equations are derived which predict the absence of mixed modes but extended ranges of bistability between homogeneous oscillatory states and hexagonal Turing patterns. Our results are confirmed by numerical simulations. The features are not confined to a neighborhood of the bifurcation point so that the conclusions of the weakly nonlinear analysis explain the observations in large portions of the parameter space at least qualitatively

Generalized Hopf bifurcation for non-smooth planar systems

Abstract: Hopf bifurcation for smooth systems is characterized by a crossing of a pair of complex conjugate eigenvalues of the linearized problem through the imaginary axis. Since this approach is not at hand for non-smooth systems, we use the geometrical characterization given by the change from an unstable to a stable focus through a centre for a basic (piecewise) linear system. In that way we find two mechanisms for the destabilizing of the basic stationary solution and for the generation of bifurcating periodic orbits: a generation switch of the stability properties or the influence of the unstable subsystem measured by the time of duration spent in the subsystem. The switch between stable and unstable subsystems seems to be a general source of destabilization observed in several mechanical systems. We expect that the features analysed for planar systems will help us to understand higher-dimensional systems as well.

An improved version of the singularity-induced bifurcation theorem

Abstract: It has been shown recently that there is a new type of codimension one bifurcation, called the singularity-induced bifurcation (SIB), arising in parameter dependent differential-algebraic equations (DAEs) of the form x/spl dot/=f and 0=g, and which occurs generically when an equilibrium path of the DAE crosses the singular surface defined by g=0 and det g/sub y/=0. The SIB refers to a stability change of the DAE owing to some eigenvalue of a related linearization diverging to infinity when the Jacobian g/sub y/ is singular. In this article an improved version (Theorem 1.1) of the SIB theorem with its simple proof is given, based on a decomposition theorem (Theorem 2.1) of parameter dependent polynomials.

New aspects in the unfolding of the nilpotent singularity of codimension three

Abstract: This paper is concerned with three-dimensional vector fields and more specifically with the study of dynamics in unfoldings of the nilpotent singularity of codimension three. The ultimate goal is to understand the dynamics and bifurcations in the unfolding of the singularity. However, it is clear from the literature that the bifurcation diagram is very complicated and a complete study is far beyond the current possibilities, not only from a theoretical point of view but also from a numerical point of view, despite recent developments of computational methods for dynamical systems. Since all complicated dynamical behaviour is known to be of small amplitude, shrinking to the singularity for parameter values tending to the bifurcation parameter, the aim in this paper is especially to focus on a different aspect that might be interesting in the study of global bifurcation problems in the presence of such a nilpotent singularity of codimension three. The notion is introduced of 'traffic regulator' and the spec...

Bifurcation diagram for compartmentalized granular gases.

Abstract: The bifurcation diagram for a vibrofluidized granular gas in N connected compartments is constructed and discussed. At vigorous driving, the uniform distribution ~in which the gas is equi-partitioned over the compartments! is stable. But when the driving intensity is decreased this uniform distribution becomes unstable and gives way to a clustered state. For the simplest case, N52, this transition takes place via a pitchfork bifurcation but for all N.2 the transition involves saddle-node bifurcations. The associated hysteresis becomes more and more pronounced for growing N. In the bifurcation diagram, apart from the uniform and the one-peaked distributions, also a number of multipeaked solutions occur. These are transient states. Their physical relevance is discussed in the context of a stability analysis.

On Stokes sets

Abstract: The main goal of these notes is to describe the combinatorial structure of the Stokes sets for polynomials in one variable, a certain bifurcation diagram in the space of monic polynomials of given degree (the precise definition is given in section 5). As it turns out, their structure is intimately connected to other bifurcation diagrams (of quadratic differentials, or of Smale functions), and to various combinatorial structures, most prominent among them being Stasheff polyhedra. These notes are expository with proofs at best sketched. A detailed exposition will appear elsewhere.

Bifurcation from relative periodic solutions

Abstract: Relative periodic solutions are ubiquitous in dynamical systems with continuous symmetry. Recently, Sandstede, Scheel and Wulff derived a center bundle theorem, reducing local bifurcation from relative periodic solutions to a finite-dimensional problem. Independently, Lamb and Melbourne showed how to systematically study local bifurcation from isolated periodic solutions with discrete spatiotemporal symmetries. In this paper, we show how the center bundle theorem, when combined with certain group theoretic results, reduces bifurcation from relative periodic solutions to bifurcation from isolated periodic solutions. In this way, we obtain a systematic approach to the study of local bifurcation from relative periodic solutions.

Chaos and the limits of predictability for the solar-wind-driven magnetosphere-ionosphere system

Abstract: The solar-wind-driven magnetosphere–ionosphere exhibits a variety of dynamical states including low-level steady plasma convection, episodic releases of geotail stored plasma energy into the ionospheric known broadly as substorms, and states of continuous strong unloading. The WINDMI model [J. P. Smith et al., J. Geophys. Res. 105, 12 983 (2000)] is a six-dimensional substorm model that uses a set of ordinary differential equations to describe the energy flow through the solar wind–magnetosphere–ionosphere system. This model has six major energy components, with conservation of energy and charge described by the coupling coefficients. The six-dimensional model is investigated by introducing reductions to derive a new minimal three-dimensional model for deterministic chaos. The reduced model is of the class of chaotic equations studied earlier [J. C. Sprott, Am. J. Phys. 68, 758 (2000)]. The bifurcation diagram remains similar, and the limited prediction time, which is in the range of three to five hours, occurs in the chaotic regime for both models. Determining all three Lyapunov exponents for the three-dimensional model allows one to determine the dimension of the chaotic attractor for the system.

Local Bifurcation Control of a Forced Single-Degree-of-Freedom Nonlinear System: Saddle-Node Bifurcation

Abstract: It is well known that saddle-node bifurcations can occur in the steady-state response of a forced single-degree-of-freedom (SDOF) nonlinear system in the cases of primary and superharmonic resonances. This discontinuous or catastrophic bifurcation can lead to jump and hysteresis phenomena, where at a certain interval of the control parameter, two stable attractors exist with an unstable one in between. A feedback control law is designed to control the saddle-node bifurcations taking place in the resonance response, thus removing or delaying the occurrence of jump and hysteresis phenomena. The structure of candidate feedback control law is determined by analyzing the eigenvalues of the modulation equations. It is shown that three types of feedback – linear, nonlinear, and a combination of linear and nonlinear – are adequate for the bifurcation control. Finally, numerical simulations are performed to verify the effectiveness of the proposed feedback control.

Baroclinic flow and the Lorenz-84 model

Abstract: The bifurcation diagram of a truncation to six degrees of freedom of the equations for quasi-geostrophic, baroclinic flow is investigated. Period doubling cascades and Shil'nikov bifurcations lead to chaos in this model. The low dimension of the chaotic attractor suggests the possibility to reduce the model to three degrees of freedom. In a physically comprehensible limit of the parameters this reduction is done explicitly. The bifurcation diagram of the reduced model in this limit is compared to the diagram of the six degrees of freedom model and agrees well. A numerical implementation of the graph transform is used to approximate the three dimensional invariant manifold away from the limit case. If the six dimensional model is reduced to a linearisation of the invariant manifold about the Hadley state, the Lorenz-84 model is found. Its parameters can then be calculated from the physical parameters of the quasi-geostrophic model. Bifurcation diagrams at physical and traditional parameter values are compared and routes to chaos in the Lorenz-84 model are described.

Local stability, hopf and resonant codimension-two bifurcation in a harmonic oscillator with two time delays

Abstract: A harmonic oscillator with two discrete time delays is considered. The local stability of the zero solution of this equation is investigated by analyzing the corresponding transcendental characteristic equation of its linearized equation and employing the Nyquist criterion. Some general stability criteria involving the delays and the system parameters are derived. By choosing one of the delays as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. The direction and stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Resonant codimension-two bifurcation is also found to occur in this model. A complete description is given to the location of points in the parameter space at which the transcendental characteristic equation possesses two pairs of pure imaginary roots, ±iω1, ±iω2 with ω1:ω2 = m:n, where m and n are positive integers. Some numerical examples are finally given for justifying the theoretical results.