Showing papers on "Bifurcation diagram published in 1998"

Bifurcation Sets of Continuous Piecewise Linear Systems with Two Zones

Abstract: Planar continuous piecewise linear vector fields with two zones are considered. A canonical form which captures the most interesting oscillatory behavior is obtained and their bifurcation sets are drawn. Different mechanisms for the creation of periodic orbits are detected, and their main characteristics are emphasized.

Time Delay Effects on Coupled Limit Cycle Oscillators at Hopf Bifurcation

Abstract: We present a detailed study of the effect of time delay on the collective dynamics of coupled limit cycle oscillators at Hopf bifurcation. For a simple model consisting of just two oscillators with a time delayed coupling, the bifurcation diagram obtained by numerical and analytical solutions shows significant changes in the stability boundaries of the amplitude death, phase locked and incoherent regions. A novel result is the occurrence of amplitude death even in the absence of a frequency mismatch between the two oscillators. Similar results are obtained for an array of N oscillators with a delayed mean field coupling and the regions of such amplitude death in the parameter space of the coupling strength and time delay are quantified. Some general analytic results for the N tending to infinity (thermodynamic) limit are also obtained and the implications of the time delay effects for physical applications are discussed.

Additive Noise Destroys a Pitchfork Bifurcation

Abstract: In the deterministic pitchfork bifurcation the dynamical behavior of the system changes as the parameter crosses the bifurcation point. The stable fixed point loses its stability. Two new stable fixed points appear. The respective domains of attraction of those two fixed points split the state space into two macroscopically distinct regions. It is shown here that this bifurcation of the dynamical behavior disappears as soon as additive white noise of arbitrarily small intensity is incorporated the model. The dynamical behavior of the disturbed system remains the same for all parameter values. In particular, the system has a (random) global attractor, and this attractor is a one-point set for all parameter values. For any parameter value all solutions converge to each other almost surely (uniformly in bounded sets). No splitting of the state space into distinct regions occurs, not even into random ones. This holds regardless of the intensity of the disturbance.

Perturbations of quadratic centers

Abstract: We study the bifurcation of limit cycles in general quadratic perturbations of plane quadratic vector fields having a center at the origin. For any of the cases, we determine the essential perturbation and compute the corresponding bifurcation function. As an application, we find the precise location of the subset of centers in Q 3 R surrounded by period annuli of cyclicity at least three. Two specific cases are considered in more detail: the isochronous center S 1 and one of the intersection points ( Q 4 + ) of Q 4 and Q 3 R . We prove that the period annuli around S 1 and Q 4 + have cyclicity two and three respectively. The proof is based on the possibility to derive appropriate Picard-Fuchs equations satisfied by the independent integrals included in the related bifurcation function.

Grazing, skipping and sliding: Analysis of the non-smooth dynamics of the DC/DC buck converter

Abstract: This paper provides an analytical insight into the observed nonlinear behaviour of a simple widely used power electronic circuit (the buck converter) and draws parallels with a wider class of piecewise-smooth systems. After introducing the buck converter model and background, the most fascinating features of its dynamical behaviour are reviewed. So-called grazing and sliding solutions are discussed and their role in determining many of the buck converter's dynamical oddities is demonstrated. In particular, a local map is studied which explains how grazing bifurcations cause sharp turning points in the bifurcation diagram of periodic orbits. Moreover, these orbits are shown to accumulate onto a sliding trajectory through a `spiralling' impact adding scenario. The structure of such a diagram is derived analytically and is shown to be closely related to the analysis of homoclinic bifurcations. The results are shown to match perfectly with numerical simulations. The sudden jump to large-scale chaos and the fingered structure of the resulting attractor are also explained.

Bifurcation and Normal Form of Nonlinear Control Systems, Part II

Abstract: The normal forms and invariants of control systems with a parameter are found. Bifurcations of equilibrium sets are classified. The changes of properties such as controllability of the linearization or stabilizability near a bifurcation point of a control system are studied.

Bifurcation and Chaos in Engineering

Abstract: 1. Dynamic Systems, Ordinary Differential Equations and Stability of Motion 2. Calculation of Flows 3. Discrete Dynamic Systems 4. Liapunov-Schmidt Reduction 5. Center Manifold Theorem and Normal Form 6. Hopf Bifurcation 7. Averaging Method in Bifurcation Theory 8. Introduction to Chaos 9. Construction of Chaotic Regions 10. Numerical Methods 11. Nonlinear Structural Dynamics.

Experimental Study of a Symmetrical Piecewise Base-Excited Oscillator

Abstract: This paper presents an experimental study on a base-excited piecewise linear oscillator with symmetrical flexible constrains of high stiffness ratio (above 20). The details of the adopted design of the oscillator, the experimental set-up, and calibration procedure are briefly discussed. The regions of chaotic motion predicted theoretically were confirmed by the experimental results arranged into bifurcation diagram Clearance, stiffness ratio, amplitude, and frequency of the external force were used as branching parameters. The discussion of the system dynamics is based on bifurcation diagrams and Lissajous curves. The investigated system tends to be periodic for large clearances and chaotic for small ones. This picture is reversed for the amplitude of the forcing change where periodic motion occurred for small values and chaos dominated for larger forcing. The same behavior is observed for increasing frequency ratio where, for values below the natural frequency, the most interesting dynamics occurs. For the investigated parameter values, the stiffness ratio variation produces only periodic motion.

Explicit Stability Zones for Cournot Game with 3 and 4 Competitors

Abstract: The dynamical system of 3 and 4 competitors in a Cournot game is studied. The stability of its fixed points (Nash-equilibria) are also investigated. The stable and unstable regions are explicitly shown. The bifurcation characteristics are found. Periodic orbits with different periods 7, 25, 18, 13, 17 etc., are detected in both cases. The study of these models is very rich in bifurcation phenomena.

Bifurcation Theory and Methods of Dynamical Systems

Abstract: Concept of bifurcation bifurcations of 2-dimensional systems polynomial Lienard systems of dimension 2 periodic solutions of periodic perturbation systems and integral manifolds bifurcation of higher dimensional systems Melnikov vector and homoclinic and heteroclinic orbits in higher dimensional systems.

Global study of a family of cubic Liénard equations

Abstract: We derive the global bifurcation diagram of a three-parameter family of cubic Lienard systems. This family seems to have a universal character in that its bifurcation diagram (or parts of it) appears in many models from applications for which a combination of hysteretic and self-oscillatory behaviour is essential. The family emerges as a partial unfolding of a doubly degenerate Bogdanov-Takens point, that is, of the codimension-four singularity with nilpotent linear part and no quadratic terms in the normal form. We give a new presentation of a local four-parameter bifurcation diagram which is a candidate for the universal unfolding of this singularity.

Singular Hopf Bifurcation in Systems with Fast and Slow Variables

Abstract: We study a general nonlinear ODE system with fast and slow variables, i.e., some of the derivatives are multiplied by a small parameter. The system depends on an additional bifurcation parameter. We derive a normal form for this system, valid close to equilibria where certain conditions on the derivatives hold. The most important condition concerns the presence of eigenvalues with singular imaginary parts, by which we mean that their imaginary part grows without bound as the small parameter tends to zero. We give a simple criterion to test for the possible presence of equilibria satisfying this condition. Using a center manifold reduction, we show the existence of Hopf bifurcation points, originating from the interaction of fast and slow variables, and we determine their nature. We apply the theory, developed here, to two examples: an extended Bonhoeffer—van der Pol system and a predator-prey model. Our theory is in good agreement with the numerical continuation experiments we carried out for the examples.

Bifurcations of heteroclinic loops

Abstract: By generalizing the Floquet method from periodic systems to systems with exponential dichotomy, a local coordinate system is established in a neighborhood of the heteroclinic loop Γ to study the bifurcation problems of homoclinic and periodic orbits Asymptotic expressions of the bifurcation surfaces and their relative positions are given The results obtained in literature concerned with the 1-hom bifurcation surfaces are improved and extended to the nontransversal case Existence regions of the 1-per orbits bifurcated from Γ are described, and the uniqueness and incoexistence of the 1-hom and 1-per orbit and the inexistence of the 2-hom and 2-per orbit are also obtained

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

Abstract: . The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns..

Multiple Bifurcations in a Polynomial Model of Bursting Oscillations

Abstract: Bursting oscillations are commonly seen to be the primary mode of electrical behaviour in a variety of nerve and endocrine cells, and have also been observed in some biochemical and chemical systems. There are many models of bursting. This paper addresses the issue of being able to predict the type of bursting oscillation that can be produced by a model. A simplified model capable of exhibiting a wide variety of bursting oscillations is examined. By considering the codimension-2 bifurcations associated with Hopf, homoclinic, and saddle-node of periodics bifurcations, a bifurcation map in two-dimensional parameter space is created. Each region on the map is characterized by a qualitatively distinct bifurcation diagram and, hence, represents one type of bursting oscillation. The map elucidates the relationship between the various types of bursting oscillations. In addition, the map provides a different and broader view of the current classification scheme of bursting oscillations.

Problems in homoclinic bifurcation with higher dimensions

Abstract: In this paper, a suitable local coordinate system is constructed by using exponential dichotomies and generalizing the Floquet method from periodic systems to nonperiodic systems. Then the Poincare map is established to solve various problems in homoclinic bifurcations with codimension one or two. Bifurcation diagrams and bifurcation curves are given.

Analysis of Hopf and Takens–Bogdanov Bifurcations in a Modified van der Pol–Duffing Oscillator

Abstract: We analyze a modified van der Pol–Duffing electronic circuit, modeled by a tridimensional autonomous system of differential equations with Z2-symmetry. Linear codimension-one and two bifurcations of equilibria give rise to several dynamical behaviours, including periodic, homoclinic and heteroclinic orbits. The local analysis provides, in first approximation, the different bifurcation sets. These local results are used as a guide to apply the adequate numerical methods to obtain a global understanding of the bifurcation sets. The study of the normal form of the Hopf bifurcation shows the presence of cusps of saddle-node bifurcations of periodic orbits. The existence of a codimension-four Hopf bifurcation is also pointed out. In the case of the Takens–Bogdanov bifurcation, several degenerate situations of codimension-three are analyzed in both homoclinic and heteroclinic cases. The existence of a Hopf–Shil'nikov singularity is also shown.

Resonant Hopf–Hopf Interactions in Delay Differential Equations

Abstract: A second-order delay differential equation (DDE) which models certain mechanical and neuromechanical regulatory systems is analyzed. We show that there are points in parameter space for which 1:2 resonant Hopf–Hopf interaction occurs at a steady state of the system. Using a singularity theoretic classification scheme [as presented by LeBlanc (1995) and LeBlanc and Langford (1996)], we then give the bifurcation diagrams for periodic solutions in two cases: variation of the delay and variation of the feedback gain near the resonance point. In both cases, period-doubling bifurcations of periodic solutions occur, and it is argued that two tori can bifurcate from these periodic solutions near the period doubling point. These results are then compared to numerical simulations of the DDE.

Uniform approximation for period-quadrupling bifurcations

Abstract: We derive a uniform approximation for semiclassical contributions of periodic orbits to the spectral density which is valid for generic period-quadrupling bifurcations in systems with a mixed phase space. These bifurcations involve three periodic orbits which coalesce at the bifurcation. In the vicinity of the bifurcation the three orbits give a collective contribution to the spectral density while the individual contributions of Gutzwiller's type would diverge at the bifurcation. The uniform approximation is obtained by mapping the action function onto the normal form corresponding to the bifurcation. This article is a continuation of previous work in which uniform approximations for generic period-m-tupling bifurcations with were derived.

Pattern Formation in Binary Fluid Convection and in Systems with Throughflow

Abstract: Linear, weakly nonlinear, and strongly nonlinear pattern forming properties of binary fluid convection and of systems with throughflow are elucidated with a broad spectrum of numerical and analytical methods and reviewed in comparison with experimental and other theoretical work. Growth dynamics, spatiotemporal behavior, stability, scaling properties, and bifurcation behavior of spatially extended stationary, traveling wave, and standing wave convective roll structures are presented. Also pulselike spatially localized traveling wave convection is investigated. The Dufour effect in gas mixtures is reviewed. 3D square and crossroll patterns and oscillations between them which occur in liquid mixtures are analyzed. In the second part the influence of an externally imposed shear flow on structure formation is investigated. The unique selection of a self sustained forwards bifurcating pattern in the absolutely unstable parameter regime is shown to be a nonlinear eigenvalue problem. Noise sustained pattern growth in the convectively unstable regime is analyzed as a result of bulk thermal fluctuations and of perturbations that are swept by the throughflow into the system via the inlet. Finally, the influence of a lateral flow on linear and nonlinear properties of binary mixture convection is studied with special emphasis on the symmetry breaking of Hopf degenerate traveling waves and their sub critical bifurcation topology.

Multi-parameter Bifurcation Analysis of Power Systems

Abstract: This paper presents a detailed bifurcation analysis of multi-parameter power systems. Equilibrium points are used to evaluate the system eigenvalues and obtain different bifurcation diagrams for two sample systems. The paper studies the influence of hard-limits, AVR droop and reactive power compensation on the local bifurcations of the test systems, using detailed generator models. The stability regions of equilibrium points and the effect that various bifurcations have on them are also studied.

Consequences of population models for the dynamics of food chains

Abstract: A class of bioenergetic ecological models is studied for the dynamics of food chains with a nutrient at the base. A constant influx rate of the nutrient and a constant efflux rate for all trophic levels is assumed. Starting point is a simple model where prey is converted into predator with a fixed efficiency. This model is extended by the introduction of maintenance and energy reserves at all trophic levels, with two state variables for each trophic level, biomass and reserve energy. Then the dynamics of each population are described by two ordinary differential equations. For all models the bifurcation diagram for the bi-trophic food chain is simple. There are three important regions; a region where the predator goes to extinction, a region where there is a stable equilibrium and a region where a stable limit cycle exists. Bifurcation diagrams for tri-trophic food chains are more complicated. Flip bifurcation curves mark regions where complex dynamic behaviour (higher periodic limit cycles as well as chaotic attractors) can occur. We show numerically that Shil'nikov homoclinic orbits to saddle-focus equilibria exists. The codimension 1 continuations of these orbits form a `skeleton' for a cascade of flip and tangent bifurcations. The bifurcation analysis facilitates the study of the consequences of the population model for the dynamic behaviour of a food chain. Although the predicted transient dynamics of a food chain may depend sensitively on the underlying model for the populations, the global picture of the bifurcation diagram for the different models is about the same.

Synchronization of chaos and its itinerancy from a network by occasional linear connection

Abstract: This paper proposes a network of continuous-time chaotic cells and considers its dynamics. The cell includes a bipolar hysteresis whose thresholds vary periodically. The cell exhibits chaos and various stable periodic orbits. We have classified these phenomena in a bifurcation diagram and have clarified basic generation mechanism of these phenomena. The network is constructed by using the Occasional Linear Connection method that connects the cells occasionally by using a sampled state of each cell. The network exhibits various phenomena: synchronization of stable periodic orbits, synchronization of chaos, and chaotic itinerancy. We have classified these phenomena and have clarified their existence condition. These results are guaranteed theoretically and are verified in the laboratory.

Bifurcation of switched nonlinear dynamical systems

Abstract: Considers non-smooth characteristics as piecewise-defined functions split by break points. The Poincare sections are naturally defined at the break points and the Poincare mapping is constructed as a composite map of local mappings. The parameter values of local bifurcations are calculated by Newton's method using these mappings. Additionally the global bifurcations are also obtained by similar method. We propose a standard method to calculate bifurcation parameter values of these systems, and show an illustrated example. This method gives a systematic approach to qualitative analysis of switched dynamical systems.

Symbolic Computation of Local Stability and Bifurcation Surfaces for Nonlinear Time-Periodic Systems

Abstract: A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincare map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

Bifurcations associated with sub-synchronous resonance

Abstract: This paper describes a set of results of detecting nonlinear phenomena appearing in a turbine generator power system with series-capacitor compensation. The analysis was based on the Floquet theory as well as the Hopf bifurcation theorem. After the first Hopf bifurcation, the stable limit cycle bifurcates to a stable torus and an unstable limit cycle which connects to a stable limit cycle by a supercritical torus bifurcation. The stable limit cycle joins with an unstable limit cycle at a cyclic fold bifurcation. This unstable limit cycle is connected to the second Hopf. It has been also numerically demonstrated that such a strange sequence of periodic orbits is created by a q-axis damper winding.