Showing papers on "Bifurcation diagram published in 1991"

Numerical analysis and control of bifurcation problems (ii): bifurcation in infinite dimensions

Abstract: A number of basic algorithms for the numerical analysis and control of bifurcation phenomena are described. The emphasis is on algorithms based on pseudoarclength continuation for ordinary differential equations. Several illustrative examples computed with the AUTO software package are included. This is Part II of the paper that appeared in the preceding issue [Doedel et al., 1991] and that mainly dealt with algebraic problems.

Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system

Abstract: A tutorial introduction in bifurcation theory is given, and the applicability of this theory to study nonlinear dynamical phenomena in a power system network is explored. The predicted behavior is verified through time simulation. Systematic application of the theory revealed the existence of stable and unstable periodic solutions as well as voltage collapse. A particular response depends on the value of the parameter under consideration. It is shown that voltage collapse is a subset of the overall bifurcation phenomena that a system may experience under the influence of system parameters. A low-dimensional center manifold reduction is applied to capture the relevant dynamics involved in the voltage collapse process. The need for the consideration of nonlinearity, especially when the system is highly stressed, is emphasized. >

Nonlinear stability and bifurcation theory

Abstract: There are now well over fifty books available on nonlinear science and chaos theory. In the past year alone, six new technical journals appeared in these areas. (Some of them may even survive). Much of the activity has been in the physics and mathematics communities. Acknowledging the latter in particulra, the authors adddress mechanicians and engineers. They hope to explain to a reader, who is assumed to possess only the minimum of mathematical background acquired by undergraduate courses, how to solve in a straightfowward manner, nonlinear stability problems. They also believe that (the problems they treat) should be understandable also for readers with little or even no knowledge in mechanics. The objects addressed are nonlinear, ordinary, and partial differential equations and iterated mappings arising as models of beams, plates, shells, linkages, railroad trucks, and the like. The book concentrates on local bifurcation and stability analysis: the problem of describing static and dynamic behaviors at parameter and state variable values near those at which loss of stability first occurs from a known branch of solutions. Global behaviors such as chaos and strange attractors are not discussed.

Patterns in the effects of infectious diseases on population growth.

Abstract: An infectious disease may reduce or even stop the exponential growth of a population. We consider two very simple models for microparasitic and macroparasitic diseases, respectively, and study how the effect depends on a contact parameter K. The results are presented as bifurcation diagrams involving several threshold values of к. The precise form of the bifurcation diagram depends critically on a second parameter ζ, measuring the influence of the disease on the fertility of the hosts. A striking outcome of the analysis is that for certain ranges of parameter values bistable behaviour occurs: either the population grows exponentially or it oscillates periodically with large amplitude.

Phase diagrams of Ising models on Husimi trees. I. Pure multisite interaction systems

Abstract: Lattice spin systems with multisite interactions have rich and interesting phase diagrams. We present some results for such systems involving Ising spins (σ=±1) using a generalization of the Bethe lattice approximation. First, we show that our approach yields good approximations for the phase diagrams of some recently studied multisite interaction systems. Second, a multisite interaction system with competing interactions is investigated and a strong connection with results from the theory of dynamical systems is made. We exhibit a full bifurcation diagram, chaos, period-3 windows, etc., for the magnetization of the base site of this system.

Hopf bifurcation on a square lattice

Abstract: A complete classification of the generic D4*T2-equivariant Hopf bifurcation problems is presented. This bifurcation arises naturally in the study of extended systems, invariant under the Euclidean group E(2), when a spatially uniform quiescent state loses stability to waves of wavenumber k not=0 and frequency omega not=0. The D4*T2 symmetry group applies when periodic boundary conditions are imposed in two orthogonal horizontal directions. The centre manifold theorem allows a reduction of the infinite dimensional problem to a bifurcation problem on C4. In normal form, the vector field on C4 commutes with an S1 symmetry, which is interpreted as a time translation symmetry. The spatial and spatio-temporal symmetries of all possible solutions are classified in terms of isotropy subgroups of D4*T2*S1.

"crossroad area–spring area" transition (i) parameter plane representation

Abstract: This paper is devoted to the bifurcation structure of a parameter plane related to one- and two-dimensional maps. Crossroad area and spring area correspond to a characteristic organization of fold and flip bifurcation curves of the parameter plane, involving the existence of cusp points (fold codimension-two bifurcation) and flip codimension-two bifurcation points. A transition "mechanism" (among others) from one area type to another one is given from a typical one-dimensional map.

Small-amplitude periodic and chaotic solutions of the complex Ginzburg-Landau equation for a subcritical bifurcation.

Abstract: We preent numerically obtained bounded solutions of the one-dimensional complex Ginzburg-Landau equation with a destabilizing cubic term and no stabilizing higher-order contributions. The boundedness results from competition between dispersion and nonlinear frequency renormalization. We find chaotic and also stationary and time-periodic states with spatial structure corresponding to a periodic array of pulses. An analytical description is presented. Possibly experimental results connected with the dispersive chaos found in binary-fluid mixtures can be explained.

Regulation of relaxed static stability aircraft

Abstract: The authors formulate and solve a regulator problem for nonlinear parameter-dependent dynamics. It is shown that the problem is solvable except at parameter values associated with bifurcation of the equilibrium equations and that such bifurcations are inherently linked to the system zero dynamics. These results are applied to the study of the regulation of the longitudinal dynamics of aircraft. It is shown how bifurcation points arise in these problems and why they affect solvability of the regulator problem. The relationships between bifurcation, system zeros, and dynamic and static stability are illustrated. >

Global bifurcation analysis of the double scroll circuit

Abstract: An in-depth analysis is made of the global 2-parameter bifurcation structures of the double scroll circuit in terms of their homoclinic, heteroclinic, and periodic orbits. Many fine details are uncovered via a 3-dimensional "unfolding" of the 2-parameter bifurcation structures. Major findings are: (i) The parameter sets which give rise to the homoclinic and heteroclinic orbits (homoclinic and heteroclinic bifurcation sets) studied in this paper are found to be all connected to each other via only one family of periodic orbits. (ii) Moreover, the structure of the windows of this family essentially determines the global structure of the periodic windows of the double scroll circuit. These bifurcation analyses are accomplished by deriving first the relevant bifurcation equations in exact analytic form and then solving these nonlinear equations by iterations. No numerical integration formula for differential equations are used.

Analysis and computation of symmetry-breaking bifurcation and scaling laws using group theoretic methods

Abstract: Group theoretic methods are used to analyse symmetry-breaking bifurcation for nonlinear equations defined on a real Hilbert space. An important result is the decomposition of the Hilbert space into orthogonal isotypic components, since the Jacobian of the nonlinear operator can be decomposed on the isotypic components. This decomposition is exploited in the detection and computation of bifurcation points. Then scaling laws that arise in many problems are considered, and a natural context is developed for the existence of a scaling law based on the symmetry of the problem. The effect of the scaling law on the bifurcation theory is explored. This theory is applied to the gravity wave problem. Also shown is the way in which the theory can extend to boundary value problems, where the natural group equivariance of the equations is destroyed by the boundary conditions.

Indeterminate jumps to resonance from a tangled saddle-node bifurcation

Abstract: We identify in this paper a new type of bifurcation which yields an unpredictable outcome and therefore seems to be an important new ingredient of nonlinear dissipative dynamics. It arises when the unstable manifold of the saddle of a saddle-node bifurcation is heteroclinically tangled with the inset of a distant saddle which is itself homoclinically tangled so that it forms a fractal basin boundary between two remote attractors. At the saddle-node fold, a slowly evolving system will thus find itself sitting precisely on a fractal basin boundary, and in the presence of even infinitesimal noise we cannot predict to which of the two remote attractors the system will jump. We show here that such an indeterminate tangled saddle-node bifurcation is a common ingredient in the resonance of softening systems.

"Period three to period two" bifurcation for piecewise linear models

Abstract: In Hommes, Nusse, and Simonovits (1990) the dynamics of a simple economic model was studied. Although this piecewise linear model is quite simple, its dynamics shows different kinds of behavior such as periodic, quasiperiodic, and chaotic behavior. In particular, a new kind of bifurcation, namely a period three to period two bifurcation, was observed numerically. This paper deals with this new bifurcation phenomenon and we show that the “period three to period two” bifurcation occurs and is a structurally stable phenomenon in a class of two-dimensional continuous, piecewise linear systems. In particular, the “period three to period two” bifurcation is a structurally stable phenomenon in economic models with Hicksian nonlinearities.

Period doubling with higher-order degeneracies

Abstract: A family of local difleomorphisms of ${\bf R}^n$ can undergo a period doubling (flip) bifurcation as an eigenvalue of a fixed point passes through $ - 1$. This bifurcation is either supercritical or subcritical, depending on the sign of a coefficient determined by higher-order terms. If this coefficient is zero, the resulting bifurcation is “degenerate.” The period doubling bifurcation with a single higher-order degeneracy is treated, as well as the more general degenerate period doubling bifurcation where a fixed point has $ - 1$ eigenvalue and any number of higher-order degeneracies. The main procedure is a Lyapunov–Schmidt reduction: period-2 orbits are shown to be in one-to-one correspondence with roots of the reduced “bifurcation function,“ which has ${\bf Z}^2$ symmetry. Illustrative examples of the occurrence of the singly degenerate period doubling in the context of periodically forced planar oscillators are also presented.

Bifurcation and chaos of the sinusoidally-driven Chua's circuit

Abstract: The effect of an external sinusoidal signal on the familiar Chua's piecewise-linear autonomous circuit is experimentally investigated. Under the influence of such a periodic excitation an immense variety of bifurcation phenomena like period doubling, period adding, quasiperiodicity, and intermittency has been observed. Also, equal periodic bifurcation, hysteresis, and coexistence of multiple attractors are reported. A two-parameter (drive amplitude–drive frequency) bifurcation diagram is constructed, which classifies the observed rich bifurcation sequences and chaotic structures.

The bifurcations of countable connections from a twisted heteroclinic loop

Abstract: Codimension-two bifurcation phenomena associated with nondegenerate heteroclinic loops are studied. The bifurcation curves of homoclinic orbits in the parameter space are characterized by the twist structure of the heteroclinic loops at the bifurcation points. Among other things, it is shown that heteroclinic orbits with any given winding number around a doubly twisted heteroclinic loop must bifurcate. Applications of these bifurcation phenomena are also discussed.

Bifurcation and Chaos: Analysis, Algorithms, Applications

Abstract: A Complete Bifurcation Scenario for the 2-d Nonlinear Laplacian with Neumann Boundary Conditions on the Unit Square.- The Effect of Fluctuations on the Transition Behavior of a Nonlinear Chemical Oscillator.- Examples of Boundary Crisis Phenomenon in Structural Dynamics.- Bifurcation, Pattern Formation and Transition to Chaos in Combustion.- On the Primary and Secondary Bifurcation of Equations Involving Scalar Nonlinearities.- Periodic Solutions Leading to Chaos in an Oscillator with Quadratic and Cubic Nonlinearities.- Turing Structures in Anisotropic Media.- Regular and Chaotic Patterns of Rayleigh-Benard Convection.- Bifurcations in Slowly Rotating Systems with Spherical Geometry.- An Elastic Model with Continuous Spectrum.- Mechanistic Requirements for Chemical Oscillations.- Envelope Soliton Chans Model for Mechanical System.- Rolling Motion of Ships Treated as Bifurcation Problem.- Normal Forms for Planar Systems with Nilpotent Linear Part.- Two Methods for the Numerical Detection of Hopf Bifurcations.- Automatic Evaluation of First and Higher-Derivative Vectors.- On the Stability of a Spinning Satellite in a Central Force Field.- Codimension Two Bifurcation in an Approximate Model for Delayed Robot Control.- Lacunary Bifurcation of Multiple Solutions of Nonlinear Eigenvalue Problems.- Branches of Stationary Solutions for Parameter-dependent Reaction-Diffusion Systems from Climate Modeling.- A Note on the Detection of Chaos in Medium Sized Time Series.- An Approach for the Analysis of Spatially Localized Oscillations.- On the Application of Invariant Manifold Theory, in particular to Numerical Analysis.- Combined Analytical-Numerical Analysis of Nonlinear Dynamical Systems.- Monotony Methods and Minimal and Maximal Solutions for Nonlinear Ordinary Differential Equations.- Interior Crisis in an Electrochemical System.- Multiple Bifurcation of Free-Convection Flow Between Vertical Parallel Plates.- Description of Chaotic Motion by an Invariant Distribution at the Example of the Driven Duffing Oscillator.- Augmented Systems for Generalized Turning Points.- Numerical Analysis of the Orientability of Homoclinic Trajectories.- Qualitative and Quantitative Behaviour of Nonlinearly Elastic Rings under Hydrostatic Pressure.- Computation of Basins of Attraction for Three Coexisting Attractors.- Controllability of Lorenz Equation.- Spatially Periodic Forcing of Spatially Periodic Oscillators.- Solution Branches at Corank-Two Bifurcation Points with Symmetry.- Two-dimensional Maps Modelling Periodically Driven Strictly Dissipative Oscillators.- On Computing Coupled Turning Points of Parameter-Dependent Nonlinear Equations.- Generating Hopf Bifurcation Formulae with MAPLE.- On a Codimension Three Bifurcation Arising in an Autonomous Electronic Circuit.- Efficient Parallel Computation of Periodic Solutions of Parabolic Partial Differential Equations.- Comparison of Bifurcation Sets of Driven Strictly Dissipative Oscillators.- Echo Waves in Reaction-Diffusion Excitable Systems.- The Local Stability of Inactive Modes in Chaotic Multi-Degree-of-Freedom Systems.- Bifurcations in Dynamic Systems with Dry Friction.- The Approximate Analytical Methods in the Study of Bifurcations and Chaos in Nonlinear Oscillators.- Periodic and Homoclinic Orbits in Conservative and Reversible Systems.- On the Dynamics of a Horizontal, Rotating, Curved Shaft.- Lyapunov Exponents and Invariant Measures of Dynamic Systems.- Computation of Hopf Branches Bifurcating from Takens-Bogdanov Points for Problems with Symmetries.

Bifurcation analysis of chemical reaction mechanisms. I, Steady state bifurcation structure

Abstract: The vocabulary and techniques of numerical bifurcation analysis are described, with an emphasis on steady state bifurcations of codimension one and two. The direct computation of bifurcation sets is shown to be of considerable utility in analyzing and comparing complex chemical reaction mechanisms. The systems chosen for analysis are the chlorite–iodide and the mixed Landolt reactions. The calculation of a simple hysteresis loop for a mechanism of the chlorite–iodide reaction using both numerical bifurcation analysis and numerical integration begins an extended comparison between the methods advocated in this paper and more familiar methods. The systematic identification of the existence of isolated branches of steady states is described for a second mechanism of the chlorite–iodide reaction. Two mechanisms for the mixed Landolt system are contrasted. It is found that the alternative negative feedback pathway mechanism, which reproduces the periodic behavior more successfully at a selected point in parame...