Showing papers on "Bifurcation diagram published in 1992"
TL;DR: In this article, saddle node bifurcation is defined as a generic instability of parameterized differential equation models and its geometry and its implications for the study of voltage collapse in electric power systems are described.
Abstract: Saddle node bifurcation is a generic instability of parameterized differential equation models. The bifurcation geometry and some implications for the study of voltage collapse in electric power systems is described. The initial direction in state space of dynamic voltage collapse can be calculated from a right eigenvector of a static power system model. The normal vector to the bifurcation set in parameter space is a simple function of a left eigenvector and is expected to be useful in emergency control near bifurcation and in computing the minimum distance to bifurcation in parameter space. >
232 citations
TL;DR: In this paper, a prototype model is exploited to reveal the origin of mixedmode oscillations, where the initial oscillatory solution is born at a supercritical Hopf bifurcation and exhibits subsequent period doubling as some parameter is varied.
Abstract: A prototype model is exploited to reveal the origin of mixed‐mode oscillations The initial oscillatory solution is born at a supercritical Hopf bifurcation and exhibits subsequent period doubling as some parameter is varied This period‐2 solution subsequently loses stability, but continues to exist−regaining stability to form the 11 mixed‐mode state (one large plus one small excursion) Other mixed‐mode states lie on isolated branches or ‘‘isolas’’ of limit cycles in the one‐parameter bifurcation diagram and are separated by regions of chaos As a second parameter is varied, the number of isola solutions increases and the ‘‘gaps’’ between them become narrower, leading to correspondingly more complete Devil’s staircases An exactly comparable scenario is shown to arise in the three variable model of the Belousov–Zhabotinsky reaction proposed recently by Gyorgyi and Field [Nature 335, 808 (1992)]
213 citations
TL;DR: In this paper, the effects of symmetry breaking O(2) to D4 on an interaction between Fourier modes with wavenumbers in the ratio 1 : 2 were studied. And the effect of introducing riblets on a wall to reduce boundary layer drag was discussed.
183 citations
TL;DR: This paper presents an ANN configuration which couples a “nonlinear principal component” network for data processing with a composite ANN based on a simple integrator scheme and this ANN is able to correctly reconstruct the bifurcation diagram of the authors' experimental data.
Abstract: Artificial neural networks (ANNs) are often used for short term discrete time predictions of experimental data. In this paper we focus on the capability of such nets to correctly identify long term behavior and, in particular, observed bifurcations. As we show, the usual discrete time mapping approach is (precisely because of its discrete nature) often incapable of reproducing observed bifurcation sequences. If the interest is only in periodic or temporally more complicated behavior, a Poincare map extracted from the experimental time series can be used to circumvent this problem. A complete dynamic picture including bifurcations of steady states can, however, only be captured by a continuous-time model. We present an ANN configuration which couples a “nonlinear principal component” network for data processing (Kramer, 1991, Usui et al., 1990) with a composite ANN based on a simple integrator scheme. This ANN is able to correctly reconstruct the bifurcation diagram of our experimental data. All t...
118 citations
TL;DR: In this article, a large class of cellular automata rules and coupled map lattices of the above type in space dimensions d=2 to 6 were studied, and a new type of non-trivial collective behavior was found, at odds with the predictions of equilibrium statistical mechanics.
Abstract: Assessing the extent to which dynamical systems with many degrees of freedom can be described within a thermodynamics formalism is a problem that currently attracts much attention. In this context, synchronously updated regular lattices of identical, chaotic elements with local interactions are promising models for which statistical mechanics may be hoped to provide some insights. This article presents a large class of cellular automata rules and coupled map lattices of the above type in space dimensions d=2 to 6. Such simple models can be approached by a mean-field approximation which usually reduces the dynamics to that of a map governing the evolution of some extensive density. While this approxima tion is exact in the d== limit, where macroscopic variables must display the time· dependent behavior of the mean·field map, basic intuition from equilibrium statistical mechanics rules out any such behavior in a low·dimensional system, since it would involve the collective motion of locally disordered elements. The models studied are chosen to be as close as possible to mean-field conditions, i.e., rather high space dimension, large connectivity, and equal·weight coupling between sites. While the mean·field evolution is never observed, a new type of non·trivial collective behavior is found, at odds with the predictions of equilibrium statistical mechanics. Both in the cellular automata models and in the coupled map lattices, macroscopic variables frequently display a non·transient, time· dependent, low·dimensional dynamics emerging out of local disorder. Striking examples are period 3 cycles in two-state cellular automata and a Hopf bifurcation for a d=5 lattice of coupled logistic maps. An extensive account of the phenomenology is given, including a catalog of behaviors, classification tables for the cellular automata rules, and bifurcation diagrams for the coupled map lattices. The observed underlying' dynamics is accompanied by an intrinsic quasi·Gaussian noise (stem, ming from the local disorder) which disappears in the infinite-size limit. The collective behaviors constitute a robust phenomenon, resisting external noise, small changes in the local dynamics, and modifi,cations of the initial and boundary conditions. Synchronous updating, high space dimension and the regularity of connections are shown to be crucial ingredients in the subtle build-up of correlations giving rise to the collective motion. The discussion stresses the need for a theoretical understanding that neither equilibrium statistical mechanics nor higher-order mean·field approxima· tions are able to provide.
100 citations
TL;DR: In this article, the model most often used by ecologists to describe interactions between predator and prey populations is analyzed with reference to the case of periodically varying parameters, and a complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique.
Abstract: The model most often used by ecologists to describe interactions between predator and prey populations is analyzed in this paper with reference to the case of periodically varying parameters. A complete bifurcation diagram for periodic solutions of period one and two is obtained by means of a continuation technique. The results perfectly agree with the local theory of periodically forced Hopf bifurcation. The two classical routes to chaos, i.e., cascade of period doublings and torus destruction, are numerically detected.
94 citations
TL;DR: The concepts of absolute and convective instability are extended to nonlinear systems with broken Galilean invariance and the classical bifurcation phenomenology is shown to be nontrivially affected by the presence of a nonremovable advection term.
Abstract: The concepts of absolute and convective instability are extended to nonlinear systems with broken Galilean invariance. As an illustrative model we describe the behavior of a flow, homogeneous in a semi-infinite domain, which undergoes a subcritical pitchfork bifurcation. The classical bifurcation phenomenology is shown to be nontrivially affected by the presence of a nonremovable advection term. In particular the existence of a hysteresis loop is shown to be restricted to the nonlinear absolute instability range. A qualitative description of the possible scenarios likely to arise in subcritically bifurcating open flows is outlined and a practical test is suggested to determine the nature of the bifurcation.
84 citations
TL;DR: In this article, the high-energy global dynamics of an undamped, strongly non-linear, two-degree-of-freedom system are considered and the mode bifurcation gives rise to a homoclinic orbit in the Poincare map of the system.
Abstract: The high-energy global dynamics of an undamped, strongly non-linear, two-degree-of-freedom system are considered. As shown in an earlier work [A.F. Vakakis and R.H. Rand, Int. J. Non-Linear Mech. 27 , 861–874 (1992)], the oscillator under consideration contains “similar” non-linear normal modes and at certain values of its structural parameters a mode bifurcation is possible. For low energies, the mode bifurcation gives rise to a homoclinic orbit in the Poincare map of the system. For high energies, large- and low-scale chaotic motions are detected, resulting from transverse intersections of the stable and unstable manifolds of an unstable antisymmetric normal mode, and from the breakdown of invariant KAM-tori. The creation of additional free subharmonic motions is studied by a subharmonic Melnikov analysis, and the stability of the subharmonic motions is examined by an averaging methodology. The main conclusion of this work is that the bifurcation of similar normal modes results in a class of large-scale free chaotic motions, which do not exist in the system before the bifurcation.
77 citations
TL;DR: This paper gives a theoretical interpretation of the bifurcations of periodic or closed orbits of electrons in atoms in magnetic fields, and asserts the existence of just five typical types of bIfurcation in conservative systems with two degrees of freedom.
Abstract: Classically chaotic systems possess a proliferation of periodic orbits. This phenomenon was observed in a quantum system through measurements of the absorption spectrum of a hydrogen atom in a magnetic field. This paper gives a theoretical interpretation of the bifurcations of periodic or closed orbits of electrons in atoms in magnetic fields. We ask how new periodic orbits can be created out of existing ones or ``out of nowhere'' as the energy changes. Hamiltonian bifurcation theory provides the answer: it asserts the existence of just five typical types of bifurcation in conservative systems with two degrees of freedom. We show an example of each type. Every case we have examined falls into one of the patterns described by the theory.
67 citations
TL;DR: An alternative continuation method for tracking unstable periodic orbits by slowly varying an available system parameter, which is a predictor-corrector method for which initially the orbit is on a chaotic attractor and can be used to track the orbit through regimes not necessarily chaotic.
Abstract: We present a continuation method for experimentally tracking unstable periodic orbits by slowly varying an available system parameter in a dynamical system. The method does not depend on an explicit model, but on the signal analysis of a measured time series. Unstable periodic orbits can be tracked through various bifurcations. We apply this to a Duffing-like circuit and compare the results to an approximate model of the circuit.
67 citations
01 Jan 1992
TL;DR: In this article, a case study in the bifurcation theory of Hamiltonian systems with or without certain discrete symmetries is presented, in which structure preserving normal form or averaging techniques are used, as well as equivariant singularity theory and theory of flat perturbations.
Abstract: Generic nonlinear oscillators with parametric forcing are considered near resonance. This can be seen as a case-study in the bifurcation theory of Hamiltonian systems with or without certain discrete symmetries. In the analysis, among other things, structure preserving normal form or averaging techniques are used, as well as equivariant singularity theory and theory of flat perturbations.
TL;DR: In this paper, the S 1 -degree and bifrucation theory are applied to provide a purely topological argument of a global Hopf bifurcation theory for functional differential equations of mixed type.
TL;DR: In this paper, the dynamics of the one-dimensional mapping which models the behavior of the single-loop modulator were studied, and the self-similarity of the dynamics and the nature of chaos in the system were explained.
Abstract: Oversampled sigma-delta modulators are finding widespread use in audio and other signal processing applications, due to their simple structure and robustness to circuit imperfections. Exact analyses of the system are complicated by the presence of a discontinuous nonlinear element—a one-bit quantizer. In this paper, we study the dynamics of the one-dimensional mapping which models the behavior of the single-loop modulator. This mapping has a discontinuity at the origin and constant slope at all other points. With slope one, the dynamics in the region of interest reduce to those of the rotation of the circle. With slope less than one, almost all system inputs give rise to globally asymptotically stable periodic orbits. We emphasize the case with slope greater than one, and explain the structure of the resultant bifurcation diagram. A symbolic dynamics based study allows us to explain the self-similarity of the dynamics and the nature of chaos in the system.
TL;DR: In this article, the tridimensional pattern selection problem for reaction-diffusion systems was studied analytically and numerically, and the results were validated with the recent experimental results.
TL;DR: Three-species food-chain models, in which the prey population exhibits group defense, are considered and it is shown that the model without delay undergoes a sequence of Hopf bifurcations, showing that a region of local stability (survival) may exist even though the positive steady states are unstable.
Abstract: Three-species food-chain models, in which the prey population exhibits group defense, are considered. Using the carrying capacity of the environment as the bifurcation parameter, it is shown that the model without delay undergoes a sequence of Hopf bifurcations. In the model with delay it is shown that using a delay as a bifurcation parameter, a Hopf bifurcation can also occur in this case. These occurrences may be interpreted as showing that a region of local stability (survival) may exist even though the positive steady states are unstable. A computer code BIFDD is used to determine the stability of the bifurcation solutions of a delay model.
TL;DR: In this paper, the authors studied the bifurcation of democratically coupled logistic maps in various space dimensions beyond the accumulation point of the direct cascade of the individual maps and gave an example of quasi-periodic collective motion for d = 5.
Abstract: Bifurcation of democratically coupled logistic maps in various space dimensions are studied beyond the accumulation point of the direct cascade of the individual maps. In dimensions d = 2 and 3, only subharmonic bifurcations between periodic collective states are observed upon increasing the control parameter. The case d = 4 displays more complicated sequences with subcritical bifurcations and attractor coexistence. In dimension five or more, even more nontrivial behaviours become possible. An example of quasi-periodic collective motion is given for d = 5. The general implications of these preliminary results are discussed.
TL;DR: In this article, the Hopf bifurcation of an anisotropic system to a quasiperiodic standing wave solution has been studied in planarly aligned nematic liquid crystals.
TL;DR: In this article, the authors apply previously obtained abstract bifurcation results to nonlinear perturbations of the periodic Schrodinger equation and show that lower or upper endpoints of the continuous spectrum are bifurbation points.
TL;DR: In this article, the effect of an external periodic excitation on the Chua's piecewise-linear circuit was examined, and a bifurcation diagram that classifies the attractors in the forcing parameters plane was presented.
Abstract: The authors examine the effect of an external periodic excitation on the Chua's piecewise-linear circuit. Under the action of such a force this circuit exhibits a large variety of bifurcation sequences, including period-doubling, period-adding, and windows in the chaos regime. In addition, at certain parameters equal periodic bifurcations, hysteresis, quasi-periodic, and intermittent behavior and coexistence of multiple attractors have also been observed. A bifurcation diagram that classifies the attractors in the forcing parameters plane is presented. >
TL;DR: In this article, the authors presented the results of a refined investigation of the dynamical behaviour of Cooperrider's complex bogie, showing that one of the solution branches in [4] and [5] was one of an asymmetric, periodic oscillation -albeit with a very small offset, but it indicates, that the asymmetric oscillation is the generic mode at speeds much lower than has hitherto been found.
Abstract: SUMMARY In this paper we present the results of a refined investigation of the dynamical behaviour of Cooperrider's complex bogie. The earlier results were presented in [4] and [7]. It was discovered, that one of the solution branches in [4] and [5] was one of an asymmetric, periodic oscillation - albeit with a very small offset, but it indicates, that the asymmetric oscillation is the generic mode at speeds much lower than has hitherto been found. The bifurcation diagram has been completed, a new type of bifurcation discovered and the other asymmetric branch determined. Furthermore we discovered chaotic motion of the bogie at much lower speeds than reported in [5] and [8][, and we present the result here. Finally we present a new solution branch, which represents an unstable, symmetric oscillation. It has the interesting property, that it turns stable in a small speed range for very high speeds. It has a smaller amplitude than the coexisting chaos. Such behaviour is not uncommon in dynamical systems, see...
01 Jan 1992
TL;DR: In this paper, a general framework of a stochastic version of bifurcation theory is proposed, exemplified by one dimensional examples which are perturbed versions of deterministic differential equations exhibiting the elementary bifurlcation scenarios.
Abstract: A general framework of a stochastic version of bifurcation theory is proposed. The concepts are exemplified by one dimensional examples which are perturbed versions of deterministic differential equations exhibiting the elementary bifurcation scenarios. As explosion in finite time is possible, local stochastic dynamical systems have to be introduced.
TL;DR: In this article, the authors introduce a symbolic approach to explore the dynamical structures of three-dimensional systems with two fixed points, which is referred to as the "cocoon" bifurcation.
Abstract: We use a geometric approach to explore the dynamical structures of three-dimensional systems with two fixed points. In particular, we study the "cocoon" bifurcation, which is a series of global and local bifurcations that create an infinite number of heteroclinic orbits (which trace out a cocoon-like tangle in space). We introduce a symbolic dynamics to the intersections of a cross-sectional plane with the manifolds of the fixed points. A one-parameter system relevant to the travelling-wave solution of the Kuramoto–Sivashinsky equation is employed to illustrate this approach. The cocoon bifurcation begins with a rapidly converging sequence of heteroclinic bifurcations (principal sequence), each of which generates a pair of heteroclinic orbits and two new pieces of the two-dimensional manifolds. These bifurcations and the manifold structures are also observed with the time delay function of chaotic scattering. At the limit of the principal sequence, a saddle-node bifurcation occurs, followed by a period doubling cascade, which destroys the elliptical orbit and results in a hyperbolic invariant set. Other heteroclinic bifurcations not in the principal sequence are responsible for the folding and eventually the fractal structures of the two-dimensional manifolds. The cocoon bifurcation is completed when all intersections of the two-dimensional manifolds are transverse. The structurally stable system after the cocoon bifurcation contains a hyperbolic invariant set (a horseshoe) imbedded in two fractal sets of the two-dimensional manifolds. All three sets have the same fractal dimension. The cocoon bifurcation is a route to fully developed chaotic scattering. After a prospective effect, called manifold reconnection, the system undergoes a secondary cocoon bifurcation. Then the system becomes a hyperbolic invariant set that is characterized by a full shift of six symbols and is imbedded in two fractal sets of two-dimensional manifolds. Higher order cocoon bifurcations may become overlapped with the homoclinic bifurcation. These results are general and do not depend on the symmetry of the system. For a physical system represented by the dynamical system, it is conjectured that the structurally stable window after the cocoon bifurcation contains the physically most probable state.
TL;DR: In this article, the steady-state solutions of a reaction-diffusion model, the Selkov scheme for glycolysis, under homogeneous Dirichlet boundary conditions are fully described.
Abstract: In this paper we study the steady-state solutions of a reaction-diffusion model, the Selkov scheme for glycolysis, under homogeneous Dirichlet boundary conditions. Near to thermodynamic equilibrium, the structure and stability of solutions are fully described. A bifurcation analysis is carried out, using the size of the region in which the reaction takes place and one diffusion coefficient as main bifurcation parameters. The analysis helps us to understand the nature of the bifurcation points, and determines the shapes and stability of the bifurcating manifolds in the neighbourhood of the constant state
TL;DR: In this paper, a degenerate codimension-2 bifurcation analysis for a two-dimensional predator-prey system is presented, where the main problem is to study parameter dependent integrals which are not algebraic.
Abstract: For a two-dimensional predator-prey system, proposed by Bazykin and depending on several parameters, a complete local bifurcation analysis with respect to all parameters is achieved. The major part of the paper is devoted to the unfolding of a degenerate codimension-2 bifurcation occurring for a one-dimensional subset of parameters. The main problem here consists in studying parameter dependent integrals which are not algebraic.
TL;DR: In this article, the amplitude and phase of the bifurcating cyclic solutions of a representative non-conservative four-dimensional autonomous system with quadratic nonlinearities were studied.
Abstract: We study motions near a Hopf bifurcation of a representative nonconservative four-dimensional autonomous system with quadratic nonlinearities. Special cases of the four-dimensional system represent the envelope equations that govern the amplitudes and phases of the modes of an internally resonant structure subjected to resonant excitations. Using the method of multiple scales, we reduce the Hopf bifurcation problem to two differential equations for the amplitude and phase of the bifurcating cyclic solutions. Constant solutions of these equations provide asymptotic expansions for the frequency and amplitude of the bifurcating limit cycle. The stability of the constant solutions determines the nature of the bifurcation (i.e., subcritical or supercritical). For different choices of the control parameter, the range of validity of the analytical approximation is ascertained using numerical simulations. The perturbation analysis and discussions are also pertinent to other autonomous systems.
TL;DR: In this paper, a wide class of problems is treated: computation of invariant sets (e.g., steady states and periodic orbits), path-following (continuation) of such sets, and the related bifurcation phenomena.
Abstract: Numerical methods are often needed if bifurcation phenomena in nonlinear dynamical systems are studied. In this paper the software system CANDYS/QA for numerical qualitative analysis is presented. A wide class of problems is treated: computation of invariant sets (e.g., steady-states and periodic orbits), path-following (continuation) of such sets, and the related bifurcation phenomena. The following bifurcation situations are detected and the corresponding critical points are calculated during path-following: turning, bifurcation, Hopf bifurcation, period-doubling, torus bifurcation points (one-parameter problems) as well as cusp and Takens-Bogdanov points (two-parameter problems). A number of newly developed methods (e.g., for computation of the Poincare map) as well as algorithms from the literature are described to demonstrate the whole procedure of a qualitative analysis by numerical means. An illustrative example analyzed by CANDYS/QA is included.
TL;DR: Results of a bifurcation analysis are given for the model of a Van der Pol-Dufffing autonomous electronic oscillator, describing the unfolding of the quartic potential F=1/4X 4 -1/2αX 2 +μX, giving rise to the elementary cusp catastrophe.
Abstract: Results of a bifurcation analysis are given for the model of a Van der Pol--Duffing autonomous electronic oscillator. The oscillator is described by three ordinary differential equations and consists of a RC oscillator resistively coupled to an LC oscillator. The steady-state problem is described by the unfolding of the quartic potential F=1/4${\mathit{X}}^{4}$-1/2\ensuremath{\alpha}${\mathit{X}}^{2}$+\ensuremath{\mu}X, giving rise to the elementary cusp catastrophe. We show how the bifurcation diagram evolves with \ensuremath{\mu} and recover a ``cross-shaped diagram'' reminiscent of the one obtained by Boissonade and De Kepper for the Belousov-Zhabotinskii chemical system [J. Phys. Chem. 84, 501 (1980)]. We also show that nonzero values of \ensuremath{\mu} result in coexisting attractors with different dynamics. Specifically, we show a limit cycle attractor in one potential well coexisting with a chaotic attractor in the other well.
TL;DR: In this paper, a simple bifurcation diagram built from several such pairwise interactions of traveling waves was proposed to explain the behavior of buoyancy-driven convection in a square container of porous medium.
TL;DR: Three-dimensional convection in a rotating fluid layer is studied near the onset of a steady-state instability using equivariant bifurcation theory, and the relative stability of the primary patterns can be determined.
Abstract: Three-dimensional convection in a rotating fluid layer is studied near the onset of a steady-state instability using equivariant bifurcation theory. The pattern selection problem is formulated as a bifurcation problem on a hexagonal lattice. Symmetry considerations determine the form of the ordinary-differential equations governing the evolution of the marginally stable modes. From the symmetry analysis the relative stability of the primary patterns can be determined. The theory is illustrated explicitly for idealized boundary conditions and the bifurcation diagrams given for all values of the Taylor and Prandtl numbers.