Showing papers on "Bifurcation diagram published in 1999"
TL;DR: In this article, a simple neural network model with two delays is considered, and the Hopf bifurcation occurs when the sum of the two delays varies and passes a sequence of critical values.
361 citations
TL;DR: This work investigates the dynamical behaviour of a simple plankton population model, which explicitly simulates the concentrations of nutrient, phytoplankton and zooplankon in the oceanic mixed layer, and uses a linear function, to represent alternative ecological assumptions.
228 citations
TL;DR: It is shown that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator-prey models.
Abstract: We describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator-prey model proposed by R. M. May [ Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, NJ, 1974]. In particular, we give a qualitative description of the bifurcation curve when two limit cycles collapse on a semistable limit cycle and disappear. Moreover, we show that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator-prey models.
179 citations
TL;DR: In this article, the authors focus on the basic mechanism causing self-replicating patterns from a global bifurcational point of view and take their clues from two model systems including the Gray-Scott (GS) model.
174 citations
TL;DR: In this paper, the effect of time delay on the collective dynamics of coupled limit cycle oscillators at Hopf bifurcation was studied and the results showed significant changes in the stability boundaries of amplitude death, phase locked and incoherent regions.
165 citations
TL;DR: In this article, the degenerate solutions of the nonlinear equation F ( e, λ, λ, u ) = 0 were studied, and several bifurcation theorems were obtained for degenerate solution.
138 citations
TL;DR: In this paper, a method to trace bifurcation sets for a piecewise-defined differential equation is proposed, where the trajectory is continuous, but it is not differentiable at break points of the characteristics.
Abstract: This paper proposes a method to trace bifurcation sets for a piecewise-defined differential equation. In this system, the trajectory is continuous, but it is not differentiable at break points of the characteristics. We define the Poincare mapping by suitable local sections and local mappings, and thereby it is possible to calculate bifurcation parameter values. As an illustrated example, we analyze the behavior of a two-dimensional nonlinear autonomous system whose state space is constrained on two half planes concerned with state dependent switching characteristics. From investigation of bifurcation diagrams, we conclude that the tangent and global bifurcations play an important role for generating various periodic solutions and chaos. Some theoretical results are confirmed by laboratory experiments.
96 citations
TL;DR: In this paper, the Yamada model for self-pulsations in (semiconductor) lasers with saturable absorber is considered and a complete picture of all possible dynamics is presented in terms of two-dimensional bifurcation diagrams, in which they find a Bogdanov-takens bifurycation as an organizing center.
94 citations
TL;DR: In this paper, a class of differential-difference equations with two delays is studied, and the stability of the bifurcating periodic solutions are determined by using the center manifold theorem and the normal form theory.
78 citations
TL;DR: In this paper, the authors show that the rapid bifurcation described by Kriegsmann [1987] is a generic bifurbation for planar symmetric piecewise-linear systems, which is responsible for the abrupt appearance of stable periodic oscillations.
Abstract: The rapid bifurcation described by Kriegsmann [1987] is shown to be a generic bifurcation for planar symmetric piecewise-linear systems. The bifurcation can be responsible for the abrupt appearance of stable periodic oscillations. Although it has some similarities with the Hopf bifurcation for smooth systems, since the stability change of an equilibrium involves the appearance of one limit cycle, the dependence of the limit cycle amplitude on the bifurcation parameter is different from the Hopf's case. To characterize this bifurcation, accurate estimates for the amplitude and period of the bifurcating limit cycle are given. The analysis is just illustrated with the application of the theoretical results to the Wien bridge oscillator. Comparisons with experimental data and Kriegsmann's analysis are also included.
64 citations
TL;DR: In this paper, a combination of bifurcation theory for two-dimensional dynamical systems and numerical simulations is used to systematically determine the possible flow topologies of the steady vortex breakdown in axisymmetric flow in a cylindrical container with rotating end-covers.
Abstract: Using a combination of bifurcation theory for two-dimensional dynamical systems and numerical simulations, we systematically determine the possible flow topologies of the steady vortex breakdown in axisymmetric flow in a cylindrical container with rotating end-covers. For fixed values of the ratio of the angular velocities of the covers in the range from −0.02 to 0.05, bifurcations of recirculating bubbles under variation of the aspect ratio of the cylinder and the Reynolds number are found. Bifurcation curves are determined by a simple fitting procedure of the data from the simulations. For the much studied case of zero rotation ratio (one fixed cover) a complete bifurcation diagram is constructed. Very good agreement with experimental results is obtained, and hitherto unresolved details are determined in the parameter region where up to three bubbles exist. For non-zero rotation ratios the bifurcation diagrams are found to change dramatically and give rise to other types of bifurcations.
01 Nov 1999
TL;DR: It is found that a switch from stability to instability may occur for certain range of system parameters and must then be followed by a switch back to stability, and it is proved that Hopf bifurcation occurs, i.e., a family of periodic solutions b ifurcates from the equilibrium when the bIfurcation parameter passes through a critical value.
Abstract: A continuously delayed neural network with strong kernel is investigated. We found that a switch from stability to instability may occur for certain range of system parameters and must then be followed by a switch back to stability. We also investigate bifurcation phenomena of this model. Using the mean time delay as a bifurcation parameter, we prove that Hopf bifurcation occurs, i.e., a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter passes through a critical value. Stability criteria for the bifurcating periodic solutions are obtained. Some computer simulations illustrate correctness of the results.
01 Jan 1999
TL;DR: In this paper, the exactness of S-shaped bifurcation curves has been proved for a class of two-point boundary value problems from combustion theory, which involves nonlinearities like e > 0.
Abstract: For a class of two-point boundary value problems we prove exactness of S-shaped bifurcation curve. Our result applies to a problem from combustion theory, which involves nonlinearities like e for a > 0.
TL;DR: This paper proposes a simple physical model using the modified van der Pol equation, which unfolds the generic synchronization behaviors of the latter kind and in which one may readily observe changes in the sychronization behaviors between the distinctive regimes as well.
Abstract: It has been known that a diffusive coupling between two limit cycle oscillations typically leads to the in-phase synchronization and also that it is the only stable state in the weak-coupling limit. Recently, however, it has been shown that the coupling of the same nature can result in the distinctive dephased synchronization when the limit cycles are close to the homoclinic bifurcation, which often occurs especially for the neuronal oscillators. In this paper we propose a simple physical model using the modified van der Pol equation, which unfolds the generic synchronization behaviors of the latter kind and in which one may readily observe changes in the sychronization behaviors between the distinctive regimes as well. The dephasing mechanism is analyzed both qualitatively and quantitatively in the weak-coupling limit. A general form of coupling is introduced and the synchronization behaviors over a wide range of the coupling parameters are explored to construct the phase diagram using the bifurcation analysis.
TL;DR: In this article, the accepted manuscript of an article for which the publisher's version is available electronically from http://dx.doi.org/10.2202/1558-3708.1047
Abstract: This is the author's accepted manuscript of an article for which the publisher's version is available electronically from http://dx.doi.org/10.2202/1558-3708.1047
TL;DR: In this article, it is shown that multiple limit cycles, hysteresis loops and catastrophic transitions may possibly accompany a Hopf bifurcation, and the theoretical argument is illustrated in Foley's liquidity cost-business cycle model.
TL;DR: In this paper, an iterative mapping that describes the nonlinear system dynamics in the continuous conduction mode is derived, and it shows that different bifurcation diagrams can be obtained from different system parameters, and that the systems generally exhibit a period-doubling route to chaos.
Abstract: Remarkably complex behaviour, namely chaos, in voltage-mode controlled DC drive systems has been investigated. An iterative mapping that describes the nonlinear system dynamics in the continuous conduction mode is derived. It shows that different bifurcation diagrams can be obtained from different system parameters, and that the systems generally exhibit a period-doubling route to chaos. Analytical modelling of period-1 and hence the period-p orbits, as well as their stability analysis using the characteristic multipliers, is presented. Thus the stable ranges of various system parameters can be determined. The theoretical results are verified by using experimental measurement.
TL;DR: In this paper, the authors studied global bifurcation phenomena for a class of nonlinear elliptic equations governed by the $h$-Hessian operator and provided new methods for establishing existence results concerning fully nonlinear Elliptic equations.
Abstract: In this paper we study global bifurcation phenomena for a class of nonlinear elliptic equations governed by the $h$-Hessian operator. The bifurcation phenomena considered provide new methods for establishing existence results concerning fully nonlinear elliptic equations. Applications to the theory of critical exponents and the geometry of $k$-convex functions are considered.
In addition, a related problem of Liouville-Gelfand type is analyzed.
TL;DR: In this article, the Hopf bifurcation and stability of nonlinear delay Lienard equations with two parameters were investigated by constructing the center manifold and using the normal form method.
Abstract: The linearized stability and Hopf bifurcation of the delay Lienard equations are studied. The linearized asymptotic stability for the null solution of the equations with two parameters is analyzed. The Hopf bifurcation and the stability of periodic solutions are investigated by constructing the center manifold and using the normal form method. Several examples for typical nonlinear delay Lienard equations are given to show the coincidence between the theoretical analysis and the numerical results.
TL;DR: In this paper, an approach for finding ferroresonance occurrence domains has been developed, where the steady state solutions are calculated by the Galerkin method and a bifurcation diagram is created using a continuation method.
TL;DR: In this article, the bifurcation structure of the two-dimensional pressure-driven flow through a curved rotating duct is studied, where the problem depends on the Reynolds number, Re=Ub/ν, the Rotation number, RΩ=bΩ/U, the aspect ratio, γ=b/h, and the radius ratio (or curvature ratio), η=ri/ro; here U is the velocity scale, b is the duct width in the spanwise direction, (ri,ro) are the inner and outer radii
Abstract: The bifurcation structure of the two-dimensional pressure-driven flow through a curved rotating duct is studied. In this study we add to the rich literature that already exists on this problem [J. Fluid Mech. 262, 353 (1994)], revealing even more intricate details of the solution structure. The problem depends on the Reynolds number, Re=Ub/ν, the Rotation number, RΩ=bΩ/U, the aspect ratio, γ=b/h, and the radius ratio (or curvature ratio), η=ri/ro; here U is the velocity scale, b is the duct width in the spanwise direction, Ω is the rotational speed, (ri,ro) are the inner and outer radii of the duct, h=ro−ri and ν is the kinematic viscosity of the fluid. For a curvature ratio η=0.960 784, continuation on Re is used to trace the bifurcation diagram for zero rotation (RΩ=0). Then, for Re=800, continuation on RΩ is used from the solutions at zero rotation (RΩ=0) to generate bifurcation diagrams for positive and negative rotational number for the purpose of studying the effect of rotation. Extended systems are...
TL;DR: In this paper, a complete bifurcation analysis for systems of two quadratic conservation laws with constant, strictly parabolic viscosity matrices was carried out by reducing to a canonical form introduced by Fiddelaers, and all such systems, except possibly those on a codimension-one variety in parameter space, give rise to nonclassical shock waves.
Abstract: We determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles, in regions of strict hyperbolicity. Whereas classical shock waves in systems of conservation laws involve a single characteristic field, nonclassical waves involve two fields in an essential way. This feature is reflected in the viscous profile differential equation, which undergoes codimension-three bifurcation of the kind studied by Dumortier et al., as opposed to the codimension-one bifurcation occurring in the classical case. We carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant, strictly parabolic viscosity matrices by reducing to a canonical form introduced by Fiddelaers. We show that all such systems, except possibly those on a codimension-one variety in parameter space, give rise to nonclassical shock waves, and we classify the number and types of their bifurcation points.
One consequence of our analysis is that weak transitional waves arise in pairs, with profiles forming a 2-cycle configuration previously shown to lead to nonuniqueness of Riemann solutions and to nontrivial asymptotic dynamics of the conservation laws. Another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws, rather than with change of type in the associated hyperbolic system.
TL;DR: In this paper, the transfer resistance of an electrocatalytic mechanism with two Hopf bifurcations was studied and it was shown that transfer resistance is always positive even when the steady-state point corresponds to a Hopf Bifurcation.
TL;DR: In this paper, the bifurcation analysis of an ethylene glycol column is studied, where the main objective is to describe the partition diagram with respect to the reboiler boilup ratio.
Abstract: An ethylene glycol column is studied in this paper, where the main objective is to describe the bifurcation diagram with respect to the reboiler boilup ratio. The bifurcation analysis reveals the existence of a unique equilibrium point at low reboiler heat inputs and three equilibrium points (two stable and one unstable) at high reboiler heat inputs. Moreover, the existence of input multiplicities at moderate and high product purity is also revealed.
TL;DR: In this article, the main aspects of an algorithm for the exact evaluation of single bifurcation points along the nonlinear equilibrium trajectory of flexible mechanisms with large 3D rotations are discussed.
13 Jan 1999
TL;DR: In this paper, the authors revisited the dynamical behavior of SISO systems and provided a control theoretic interpretation of the adaptive control question and solution approach, for which they showed a rich variety of potential asymptotic dynamics.
Abstract: It is well known that linear SISO systems that can be rendered passive through constant output feedback can be adaptively stabilized through a single gain adaptation law. We revisit the dynamical behavior of such systems and exhibit through a bifurcation analysis a rich variety of potential asymptotic dynamics, for which we provide a control theoretic interpretation. This in turn leads us to question the actual adaptive control question and solution approach.
TL;DR: In this paper, the authors describe a new algorithm for the calculation of Liapunov quantities for two-dimensional systems in terms of the complex form of the system, which is more transparent conceptually and more efficient computationally.
Abstract: We describe a new algorithm for the calculation of Liapunov quantities for two-dimensional systems in terms of the complex form of the system. It is more transparent conceptually and more efficient computationally. We demonstrate the advantages of this approach by deriving the integrability conditions and information about the bifurcation of limit cycles for some cubic systems.
TL;DR: In this article, an ordinary differential equation controlled by a stochastic process is studied and the existence and uniqueness of solutions on associated local times (Trotter and Ray-Knight theorems) and on time and direction of bifurcation.
Abstract: We study an ordinary differential equation controlled by a stochastic process. We present results on existence and uniqueness of solutions, on associated local times (Trotter and Ray-Knight theorems) and on time and direction of bifurcation. A relationship with Lipschitz approximations to Brownian paths is also discussed.
TL;DR: In this paper, the existence of a sequence of time delays 0 t t???, with t a` as 0 1 k k a`, such that a local Hopf bifurcation occurs at t s t, provided k < ǫ < w x w x.