Showing papers on "Bifurcation diagram published in 2005"
TL;DR: In this article, a simple homotopy is constructed by the modified Lindstedt-Poincare method, by the solution and the coefficient of linear term are expanded into series of the embedding parameter.
Abstract: A simple homotopy is constructed, by the modified Lindstedt-Poincare method(He,J.H. International Journal of Non-Linear Mechanics , 37, 2002, 309-314 ), the solution and the coefficient of linear term are expanded into series of the embedding parameter. Only one iteration leads to accurate solution.
907 citations
TL;DR: In this article, a new three-dimensional continuous quadratic autonomous chaotic system, modified from the Lorenz system, was reported, in which each equation contains a single quadralatic crossproduct term, which is different from the original Lorenz, Rossler, Chen, Lu systems.
Abstract: This paper reports a new three-dimensional continuous quadratic autonomous chaotic system, modified from the Lorenz system, in which each equation contains a single quadratic cross-product term, which is different from the Lorenz, Rossler, Chen, Lu systems. Basic properties of the new system are analyzed by means of Lyapunov exponent spectrum and bifurcation diagrams. Analysis results show that this system has complex dynamics with some interesting characteristics.
260 citations
TL;DR: In this article, a delay-differential equation was used to model a bidirectional associative memory (BAM) neural network with three neurons and its dynamics were studied in terms of local analysis and Hopf bifurcation analysis.
217 citations
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01 Jun 2005
TL;DR: In this article, the authors introduce the Steady State Bifurcation Theory (SSB) and introduce the reduction procedure and stability of the stability of a single state in an infinite-dimensional case.
Abstract: # Introduction to Steady State Bifurcation Theory # Introduction to Dynamic Bifurcation # Reduction Procedures and Stability # Steady State Bifurcation # Dynamic Bifurcation Theory: Finite Dimensional Case # Dynamic Bifurcation Theory: Infinite Dimensional Case # Bifurcations for Nonlinear Elliptic Equations # Reaction-Diffusion Equations # Pattern Formation and Wave Equations # Fluid Dynamics
196 citations
TL;DR: In this paper, the dynamics of one-dimensional Chapman-Jouguet detonations driven by chain-branching kinetics were studied using numerical simulations, and it was shown that the route to higher instability follows the Feigenbaum route of a period-doubling cascade.
Abstract: The dynamics of one-dimensional Chapman–Jouguet detonations driven by chain-branching kinetics is studied using numerical simulations. The chemical kinetic model is based on a two-step reaction mechanism, consisting of a thermally neutral induction step followed by a main reaction layer, both governed by Arrhenius kinetics. Results are in agreement with previous studies that detonations become unstable when the induction zone dominates over the main reaction layer. To study the nonlinear dynamics, a bifurcation diagram is constructed from the computational results. Similar to previous results obtained with a single-step Arrhenius rate law, it is shown that the route to higher instability follows the Feigenbaum route of a period-doubling cascade. The corresponding Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be close to the universal value of 4.669. The present parametric analysis determines quantitatively the relevant non-dimensional parameter χ, defined...
155 citations
TL;DR: In this article, a simple but effective iteration method is proposed to search for limit cycles or bifurcation curves of delay-differential equations, and an example is given to illustrate its convenience and effectiveness.
146 citations
TL;DR: A new four-dimensional continuous autonomous chaotic system, in which each equation in the system contains a 3-term cross product.
Abstract: This paper reports a new four-dimensional continuous autonomous chaotic system, in which each equation in the system contains a 3-term cross product. Basic properties of the system are analyzed by means of Lyapunov exponents and bifurcation diagrams.
138 citations
TL;DR: In this article, the dynamics of the harmonically excited parametric pendulum when it exhibits rotational orbits are considered and analytical and numerical analysis techniques are employed to explore the dynamic responses to different parameters and initial conditions.
Abstract: The authors consider the dynamics of the harmonically excited parametric pendulum when it exhibits rotational orbits. Assuming no damping and small angle oscillations, this system can be simplified to the Mathieu equation in which stability is important in investigating the rotational behaviour. Analytical and numerical analysis techniques are employed to explore the dynamic responses to different parameters and initial conditions. Particularly, the parameter space, bifurcation diagram, basin of attraction and time history are used to explore the stability of the rotational orbits. A series of resonance tongues are distributed along the non-dimensionalied frequency axis in the parameter space plots. Different kinds of rotations, together with oscillations and chaos, are found to be located in regions within the resonance tongues.
117 citations
TL;DR: In this paper, the elastic strain energy of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated.
Abstract: Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated. Two different non-linear strain–displacement relationships, from the Donnell's and Novozhilov's shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometric imperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclength continuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio among their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under static and dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behaviour have been observed.
110 citations
TL;DR: The general bifurcation diagram in the parameter plane of time delay tau and feedback strength K allows one to explain the phenomena that have been discovered in some previous works and has essentially a multileaf structure that constitutes multistability.
Abstract: We study the effect of time delayed feedback control in the form proposed by Pyragas on deterministic chaos in the R\"ossler system. We reveal the general bifurcation diagram in the parameter plane of time delay $\ensuremath{\tau}$ and feedback strength $K$ which allows one to explain the phenomena that have been discovered in some previous works. We show that the bifurcation diagram has essentially a multileaf structure that constitutes multistability: the larger the $\ensuremath{\tau}$, the larger the number of attractors that can coexist in the phase space. Feedback induces a large variety of regimes nonexistent in the original system, among them tori and chaotic attractors born from them. Finally, we estimate how the parameters of delayed feedback influence the periods of limit cycles in the system.
105 citations
TL;DR: In this paper, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analyzed using basic nonlinear dynamics and chaos theory.
Abstract: To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.
TL;DR: In this paper, a ring of identical elements with time delayed, nearest-neighbour coupling is considered and the individual elements are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback.
Abstract: We consider a ring of identical elements with time delayed, nearest-neighbour coupling. The individual elements are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. The bifurcation and stability of nontrivial asynchronous oscillations from the trivial solution are analysed using equivariant bifurcation theory and centre manifold construction.
TL;DR: An SIRS epidemic model, with a generalized nonlinear incidence as a function of the number of infected individuals, is developed and analyzed, and it is shown that $\R$, called the basic reproductive number, is independent of the functional form of the incidence.
Abstract: An SIRS epidemic model, with a generalized nonlinear incidence as a function of the number of infected individuals, is developed and analyzed. Extending previous work, it is assumed that the natural immunity acquired by infection is not permanent but wanes with time. The nonlinearity of the functional form of the incidence of infection, which is subject only to a few general conditions, is biologically justified. The stability analysis of the associated equilibria is carried out, and the threshold quantity ($\R$) that governs the disease dynamics is derived. It is shown that $\R$, called the basic reproductive number, is independent of the functional form of the incidence. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms are derived for the different types of bifurcation that the model undergoes, including Hopf, saddle-node, and Bogdanov--Takens. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such a...
TL;DR: In this paper, the stability and transition of symmetric flow past two circular cylinders arranged in tandem are investigated numerically and the effect of the gap spacing between the two cylinders on the stability of the flow is investigated.
Abstract: The instability and transition of flow past two circular cylinders arranged in tandem are investigated numerically A steady symmetric flow is realized at small Reynolds numbers, but the flow becomes unstable above a critical Reynolds number and makes a transition to an oscillatory flow We obtained the symmetric flow numerically and analyze its stability by applying linear stability theory The nonlinear oscillatory flow arising from the instability is obtained not only by numerical simulation but also by direct numerical calculation of the equilibrium solution, and the bifurcation diagram for the nonlinear equilibrium solution is depicted We focused our attention on the effect of the gap spacing between the two cylinders on the stability and transition of the flow The transition of the flow from a steady state to an oscillatory state is clarified to occur due to a supercritical or subcritical Hopf bifurcation depending upon the gap spacing We found that there is a certain range of the gap spacing where physical quantities such as the drag and lift coefficients and the Strouhal number show an abrupt change when the gap spacing is continuously changed We identified the origin of the abrupt change as the existence of multiple stable solutions for the flow
TL;DR: A criterion without using eigenvalues is proposed for maps of arbitrary dimension and demonstrated that the proposed criterion is preferable to the classical Hopf bifurcation criterion in theoretical analysis and practical applications.
Abstract: The classical Hopf bifurcation criterion is stated in terms of the properties of eigenvalues. In this paper, a criterion without using eigenvalues is proposed for maps of arbitrary dimension. The parameter mechanism of Hopf bifurcation may be explicitly formulated on the basis of the criterion. A numerical example demonstrates that the proposed criterion is preferable to the classical Hopf bifurcation criterion in theoretical analysis and practical applications.
TL;DR: In this paper, the torus T 1 and T 2 bifurcation of a three-degree-of-freedom vibro-impact system is considered and the period n -1 motion is determined and its Poincare map is established.
Abstract: Hopf–Hopf bifurcation of a three-degree-of-freedom vibro-impact system is considered in this paper. The period n - 1 motion is determined and its Poincare map is established. When two pairs of complex conjugate eigenvalues of the Jacobian matrix of the map at fixed point cross the unit circle simultaneously, the six-dimensional Poincare map is reduced to its four-dimensional normal form by the center manifold and the normal form methods. Two-parameter unfoldings and bifurcation diagrams near the critical point are analyzed. It is proved that there exist the torus T 1 and T 2 bifurcation under some parameter combinations. Numerical simulation results reveal that the vibro-impact system may present different types of complicated invariant tori T 1 and T 2 as two controlling parameters varying near Hopf–Hopf bifurcation points. Investigating torus bifurcation in vibro-impact system has important significance for studying global dynamical behavior and routes to chaos via quasi-period bifurcation.
TL;DR: In this article, the authors studied the hopf bifurcation of an Internet congestion control algorithm, namely, Random Exponential Marking (REM) algorithm, with communication delay.
Abstract: The purpose of this paper is to study bifurcation of an Internet congestion control algorithm, namely REM (Random Exponential Marking) algorithm, with communication delay. By choosing the delay constant as a bifurcation parameter, we prove that REM algorithm exhibits Hopf bifurcation. The formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by applying the center manifold theorem and the normal form theory. Finally, a numerical simulation is present to verify the theoretical results.
TL;DR: In this article, the authors considered the Van der Pol equation with delayed feedback and a modified version where a delayed term provided the damping, and the stability and direction of the Hopf bifurcation were determined by applying the normal form theory and the center manifold theorem.
TL;DR: Nonautonomous bifurcation theory studies the change of attractors of nonautonomous systems which are introduced here with the process formalism as well as the skew product formalism.
Abstract: Nonautonomous bifurcation theory studies the change of attractors of nonautonomous systems which are introduced here with the process formalism as well as the skew product formalism. We present a t...
TL;DR: In this paper, conditions for existence of the foci and centres are proposed and the focus-centre problem and Hopf bifurcation are considered, and appropriate examples are given to ilustrate the Hopf Bifurcation theorem.
Abstract: The objective of the paper is to obtain results on the behavior of a specific plane discontinuous dynamical system in the neighbourhood of the singular point. A new technique of investigation is presented. Conditions for existence of the foci and centres are proposed. The focus-centre problem and Hopf bifurcation are considered. Appropriate examples are given to ilustrate the bifurcation theorem.
TL;DR: In this paper, it was shown that entanglement in nonlinear bipartite systems can be associated with a fixed point bifurcation in the classical description, which corresponds to a quantum stationary state, usually a ground state.
Abstract: How do the classical dynamics of a composite system relate to the entanglement characteristics of the corresponding quantum system? We show that entanglement in nonlinear bipartite systems can be associated with a fixed point bifurcation in the classical description. In a non dissipative system a fixed point corresponds to a quantum stationary state, usually a ground state. Using the example of coupled giant spins we show that, when the fixed point undergoes a supercritical pitchfork bifurcation, the corresponding quantum state achieves a maximum amount of entanglement. By way of contrast, we consider a molecular BEC system that experiences a different kind of bifurcation and does not exhibit a peak in the entanglement corresponding to the bifurcation parameter.
TL;DR: Numerical simulation results confirm that the new feedback controller using time delay is efficient in controlling Hopf bifurcation and can be extended to study higher dimensional delay differential equations.
Abstract: In this note, we consider Hopf bifurcation control for an Internet congestion model with a single route accessed by a single source. It has been shown that the system without control cannot guarantee a stationary sending rate. As the positive gain parameter of the system passes a critical point, Hopf bifurcation occurs. To control the Hopf bifurcation, a time-delayed feedback controller using polynomial function is proposed to delay the onset of undesirable Hopf bifurcation. Numerical simulation results confirm that the new feedback controller using time delay is efficient in controlling Hopf bifurcation. This approach can be extended to study higher dimensional delay differential equations.
TL;DR: In this article, the authors studied chaotic anticontrol and chaos synchronization of brushless DC motor system and traced the parameter of the system via adaptive control and random optimization method, using numerical results such as phase diagram, bifurcation diagram, and Lyapunov exponent.
Abstract: Chaotic anticontrol and chaos synchronization of brushless DC motor system are studied in this paper. Nondimensional dynamic equations of three time scale brushless DC motor system are presented. Using numerical results, such as phase diagram, bifurcation diagram, and Lyapunov exponent, periodic and chaotic motions can be observed. Then, chaos synchronization of two identical systems via additional inputs and Lyapunov stability theory are studied. And further, the parameter of the system is traced via adaptive control and random optimization method.
TL;DR: In this article, the existence, stability properties, dynamical evolution and bifurcation diagram of localized patterns and hole solutions in one-dimensional extended systems are studied from the point of view of front interactions.
Abstract: The existence, stability properties, dynamical evolution and bifurcation diagram of localized patterns and hole solutions in one-dimensional extended systems are studied from the point of view of front interactions. An adequate envelope equation is derived from a prototype model that exhibits these particle-like solutions. This equation allows us to obtain an analytical expression for the front interaction, which is in good agreement with numerical simulations.
TL;DR: In this paper, a food web consisting of two independent preys and a predator is modeled incorporating modified Holling type-II functional response, and necessary and sufficient conditions for persistence of the food web are obtained.
TL;DR: In this article, the authors apply the multiple-scale method to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point, illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force.
Abstract: The Multiple-Scale Method is applied directly to a one-dimensional continuous model to derive the equations governing the asymptotic dynamic of the system around a bifurcation point. The theory is illustrated with reference to a specific example, namely an internally constrained planar beam, equipped with a lumped viscoelastic device and loaded by a follower force. Nonlinear, integro-differential equations of motion are derived and expanded up to cubic terms in the transversal displacements and velocities of the beam. They are put in an operator form incorporating the mechanical boundary conditions, which account for the lumped viscoelastic device; the problem is thus governed by mixed algebraic-integro-differential operators. The linear stability of the trivial equilibrium is first studied. It reveals the existence of divergence, Hopf and double-zero bifurcations. The spectral properties of the linear operator and its adjoint are studied at the bifurcation points by obtaining closed-form expressions. Notably, the system is defective at the double-zero point, thus entailing the need to find a generalized eigenvector. A multiple-scale analysis is then performed for the three bifurcations and the relevant bifurcation equations are derived directly in their normal forms. Preliminary numerical results are illustrated for the double-zero bifurcation.
TL;DR: In this paper, the authors considered planar cubic systems with a unique rest point of center-focus type and constant angular velocity and obtained an affine classification in three families and their corresponding phase portraits on the Poincare sphere.
09 May 2005
TL;DR: A framework based on a differential manifold approach that combines identification and tracing of both saddle node and Hopf bifurcation margin boundaries without calculating any eigenvalues is presented, paving the way for online voltage and oscillatory stability assessment.
Abstract: This paper presents a framework based on a differential manifold approach that combines identification and tracing of both saddle node and Hopf bifurcation margin boundaries without calculating any eigenvalues. For a given base case, we first identify either the saddle node or Hopf bifurcation. The Hopf bifurcation is easily detected by observing the sign change of scalar index in the tangent vector without eigenvalue calculation. Based on manifold and bifurcation theory, a unified formulation for a variety of bifurcation related voltage and oscillatory stability margin boundary tracing in multiparameter space is proposed. The bifurcation-related margin boundary could be traced along any control scenario in multicontrol parameter space combined with any given loading scenario. This is achieved by moving from one boundary point to the next without retracing the entire PV curve. This paves the way for online voltage and oscillatory stability assessment. The unified boundary predictor-corrector-identifier tracing framework is originally employed to trace both voltage collapse and oscillatory stability margin boundaries, which are limited by the saddle node and Hopf bifurcation, respectively. The manifold-based methodologies presented in this paper facilitate the development of fast margin monitoring and control algorithms.
TL;DR: In this article, the analysis of the dynamics of a ferroresonant circuit is presented using continuation techniques and bifurcation theory, and a detailed picture is drawn of various transitions between the individual periodic steady-state regimes of the circuit.
Abstract: In this contribution, the analysis of the dynamics of a ferroresonant circuit are presented using continuation techniques and bifurcation theory. Despite its great simplicity, this circuit can assume a diverse range of steady-state regimes including fundamental and subharmonic ferroresonance, quasiperiodic oscillations, and chaos. The system dynamics are explored through the continuation of periodic solutions of the associated circuit equations. A detailed picture is drawn of various transitions between the individual periodic steady-state regimes of the circuit. Bifurcation points are computed, revealing a clearly defined succession of periodic steady-state regimes, including the Feigenbaum route to chaos through a cascade of period doublings. The analysis presented is performed using the freely available software package XPPAUT. The contribution of this paper is to provide a detailed description of how to define the circuit equations in XPPAUT and how to conduct the interactive bifurcation analysis. The proposed approach is shown to be both computationally efficient and robust, as it eliminates the need for numerically critical and long lasting transient simulations.
TL;DR: In this paper, the authors explored the bifurcation of limit cycles for planar polynomial systems with even number of degrees and obtained the maximum number of limit cycle configurations for different sets of controlled parameters.
Abstract: This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H (6) ⩾ 35 = 6 2 − 1, where H (6) is the Hilbert number for sixth-degree polynomial systems.