Showing papers on "Bifurcation diagram published in 2000"
TL;DR: It is analytically shown that transients from a state of incoherence firing can be immediate and the stability of incoherent firing is analyzed in terms of the noise level and transmission delay, and a bifurcation diagram is derived.
Abstract: An integral equation describing the time evolution of the population activity in a homogeneous pool of spiking neurons of the integrate-and-fire type is discussed. It is analytically shown that transients from a state of incoherent firing can be immediate. The stability of incoherent firing is analyzed in terms of the noise level and transmission delay, and a bifurcation diagram is derived. The response of a population of noisy integrate-and-fire neurons to an input current of small amplitude is calculated and characterized by a linear filter L. The stability of perfectly synchronized "locked" solutions is analyzed.
470 citations
TL;DR: BIFurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input.
Abstract: Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying t...
350 citations
TL;DR: A system of delay differential equations representing a model for a pair of neurons with time-delayed connections between the neurons and time delayed feedback from each neuron to itself is studied and it is shown that the trivial fixed point may lose stability via a pitchfork bifurcation, a Hopf bifURcation, or one of three types of codimension-two bIfurcations.
Abstract: A system of delay differential equations representing a model for a pair of neurons with time-delayed connections between the neurons and time delayed feedback from each neuron to itself is studied. Conditions for the linear stability of the trivial solution of this system are represented in a parameter space consisting of the sum of the time delays between the elements and the product of the strengths of the connections between the elements. It is shown that the trivial fixed point may lose stability via a pitchfork bifurcation, a Hopf bifurcation, or one of three types of codimension-two bifurcations. Multistability near these latter bifurcations is predicted using center manifold analysis and confirmed using numerical simulations.
244 citations
TL;DR: The origin of the current reversal is identified as a bifurcation from a chaotic to a periodic regime, and trajectories revealing intermittent chaos and anomalous deterministic diffusion are observed.
Abstract: We address the problem of the classical deterministic dynamics of a particle in a periodic asymmetric potential of the ratchet type. We take into account the inertial term in order to understand the role of the chaotic dynamics in the transport properties. By a comparison between the bifurcation diagram and the current, we identify the origin of the current reversal as a bifurcation from a chaotic to a periodic regime. Close to this bifurcation, we observed trajectories revealing intermittent chaos and anomalous deterministic diffusion.
212 citations
TL;DR: In this article, an autonomous free-running Cuk converter is studied and it is shown that the system loses stability via a supercritical Hopf bifurcation, and the boundary of stability is derived and local trajectories of motion studied.
Abstract: An autonomous free-running Cuk converter is studied in this paper. Analysis of the describing nonlinear state equations shows that the system loses stability via a supercritical Hopf bifurcation. The boundary of stability is derived and local trajectories of motion studied. Cycle-by-cycle simulations of the actual system reveal the typical bifurcation from a stable equilibrium state to chaos, via limit cycles, and quasi-periodic orbits. Experimental measurements confirm the bifurcation scenarios. The occurrence of such kinds of bifurcation in autonomous dc/dc converters has been rarely known in power electronics.
135 citations
TL;DR: In this article, a delay differential equation with self-connections and two delays is considered, and conditions ensuring the stability of the periodic cycles are given in terms of local stability and bifurcation analysis.
96 citations
TL;DR: In this paper, a system of ODEs which describes the transmission dynamics of childhood diseases is considered, and a three-parameter unfolding of the normal form is studied to capture possible complex dynamics of the original system which is subjected to certain constraints on the state space due to biological considerations.
93 citations
TL;DR: In this article, the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single Hopf bifurcation oscillator were investigated and a host of complex temporal phenomena such as phase slips, frequency suppression, multiple periodic states and chaos were observed in a large number of coupled limit cycle oscillators.
89 citations
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31 Jul 2000
TL;DR: In this paper, the Liapunov-Schmidt method is used to detect and compute Bifurcation Points and Branch Switching at Simple Bifurlcation Points, and Hopf/Steady State Mode Interactions.
Abstract: 1. Reaction-Diffusion Equations.- 2. Continuation Methods.- 3. Detecting and Computing Bifurcation Points.- 4. Branch Switching at Simple Bifurcation Points.- 5. Bifurcation Problems with Symmetry.- 6. Liapunov-Schmidt Method.- 7. Center Manifold Theory.- 8. A Bifurcation Function for Homoclinic Orbits.- 9. One-Dimensional Reaction-Diffusion Equations.- 10. Reaction-Diffusion Equations on a Square.- 11. Normal Forms for Hopf Bifurcations.- 12. Steady/Steady State Mode Interactions.- 13. Hopf/Steady State Mode Interactions.- 14. Homotopy of Boundary Conditions.- 15. Bifurcations along a Homotopy of BCs.- 16. A Mode Interaction on a Homotopy of BCs.- List of Figures.- List of Tables.
79 citations
TL;DR: This work carries out a two-parameter bifurcation analysis of a model of the Colpitts oscillator and shows that the birth of the harmonic cycle is associated with a Hopf bIfurcation and the effects of idealization in the model are discussed.
Abstract: In this work we consider the Colpitts oscillator as a paradigm for sinusoidal oscillation and we investigate its nonlinear dynamics In particular, we carry out a two-parameter bifurcation analysis of a model of the oscillator This analysis is conducted by combining numerical continuation techniques and normal form theory First, we show that the birth of the harmonic cycle is associated with a Hopf bifurcation and we discuss the effects of idealization in the model Various families of limit cycles are identified and their bifurcations are analyzed in detail In particular, we demonstrate that the bifurcation diagram in the parameter space is organized by an infinite family of homoclinic bifurcations Finally, local and global coexistence phenomena are described
72 citations
TL;DR: This paper is concerned with the study of nonlinear phenomena in a closed loop voltage-controlled DC–DC Buck–Boost converter when suitable parameters are varied and it is shown that the winding number plotted as a function of the bifurcation parameter is a devil's staircase.
Abstract: This paper is concerned with the study of nonlinear phenomena in a closed loop voltage-controlled DC–DC Buck–Boost converter when suitable parameters are varied. The dynamics is analyzed using both the continuous-time model and the numerically computed stroboscopic map. The analysis of the one-dimensional bifurcation diagram shows that Neimarck–Sacker bifurcation occurs at certain values of the parameters. Phase-locking periodic windows, the period-adding sequence, and transition from quasiperiodicity to period-doubling via torus breakdown are also obtained. The two-dimensional bifurcation diagram is carefully computed. This shows that phase-locking orbits produce so-called Arnold tongues in the parameter space. It is shown that the winding number plotted as a function of the bifurcation parameter is a devil's staircase. As typically occurs in general circle maps, the fine structures of the Arnold tongues and the devil's staircase show self-similarity.
TL;DR: Global analysis of the bifurcation structure suggested that generation of these regions is associated with degenerate Hopf biforcations, and identified parameter regions in which either two stable periodic solutions with different amplitudes and periods and a stable equilibrium point or twostable periodic solutions coexist.
Abstract: The Hodgkin-Huxley equations (HH) are parameterized by a number of parameters and shows a variety of qualitatively different behaviors depending on the parameter values. We explored the dynamics of the HH for a wide range of parameter values in the multiple-parameter space, that is, we examined the global structure of bifurcations of the HH. Results are summarized in various two-parameter bifurcation diagrams with I
ext (externally applied DC current) as the abscissa and one of the other parameters as the ordinate. In each diagram, the parameter plane was divided into several regions according to the qualitative behavior of the equations. In particular, we focused on periodic solutions emerging via Hopf bifurcations and identified parameter regions in which either two stable periodic solutions with different amplitudes and periods and a stable equilibrium point or two stable periodic solutions coexist. Global analysis of the bifurcation structure suggested that generation of these regions is associated with degenerate Hopf bifurcations.
TL;DR: In this article, it was shown that in the neighbourhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area-preserving map, there is a ''twistless'' torus.
Abstract: We show that in the neighbourhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area-preserving map, there is generically a bifurcation that creates a `twistless' torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created and eventually collides with the saddle-centre bifurcation that creates the period-three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the non-degeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.
TL;DR: In this article, the authors considered discretization of parameter-dependent delay differential equations of the form x′(t) = f(x(t),x (t−τ),λ), λ∈ R, and they showed that, if the delay differential equation undergoes a Hopf bifurcation, then the discrete scheme undergoing the same type.
TL;DR: In this article, an eigenmode linear stability analysis is performed for the flow between two concentric cylinders, with the inner one rotating and in the presence of an axial, stable density stratification.
Abstract: In this article we present new experimental and theoretical results which were obtained for the flow between two concentric cylinders, with the inner one rotating and in the presence of an axial, stable density stratification. This system is characterized by two control parameters: one destabilizing, the rotation rate of the inner cylinder; and the other stabilizing, the stratification. Two oscillatory linear stability analyses assuming axisymmetric flow conditions are presented. First an eigenmode linear stability analysis is performed, using the small-gap approximation. The solutions obtained give insight into the instability mechanisms and indicate the existence of a confined internal gravity wave mode at the onset of instability. In the second stability analysis, only diffusion is neglected, predicting accurately the instability threshold as well as the critical pulsation for all the stratifications used in the experiments. Experiments show that the basic, purely azimuthal flow (circular Couette flow) is destabilized through a supercritical Hopf bifurcation to an oscillatory flow of confined internal gravity waves, in excellent agreement with the linear stability analysis. The secondary bifurcation, which takes the system to a pattern of drifting non-axisymmetric vortices, is a saddle-node bifurcation. The proposed bifurcation diagram shows a global bifurcation, and explains the discrepancies between previous experimental and numerical results. For slightly larger values of the rotation rate, weakly turbulent spectra are obtained, indicating an early appearance of weak turbulence: stationary structures and defects coexist. Moreover, in this regime, there is a large distribution of structure sizes. Visualizations of the next regime exhibit constant-wavelength structures and fluid exchange between neighbouring cells, similar to wavy vortices. Their existence is explained by a simple energy argument. The generalization of the bifurcation diagram to hydrodynamic systems with one destabilizing and one stabilizing control parameter is discussed. A qualitative argument is derived to discriminate between oscillatory and stationary onset of instability in the general case.
TL;DR: In this paper, the bifurcation diagram corresponding to stationary solutions of the nonlinear Schrodinger equation describing a superflow around a disc is numerically computed using continuation techniques.
TL;DR: An artificial neural network consisting of three neurons with nonlinear, positive and bounded response functions of the neurons is considered, which passes from stable to periodic and then chaotic regimes.
TL;DR: In this article, an amplitude equation for a reaction-diffusion system with a Hopf bifurcation coupled to one or more slow real eigenmodes was derived, which is useful even for systems where the actual bifurlcation underlying the description cannot be realized.
TL;DR: In this article, a detailed analysis of the bound-state spectrum of HOCl (hypoclorous acid) in the ground electronic state is presented, and exact quantum mechanical calculations (filter diagonalization) are performed employing an ab initio potential energy surface, which has been constructed using the multireference configuration-interaction method and a quintuple-zeta one-particle basis set.
Abstract: A detailed analysis of the bound-state spectrum of HOCl (hypoclorous acid) in the ground electronic state is presented. Exact quantum mechanical calculations (filter diagonalization) are performed employing an ab initio potential energy surface, which has been constructed using the multireference configuration-interaction method and a quintuple-zeta one-particle basis set. The wave functions of all bound states up to the HO+Cl dissociation threshold are visually inspected in order to assign the spectrum in a rigorous way and to elucidate how the spectrum develops with energy. The dominant features are (1) a 2:1 anharmonic resonance between the bending mode and the OCl stretching mode, which is gradually tuned in as the energy increases, and (2) a saddle-node bifurcation, i.e., the sudden birth of a new family of states. The bifurcation is further investigated in terms of the structure of the classical phase space (periodic orbits, continuation/bifurcation diagram). It is also discussed how the spectrum of...
TL;DR: In this article, the bifurcation behavior of an articulated loading platform subjected to harmonic excitation is investigated by the incremental harmonic balance (IHB) method, where the elements of the Jacobian matrix and the residue vector arising in the IHB formulations are derived in closed form.
TL;DR: In this article, experimental results obtained by operating the Bray−Liebhafsky (BL) reaction in the CSTR are presented, and the dynamic behavior of the reaction is examined at several operation points in the concentration phase space, by varying different parameters, including specific flow rate, temperature, and mixed inflow concentrations of the feed species, one at a time.
Abstract: Experimental results obtained by operating the Bray−Liebhafsky (BL) reaction in the CSTR are presented. The dynamic behavior of the BL reaction is examined at several operation points in the concentration phase space, by varying different parameters, the specific flow rate, temperature, and mixed inflow concentrations of the feed species, one at a time. Different types of bifurcation leading to simple periodic orbits, supercritical and subcritical Hopf bifurcations, saddle node infinite period bifurcation (SNIPER), saddle loop infinite period bifurcation, and jug handle bifurcation, are observed. Moreover, complex dynamic behavior, including transition from simple periodic oscillations to complex mixed-mode oscillations and chaos, and bistability are also discovered.
TL;DR: Numerically different cases of bistability between steady, periodic, and quasi-periodic regimes are studied and the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bIfurcation diagrams of the original laser equations and the slow time amplitude equation.
Abstract: Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities are starting from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi-periodic states. We determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double Hopf bifurcation points. We study numerically different cases of bistability between steady, periodic, and quasi-periodic regimes. Finally, the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation.
TL;DR: Guckenheimer et al. as discussed by the authors discussed the mathematical analysis of a codimension two bifurcation determined by the coincidence of a subcritical Hopf bifurbation with a homoclinic orbit of the Hopf equilibrium.
TL;DR: In this article, the authors discuss voltage collapse indices with the help of bifurcation theory and identify a well-behaved eigenvalue as a function of load increase.
TL;DR: In this article, a weakly nonlinear stability analysis is proposed to investigate the structural instability of the pitchfork bifurcation for a symmetric flow in a slightly asymmetric channel, and an amplitude equation is derived by including the effect of the imperfection of the system, and its coefficients are evaluated numerically.
Abstract: Flow in a symmetric channel with a sudden expansion makes a transition from a symmetric flow to an asymmetric one due to a symmetry-breaking pitchfork bifurcation on a gradual increase of the Reynolds number if the system is perfectly symmetric. However, an unavoidable infinitesimal imperfection of the system may render the pitchfork bifurcation imperfect. A weakly nonlinear stability analysis is proposed to investigate the structural instability of the bifurcation for such a flow. As a result, an amplitude equation for a disturbance is derived by including the effect of the imperfection of the system, and its coefficients are evaluated numerically. The equilibrium amplitude of the disturbance is calculated from the amplitude equation and compared with the experimental results for the flow in a channel that is presumed symmetric and also with the numerical solution of the full nonlinear equations for the flow in a slightly asymmetric channel.
28 May 2000
TL;DR: A one-dimensional map is derived explicitly of a system interrupted by own state and a periodic interval that has prospects of occurrence of border-collision bifurcation and the existence of regions of periodic solution within two-parameter space is shown.
Abstract: This paper considers a system interrupted by own state and a periodic interval. We know this system has prospects of occurrence of border-collision bifurcation. To analyze properties of the dynamics, we derive a one-dimensional map explicitly. We show some theorems and the existence of regions of periodic solution within two-parameter space. Some theoretical results are verified by laboratory experiments.
TL;DR: In this article, the authors presented a method for constructing a map of parameter regions with qualitatively different bifurcation diagrams for an adiabatic reverse-flow reactor (RFR), the direction of feed to which is changed periodically.
TL;DR: In this article, the vibrational dynamics of HOCl were extended to a 2D system by freezing the HO bond length to its equilibrium value, and all of the calculated bound states of the 2D HOCl system, as well as the first 40 resonances were assigned with a Fermi polyad quantum number.
Abstract: This work is aimed at extending recent studies dealing with the highly excited vibrational dynamics of HOCl [J. Chem. Phys. 111, 6807 (1999); J. Chem. Phys. 112, 77 (2000)], by taking advantage of the fact that the OH-stretch remains largely decoupled from the two other degrees of freedom up to and above the dissociation threshold. The molecule is thus reduced to a two-dimensional (2D) system by freezing the OH bond length to its equilibrium value. All of the calculated bound states of the 2D system, as well as the first 40 resonances, can be assigned with a Fermi polyad quantum number. The bifurcation diagram of the principal families of periodic orbits (POs) is extended to higher energies compared to 3D studies. In particular, the birth of “inversion” states (states exploring two equivalent wells connected through the linear HOCl configuration) is related to a period-doubling bifurcation of the families of bending POs, while “dissociation” states (states for which the energy flows back and forth along t...
TL;DR: In this paper, weakly non-linear properties and phase space analysis of a low-order model of a self-sustaining process from the Navier-Stokes equations for a sinusoidal shear flow are investigated.
Abstract: Various experiments have outlined generic properties of the subcritical transition to turbulence in plane Couette flow. A low order model of a self-sustaining process has been derived by Waleffe [11] from the Navier-Stokes equations for a sinusoidal shear flow. This paper investigates the weakly non-linear properties and the phase space analysis of this model, including the dependence on the model parameters. It is shown that the asymptotic dynamics essentially reduces to a bidimensional manifold, that many trajectories exhibit long transients, and that a statistical description of the nonlinear response to finite amplitude perturbations is needed in order to recover the bifurcation diagram from an experimental point of view. Comparison with recent experimental results obtained in the plane Couette flow finally outlines the relevance of this kind of approach.
TL;DR: In this paper, a class of cubic Hamiltonion systems with the higher-order perturbed term of degree n = 5, 7, 9, 11, 13 is investigated, and it is shown that there exist at least 13 limit cycles with the distribution C 1 9 ⊃2[C 2 3 ⊂2 C 2 2 ] (let C m k denote a nest of limit cycles which encloses m singular points, and the symbol ''⊂'' is used to show the enclosing relations between limit cycles, while the sign ''+'' is divided
Abstract: A class of cubic Hamiltonion system with the higher-order perturbed term of degree n =5, 7, 9, 11, 13 is investigated. We find that there exist at least 13 limit cycles with the distribution C 1 9 ⊃2[ C 2 3 ⊃2 C 2 2 ] (let C m k denote a nest of limit cycles which encloses m singular points, and the symbol `⊂' is used to show the enclosing relations between limit cycles, while the sign `+' is used to divide limit cycles enclosing different critical points. Denote simply C m k + C m k =2 C m k , etc.) in the Hamiltonian system under the perturbed term of degree 7, and give the complete bifurcation diagrams and classification of the phase portraits by using bifurcation theory and qualitative method and numerical simulations. These results in this paper are useful for the study of the weaken Hilbert 16th problem.