Showing papers on "Infinite-period bifurcation published in 1986"
TL;DR: In this article, the singular Hopf bifurcation from a basic steady state to these relaxation oscillations was studied. But the authors focused on the symmetry of the harmonic oscillations near the Hopf point and did not consider the reverse transition from stable steady states to periodic solutions.
Abstract: Relaxation oscillations characterized by two quite different time scales are described by mathematical models of the form $x_t = f(x,y,\lambda ,\varepsilon )$ and $y_t = \varepsilon g(x,y,\lambda ,\varepsilon )$ where $\varepsilon \ll 1$ and $\lambda $ is the control parameter. In this paper, we study the singular Hopf bifurcation from a basic steady state to these relaxation oscillations. Our bifurcation analysis shows how the harmonic oscillations near the bifurcation point progressively change to become pulsed, triangular oscillations.In the second part of the paper, we present a numerical study of the FitzHugh–Nagumo equations for nerve conduction. We first observe that the numerical results are in good agreement with the analytical predictions. We then consider the switching from a stable steady state to a stable periodic solution, or the reverse transition. Our purpose is to explain the annihilation experiments described in the nerve conduction literature.
180 citations
TL;DR: The perturbation is shown to always suppress the bifurcation, shifting the bIfurcation point and stabilizing the behavior at the original bifURcation point, resulting in a closely spaced set of peaks in the response spectrum.
Abstract: We consider the effect on a generic period-doubling bifurcation of a periodic perturbation, whose frequency ${\ensuremath{\omega}}_{1}$ is near the period-doubled frequency ${\ensuremath{\omega}}_{0}$/2. The perturbation is shown to always suppress the bifurcation, shifting the bifurcation point and stabilizing the behavior at the original bifurcation point. We derive an equation characterizing the response of the system to the perturbation, analysis of which reveals many interesting features of the perturbed bifurcation, including (1) the scaling law relating the shift of the bifurcation point and the amplitude of the perturbation, (2) the characteristics of the system's response as a function of bifurcation parameter, (3) parametric amplification of the perturbation signal including nonlinear effects such as gain saturation and a discontinuity in the response at a critical perturbation amplitude, (4) the effect of the detuning (${\ensuremath{\omega}}_{1}$-${\ensuremath{\omega}}_{0}$/2) on the bifurcation, and (5) the emergence of a closely spaced set of peaks in the response spectrum. An important application is the use of period-doubling systems as small-signal amplifiers, e.g., the superconducting Josephson parametric amplifier.
58 citations
41 citations
TL;DR: It is demonstrated that a simple revised Oregonator type and bromine-hydrolysis-controlled model cannot describe the SNIPER bifurcation behavior of the system in a CSTR.
Abstract: This paper presents the bifurcation diagram of one of the simplest Belousov-Zhabotinskii (BZ) type oscillators (the bromate-cerium-oxalic acid system) in a continuous flow stirred tank reactor (CSTR). The results support that the transitions from steady states to periodic orbits, and vice versa, proceed via saddle node infinite period (SNIPER) bifurcations. This finding agrees with the results in batch when the produced bromine was removed by an inert gas stream (Noszticzius et a].), Computations demonstrate that a simple revised Oregonator type and bromine-hydrolysis-controlled model cannot describe the SNIPER bifurcation behavior of the system in a CSTR.
32 citations
TL;DR: In this article, the behavior of center-indices is analyzed for two-parameter flows and the integer sum of centerindices along a one-dimensional curve in parameter space is called the H-index.
Abstract: The behavior of center-indices, as introduced by J. Mallet-Paret & J. Yorke, is analyzed for two-parameter flows. The integer sum of center-indices along a one-dimensional curve in parameter space is called the H-index. A nonzero H-index implies global Hopf bifurcation. The index H is not a homotopy invariant. This fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero, which we call B-points. To each B-point we assign an integer B-index, such that the H-index relates to the B-indices by a formula such as occurs in the calculus of residues.
28 citations
TL;DR: In this article, it was shown that there is an infinite period bifurcation creating stable limit cycles at finite amplitude, and the conditions under which such a bifurbation coming out of a separatrix loop from a saddle point in the plane can occur are first derived under the assumption that the Reynolds number is small.
Abstract: The discussion in a previous paper on roll waves is completed by showing how the limit cycles created at small amplitude by a Hopf bifurcation are destroyed. It is shown that there is an infinite period bifurcation creating stable limit cycles at finite amplitude. The conditions under which such a bifurcation coming out of a separatrix loop from a saddle point in the plane can occur are first derived (under the assumption that the Reynolds number is small). The complete evolution of the limit cycles is then deduced. In the subcritical case it is found that there is just one stable limit cycle, created at small amplitude by a Hopf bifurcation and destroyed at finite amplitude by an infinite period bifurcation. In the supercritical case it is shown that there are two limit cycles, one unstable (created by a Hopf bifurcation) and the other stable (created by the infinite period bifurcation) which finally merge and are then both destroyed. The discontinuous roll wave solutions derived by R. F. Dressler ( Communs pure appl. Math. 2, 49-194 (1949)) are compared with the continuous solutions for large values of the Reynolds number. It is shown that there is a difference in the jump condition between Dressler’s solutions and the present ones. It is then shown that this difference could be resolved by a slight modification to the dissipation term, which leaves the basic form of the continuous solutions unaltered. Finally it is then shown that both sets of waves are similar in that they both terminate with a solitary wave.
28 citations
TL;DR: In this paper, the authors derived conditions under which a limit cycle, created by a Hopf bifurcation, can be destroyed (at finite amplitude) by an infinite period bifurbation, this latter appearing out of a homoclinic orbit formed by the separatrices of a saddle-point equilibrium state.
Abstract: By considering a model equation we are able to derive conditions under which a limit cycle, created (at small amplitude) by a Hopf bifurcation, can be destroyed (at finite amplitude) by an infinite period bifurcation, this latter appearing out of a homoclinic orbit formed by the separatrices of a saddle-point equilibrium state. Further, we are able to extend the methods used for showing the existence of an infinite period bifurcation to calculate the amplitude of the limit cycle over its whole range of existence. These ideas are then applied to an equation arising in the theory of roll waves down an open inclined channel, extending previous work to include the case when the Reynolds number is large with the Froude number close to its critical value for the temporal instability of the uniform flow. Here the governing equation reduces to one similar in form to the model equation.
23 citations
TL;DR: In this article, a general theory for the convergence of predictor-solver continuation methods near singular points of a nonlinear equation is developed, which provides a uniform treatment of continuation for simple and multiple bifurcation problems.
Abstract: The performance of predictor-solver continuation methods near bifurcation points is considered. A general theory is developed for the convergence of two commonly used solvers, namely Newton’s method and the chord method, near singular points of a nonlinear equation. This theory provides a uniform treatment of continuation for simple bifurcation, multiple bifurcation, and multiple limit point bifurcation problems. For these types of bifurcation it is shown that there are conical domains of attraction, for the above iterative methods, centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations.
18 citations
TL;DR: In this paper, the authors studied the finite amplitude periodic forcing of chemical oscillators and showed that the bifurcation diagram of the autonomous system with respect to this parameter affects the qualitative response of the full forced system.
12 citations
TL;DR: In this article, a stability analysis of the steady state straight line downhill motion of a tractor-semitrailer vehicle is given using the methods of bifurcation theory.
Abstract: SUMMARY A stability analysis of the steady state straight line downhill motion of a tractor-semitrailer vehicle is given. In order to have, basically, also asymptotically stable motions of the system an active driver (controller) had to be included in the system. Using the methods of bifurcation theory we give a nonlinear stability analysis of the controlled system tractor-semitrailer-driver. Due to the controller in this system only Hopf-bifurcations occur at loss of stability. I.e. all instabilities are oscillatory. However, both stable and unstable limit cycles after bifurcation are found.
6 citations
TL;DR: In this article, a description of the bifurcation of periodic solutions occurring when a Hamiltonian system of two degrees of freedom passes through non-semisimple 1∶−1 resonance at an equilibrium is given.
Abstract: In this paper a description is given of the bifurcation of periodic solutions occurring when a Hamiltonian system of two degrees of freedom passes through nonsemisimple 1∶−1 resonance at an equilibrium. A bifurcation like this is found in the planar circular restricted problem of three bodies at the Lagrange equilibriumL
4 when the mass parameter passes through the critical value of Routh.
TL;DR: In this paper, a perturbation and asymptotic method is presented for analyzing a class of subcritical bifurcation problems whose solutions possess minimum transition values, and the effects of imperfections on the response of the systems are also investigated.
Abstract: Perturbation and asymptotic methods are presented for analyzing a class of subcritical bifurcation problems whose solutions possess minimum transition values. These minimum transition values are determined. In addition, the dynamics of the transitions from the basic state to the larger amplitude bifurcation states are obtained. The effects of imperfections on the response of the systems are also investigated. The method is presented for two model problems. However, it is valid for a wide class of problems in elastic and hydrodynamic stability, in reaction-diffusion systems and in other applications. In the first problem we obtain subcritical steady bifurcation states for a one-dimensional nonlinear diffusion problem. In the second problem we consider the subcritical Hopf bifurcation of periodic solutions for a higher order van der Pol–Duffing oscillator.
TL;DR: In this paper, the stability and bifurcation behavior of a double-wedge aerofoil performing a pitching motion at high angles of attack were investigated, and the existence of degenerate Hopf bifurbation due to the violation of Hopf s transversality condition at certain critical values of the system parameters was shown.
Abstract: This paper investigates the stability and bifurcation behavior of a double-wedge aerofoil performing a pitching motion at high angles of attack. When a pair of complex conjugate eigenvalues crosses the imaginary axis of the eigenvalue plane, the trivial solution loses stability giving rise to a periodic solution, known as Hopf bifurcation, provided certain transversality conditions are not violated. The existence of degenerate Hopf bifurcation due to the violation of Hopf s transversality condition at certain critical values of the system parameters is shown. The behavior of the pitching motion near these critical values is examined by unfolding the degeneracies. For the supersonic double-wedge aerofoil, various parameters defining the bifurcation paths were numerically evaluated.
TL;DR: In this paper, the local normal form of a dynamical system undergoing a bifurcation of co-dimension two, weakly perturbed by noise, is used to derive analytical expressions for the mean exit times from the domains of the attracting fixed point or limit cycle.
Abstract: The local normal form of a dynamical system undergoing a bifurcation of co-dimension two, weakly perturbed by noise, is used to derive analytical expressions for the mean exit times from the domains of the attracting fixed point or limit cycle.
TL;DR: In this article, a degenerate dynamic bifurcation phenomenon exhibited by autonomous systems is analyzed in detail, where a key coefficient vanishes at a critical point where Hopf's transversality condition is also violated simultaneously.
Abstract: A degenerate dynamic bifurcation phenomenon exhibited by autonomous systems is analyzed in detail. The situation arises when a key coefficient vanishes at a critical point where Hopf's transversality condition is also violated simultaneously. It is demostrated analytically that, unlike the phenomenon of Hopf bifurcation, in this case the existence of bifurcating family of limit cycles cannot be guaranteed. Indeed, an existence condition emerges as an integral part of the analysis. Under this existence condition, several topologically distinct phenomena may arise, and the conditions giving rise to such cases are discussed. the asymptotic equations of the bifurcating paths, the family of limit cycles and frequency-amplitude relationships are given in general, explicitforms which can be used in the analyses of specific problems directly.
TL;DR: In this paper, the numerical computation of progressive free surface gravity waves on a horizontal bed was studied and two transition values Q1 and Q2 with corresponding transition bifurcation branches that classify waves into three disjoint branch sets B1, B2 and B3 were determined.
Abstract: We are concerned with the numerical computation of progressive free surface gravity waves on a horizontal bed. They are regarded as families of bifurcation branches (λ,A)Q of constant discharge Q.
Numerically we determine two transition values Q1 and Q2 with corresponding transition bifurcation branches that classify waves into three disjoint branch sets B1, B2 and B3. Their members are families of waves (λ,A)Q satisfying the conditions 0
TL;DR: In this paper, the stability of steady-state solutions is established based on the theory in [9] and the critical curve which describes the transition of steady state solution from stable to unstable is found.