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Bifurcation diagram
About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.
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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: In this article, the existence of a self-sustained periodic solution for a hyperbolic partial differential equation with boundary conditions given on the rectangle shown in Fig. 1.
Abstract: The existence of a self-sustained periodic solution in the autonomous equation m'(t) — au' (t — h) + /3u(t) + ay u(t — h) = «/(w(r)) is proved under appropriate assumptions on a, ft, y, f and h. The method of proof consists in converting this equation into an equivalent nonlinear integral equation and demonstrating the convergence of an appropriate iteration scheme. In this paper we consider the equation m'(t) — au'(r — h) + /3w(r) + ayu(r — h) = ef(u(T)), (1) where h > 0 and a = aa(l + e). The existence of a periodic solution will be proved for small t > 0 under appropriate assumptions on the parameters a0, (3, y, and /. The left-hand side of this equation is a linear difference-differential operator of neutral type (for a definition see [1]). The existence of periodic solutions for functional-differential equations which include difference-differential equations of retarded type but not neutral type has been discussed by Krasovskii [2], Shimanov [3], [4], and Hale [5]. In all these cases, the equations are of the forced type where the right-hand side is a 2?r-periodic function of r. Equation (1), which we consider, is autonomous, and we look for a self-sustained oscillation. Difference-differential equations of the type (1) arise from electrical networks such as the one shown in Fig. 1. The equations of this network are Udi/dt) = -dv/dx, 0 < ^ < J (2) C{dv/dt) = —di/dx, E — v(0, t) — Ri(0, t) = 0, Ci(dv(l, t)/dt) = i{ 1, t) — g(v( 1, t)), where L, C are the specific inductance and capacitance in the transmission line. The question of the existence of a periodic solution of some unknown period T can be posed by giving the additional boundary conditions v(x, 0) = v(x, T), i(x, 0) = i(x, T). Thus, we have a boundary-value problem for a hyperbolic partial differential equation with boundary conditions given on the rectangle shown in Fig. 2. Of course, in general, a boundary-value problem for a hyperbolic equation is not well posed. The difference here is that the boundary t — T is free to be chosen. •Received November 5, 1965; revised manuscript received March 4, 1966. The results reported in this paper were obtained in the couse of research jointly sponsored by the Air Force Office of Scientific Research (Contract AF 49(638)-1139) and IBM. 216 It. K. BRAYTON [Vol. XXIV, No. 3
146 citations
TL;DR: In this paper, the bifurcation of limit cycles in general quadratic perturbations of plane vector fields having a center at the origin is studied and the essential perturbation is determined.
Abstract: We study the bifurcation of limit cycles in general quadratic perturbations of plane quadratic vector fields having a center at the origin. For any of the cases, we determine the essential perturbation and compute the corresponding bifurcation function. As an application, we find the precise location of the subset of centers in Q 3 R surrounded by period annuli of cyclicity at least three. Two specific cases are considered in more detail: the isochronous center S 1 and one of the intersection points ( Q 4 + ) of Q 4 and Q 3 R . We prove that the period annuli around S 1 and Q 4 + have cyclicity two and three respectively. The proof is based on the possibility to derive appropriate Picard-Fuchs equations satisfied by the independent integrals included in the related bifurcation function.
146 citations
TL;DR: Numerical results demonstrate that the computer virus model using an SIRS model and the threshold value R 0 determining whether the disease dies out is obtained and has periodic solution when time delay is larger than a critical values.
143 citations
TL;DR: In this article, the Moore-Greitzer model for compressor instability is reduced to a set of three ordinary differential equations, which are approached from the point of view of bifurcation theory.
Abstract: With a one-mode truncation it is possible to reduce the Moore–Greitzer model for compressor instability to a set of three ordinary differential equations These are approached from the point of view of bifurcation theory Most of the bifurcations emerge from a degenerate Takens–Bogdanov bifurcation point The bifurcation sets are completed using the numerical branch tracking scheme AUTO Despite the severity of the truncation, the agreement with experimental results is excellent
143 citations