Search results for "Infinite-period bifurcation"
showing 5 items of 5 documents
Remarks on the economic interpretation of Hopf bifurcations
1999
Abstract The Hopf bifurcation theorem has become a frequently used tool in the study of nonlinear dynamical economic systems. In this paper, it is shown that phenomena like multiple limit cycles, hysteresis loops and catastrophic transitions may possibly accompany a Hopf bifurcation. The theoretical argument is illustrated in Foley's liquidity cost–business cycle model.
Coexistence of periods in a bifurcation
2012
Abstract A particular type of order-to-chaos transition mediated by an infinite set of coexisting neutrally stable limit cycles of different periods is studied in the Varley–Gradwell–Hassell population model. We prove by an algebraic method that this kind of transition can only happen for a particular bifurcation parameter value. Previous results on the structure of the attractor at the transition point are here simplified and extended.
The period function of reversible quadratic centers
2006
Abstract In this paper we investigate the bifurcation diagram of the period function associated to a family of reversible quadratic centers, namely the dehomogenized Loud's systems. The local bifurcation diagram of the period function at the center is fully understood using the results of Chicone and Jacobs [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433–486]. Most of the present paper deals with the local bifurcation diagram at the polycycle that bounds the period annulus of the center. The techniques that we use here are different from the ones in [C. Chicone, M. Jacobs, Bifurcation of critical periods for plane vecto…
Lusternik-Schnirelmann Critical Values and Bifurcation Problems
1987
We present a method to calculate bifurcation branches for nonlinear two point boundary value problems of the following type $$ \{ _{u(a) = u(b) = 0,}^{ - u'' = \lambda G'(u)} $$ (1.1) where G : R → R is a smooth mapping. This problem can be formulated equivalently as $$ g' \left(u \right)= \mu u, $$ (1.2) where $$ g \left(u \right)= \overset{b} {\underset{a} {\int}} G \left(u \left(t \right) \right) dt $$ (1.3) and μ = 1/λ. Solutions of this problem can be found by locating the critical points of the functional g : H → R on the spheres \(S_r= \lbrace x \in H \mid \;\parallel x \parallel =r \rbrace, r >0.\) (The Lagrange multiplier theorem.)
Desingularization Theory and Bifurcation of Non-elementary Limit Periodic Sets
1998
In the study of the Bogdanov-Takens unfolding, we introduced in 4.3.5.2 the following formulas of rescaling in the phase-space and in the parameter space: $$ x = {r^2}\bar x,y = {r^3}\bar y,\mu = - {r^4},\nu = {r^2}\bar \nu . $$