Search results for "Saddle-node bifurcation"
showing 10 items of 11 documents
Two-photon laser dynamics.
1995
Degenerate as well as nondegenerate three-level two-photon laser (TPL) models are derived. In the limit of equal cavity losses for both fields, it is shown that the nondegenerate model reduces to the degenerate one. We also demonstrate the isomorphism existing between our degenerate TPL model and that of a dressed-state TPL. All these models contain ac-Stark and population-induced shifts at difference from effective Hamiltonian models. The influence of the parameters that control these shifts on the nonlinear dynamics of a TPL is investigated. In particular, the stability of the periodic orbits that arise at the Hopf bifurcation of the system and the extension of the self-pulsing domains of…
A free boundary problem stemmed from combustion theory. Part II: Stability, instability and bifurcation results
2002
AbstractWe deal with a free boundary problem, depending on a real parameter λ, in a infinite strip in R2, providing stability, instability and bifurcation.
Pattern formation driven by cross–diffusion in a 2D domain
2012
Abstract In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.
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.
A SUBCRITICAL BIFURCATION FOR A NONLINEAR REACTION–DIFFUSION SYSTEM
2010
In this paper the mechanism of pattern formation for a reaction-diffusion system with nonlinear diffusion terms is investigated. Through a linear stability analysis we show that the cross-diffusion term allows the pattern formation. To predict the form and the amplitude of the pattern we perform a weakly nonlinear analysis. In the supercritical case the Stuart-Landau equation is found, which rules the evolution of the amplitude of the most unstable mode. With the increasing distance from the bifurcation value of the cross-diffusion parameter, the weakly nonlinear analysis fails and a Fourier–Galerkin approach is adopted. In the subcritical case the weakly nonlinear analysis must be pushed u…
Transitions in a stratified Kolmogorov flow
2016
We study the Kolmogorov flow with weak stratification. We consider a stabilizing uniform temperature gradient and examine the transitions leading the flow to chaotic states. By solving the equations numerically we construct the bifurcation diagram describing how the Kolmogorov flow, through a sequence of transitions, passes from its laminar solution toward weakly chaotic states. We consider the case when the Richardson number (measure of the intensity of the temperature gradient) is $$Ri=10^{-5}$$ , and restrict our analysis to the range $$0<Re<30$$ . The effect of the stabilizing temperature is to shift bifurcation points and to reduce the region of existence of stable drifting states. The…
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…
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: $$ x = {r^2}\bar x,y = {r^3}\bar y,\mu = - {r^4},\nu = {r^2}\bar \nu . $$
Control of a Non-isothermal CSTR by Type-2 Fuzzy Logic Controllers
2009
The paper describes the application of a type-2 fuzzy logic controller (FLC) to a non-isothermal continuous stirred tank reactor (CSTR) characterized by the presence of saddle node and Hopf bifurcations. Its performance is compared with a type-1 fuzzy logic controller performance. A full analysis of the uncontrolled CSTR dynamic was carried out and used for the feedback-feedforward fuzzy controllers development. Simulation results confirm the effectiveness and the robustness of the type-2 FLCs which outperform their type-1 counterparts, particularly when uncertainties are present in the system.