Search results for "bifurcation theory"

Scenario of the Birth of Hidden Attractors in the Chua Circuit

2017

Recently it was shown that in the dynamical model of Chua circuit both the classical selfexcited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of selfexcited and hidden attractors is studied. It is shown a pitchfork bifurcation in which a pair of symmetric attractors coexists and merges into one symmetric attractor through an attractormerging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed.

Efficient computation of stable bifurcating branches of nonlinear eigenvalue problems

1983

Ober Ein Rayleigh-Ritz-Verfahren zur Bestimmung Kritischer Werte

1980

This paper is concerned with the existence of critical points for a functional f defined on the level set of a second functional g. Existence of nontrivial solutions for the nonlinear eigenvalue-problem f′(u) = λg′(u) and convergence for a nonlinear analogue to the Rayleigh-Ritz-Method is proven. The results are applied to a nonlinear ordinary eigenvalue problem where it is shown that the lowest point in the continuous spectrum of the associated linearized operator is a bifurcation point of infinite multiplicity.

Instability and bistability during the growth of a corrosion scale on metals and alloys

1986

This paper summarizes the main results for the interpretation of the self organized corrosion scales observed in oxidation or sulfidation of some metals or alloys. It consists also of a reconsideration of the classical theoretical concepts used in Reactivity of Solids. It proposes new theoretical tools that have been fruitfully utilized in other topics : non linear and coupled processes, stability analysis and bifurcation theory. Some examples are developed, where the corrosion kinetics at high temperature are interpreted in term of chemical bistable system able to oscillate spontaneously and mechanochemical couplings are also taken into account. In according with experimental results, all …

Formation of Coherent Structures in Kolmogorov Flow with Stratification and Drag

2014

We study a weakly stratified Kolmogorov flow under the effect of a small linear drag. We perform a linear stability analysis of the basic state. We construct the finite dimensional dynamical system deriving from the truncated Fourier mode approximation. Using the Reynolds number as bifurcation parameter we build the corresponding diagram up to Re=100. We observe the coexistence of three coherent structures.

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.

Anharmonic effects on the dynamic behavior’s of Klein Gordon model’s

2021

Abstract This work completes and extends the Ref. Tchakoutio Nguetcho et al. (2017), in which we have focused our attention only on the dynamic behavior of gap soliton solutions of the anharmonic Klein-Gordon model immersed in a parameterized on-site substrate potential. We expand our work now inside the permissible frequency band. These considerations have crucial effects on the response of nonlinear excitations that can propagate along this model. Moreover, working in the allowed frequency band is not only interesting from a physical point of view, it also provides an extraordinary mathematical model, a new class of differential equations possessing vital parameters and vertical singular …

Techniques in the Theory of Local Bifurcations: Cyclicity and Desingularization

1993

A fundamental open question of the bifurcation theory of vector fields in dimension 2 is whether the number of locally bifurcating limit cycles in an analytic unfolding is bounded, or more precisely, whether any limit periodic set has finite cyclicity. In these notes we introduce several techniques for attacking this question: asymptotic expansion of return maps, ideal of coefficients, desingularization of parametrized families. Moreover, because of their practical interest, we present some partial results obtained by these techniques.

Quantum Criticality in a Bosonic Josephson Junction

2011

In this paper we consider a bosonic Josephson junction described by a two-mode Bose-Hubbard model, and we thoroughly analyze a quantum phase transition occurring in the system in the limit of infinite bosonic population. We discuss the relation between this quantum phase transition and the dynamical bifurcation occurring in the spectrum of the Discrete Self Trapping equations describing the system at the semiclassical level. In particular, we identify five regimes depending on the strength of the effective interaction among bosons, and study the finite-size effects arising from the finiteness of the bosonic population. We devote a special attention to the critical regime which reduces to th…