Search results for " DYNAMICAL SYSTEM"
showing 10 items of 188 documents
An operator-like description of love affairs
2010
We adopt the so--called \emph{occupation number representation}, originally used in quantum mechanics and recently considered in the description of stock markets, in the analysis of the dynamics of love relations. We start with a simple model, involving two actors (Alice and Bob): in the linear case we obtain periodic dynamics, whereas in the nonlinear regime either periodic or quasiperiodic solutions are found. Then we extend the model to a love triangle involving Alice, Bob and a third actress, Carla. Interesting features appear, and in particular we find analytical conditions for the linear model of love triangle to have periodic or quasiperiodic solutions. Numerical solutions are exhibi…
Moment Equations for a Spatially Extended System of Two Competing Species
2005
The dynamics of a spatially extended system of two competing species in the presence of two noise sources is studied. A correlated dichotomous noise acts on the interaction parameter and a multiplicative white noise affects directly the dynamics of the two species. To describe the spatial distribution of the species we use a model based on Lotka-Volterra (LV) equations. By writing them in a mean field form, the corresponding moment equations for the species concentrations are obtained in Gaussian approximation. In this formalism the system dynamics is analyzed for different values of the multiplicative noise intensity. Finally by comparing these results with those obtained by direct simulat…
Attracteur de polarisation dans les fibres optiques
2004
Nous etudions la dynamique non lineaire des etats de polarisation de deux ondes coherentes ou incoherentes se propageant dans une fibre optique. Nous presentons en particulier un effet original d'attraction de la polarisation.
Noise delayed decay of unstable states: theory versus numerical simulations
2004
We study the noise delayed decay of unstable nonequilibrium states in nonlinear dynamical systems within the framework of the overdamped Brownian motion model. We give the exact expressions for the decay times of unstable states for polynomial potential profiles and obtain nonmonotonic behavior of the decay times as a function of the noise intensity for the unstable nonequilibrium states. The analytical results are compared with numerical simulations.
Group Foundations of Quantum and Classical Dynamics : Towards a Globalization and Classification of Some of Their Structures
1987
This paper is devoted to a constructiveand critical analysis of the structure of certain dynamical systems from a group manifold point of view recently developed. This approach is especially suitable for discussing the structure of the quantum theory, the classical limit, the Hamilton-Jacobi theory and other problems such as the definition and globalization of the Poincare-Cartan form which appears in the variational approach to higher order dynamical systems. At the same time, i t opens a way for the classification of all hamiltonian and lagrangian systems associated with suitably defined dynamical groups. Both classical and quantum dynamics are discussed, and examples of all the different…
Infinite orbit depth and length of Melnikov functions
2019
Abstract In this paper we study polynomial Hamiltonian systems d F = 0 in the plane and their small perturbations: d F + ϵ ω = 0 . The first nonzero Melnikov function M μ = M μ ( F , γ , ω ) of the Poincare map along a loop γ of d F = 0 is given by an iterated integral [3] . In [7] , we bounded the length of the iterated integral M μ by a geometric number k = k ( F , γ ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + ϵ ω with arbitrary high length first nonzero Melnikov function M μ along…
Blenders near polynomial product maps of $\mathbb C^2$
2021
In this paper we show that if $p$ is a polynomial which bifurcates then the product map $(z,w)\mapsto(p(z),q(w))$ can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets can be chosen to be of two types : repelling or saddle. As a consequence, such product map belongs to the closure of the interior of two different sets : the bifurcation locus of $H_d(\mathbb P^2)$ and the set of endomorphisms having an attracting set of non-empty interior. In an independent part, we use perturbations of H\'enon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets.
Pseudo-abelian integrals: Unfolding generic exponential case
2009
The search for bounds on the number of zeroes of Abelian integrals is motivated, for instance, by a weak version of Hilbert's 16th problem (second part). In that case one considers planar polynomial Hamiltonian perturbations of a suitable polynomial Hamiltonian system, having a closed separatrix bounding an area filled by closed orbits and an equilibrium. Abelian integrals arise as the first derivative of the displacement function with respect to the energy level. The existence of a bound on the number of zeroes of these integrals has been obtained by A. N. Varchenko [Funktsional. Anal. i Prilozhen. 18 (1984), no. 2, 14–25 ; and A. G. Khovanskii [Funktsional. Anal. i Prilozhen. 18 (1984), n…
Vanishing Abelian integrals on zero-dimensional cycles
2011
In this paper we study conditions for the vanishing of Abelian integrals on families of zero-dimensional cycles. That is, for any rational function $f(z)$, characterize all rational functions $g(z)$ and zero-sum integers $\{n_i\}$ such that the function $t\mapsto\sum n_ig(z_i(t))$ vanishes identically. Here $z_i(t)$ are continuously depending roots of $f(z)-t$. We introduce a notion of (un)balanced cycles. Our main result is an inductive solution of the problem of vanishing of Abelian integrals when $f,g$ are polynomials on a family of zero-dimensional cycles under the assumption that the family of cycles we consider is unbalanced as well as all the cycles encountered in the inductive proce…
Flexible periodic points
2014
We define the notion of ${\it\varepsilon}$-flexible periodic point: it is a periodic point with stable index equal to two whose dynamics restricted to the stable direction admits ${\it\varepsilon}$-perturbations both to a homothety and a saddle having an eigenvalue equal to one. We show that an ${\it\varepsilon}$-perturbation to an ${\it\varepsilon}$-flexible point allows us to change it to a stable index one periodic point whose (one-dimensional) stable manifold is an arbitrarily chosen $C^{1}$-curve. We also show that the existence of flexible points is a general phenomenon among systems with a robustly non-hyperbolic two-dimensional center-stable bundle.