Smale Type Research Articles

Numerical analysis of behaviour of the Cucker–Smale type models with fractional operators

The behaviour of fractional-order multi-agent discrete-time systems is examined by numerical analysis. We consider the Cucker–Smale type models with fractional operators. The asymptotic stability of a class of linear fractional discrete-time systems as a tool providing a consensus in the nonlinear fractional discrete-time systems is investigated. The proposed results are based on linearization method of the considered multi-agent system. Interesting examples illustrate the effectiveness of the derived results.

Phase Transitions in a Kinetic Flocking Model of Cucker--Smale Type

We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a "disordered" to an "ordered" state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition.

Flocking of multi-agent systems with multiplicative and independent measurement noises

We investigate the effect of noise and group density on the emergence of flocking in a stochastic Cucker–Smale type system. We consider the independent and multiplicative measurement noise and derive sufficient conditions for flocking and non-flocking in terms of the noise intensity and group density. We report that, in noise environments, it is rather easy for the group with high density to form and maintain ordered group motion. Moreover, we find that, under the perturbation of multiplicative noise, systems with weak coupling can tolerate relatively strong perturbation of noise.

Metastability in a class of hyperbolic dynamical systems perturbed by heavy-tailed Lévy type noise

We consider a finite dimensional deterministic dynamical system with finitely many local attractors Kι, each of which supports a unique ergodic probability measure Pι, perturbed by a multiplicative non-Gaussian heavy-tailed Lévy noise of small intensity ε > 0. We show that the random system exhibits a metastable behavior: there exists a unique ε-dependent time scale on which the system reminds of a continuous time Markov chain on the set of the invariant measures Pι. In particular our approach covers the case of dynamical systems of Morse–Smale type, whose attractors consist of points and limit cycles, perturbed by multiplicative α-stable Lévy noise in the Itô, Stratonovich and Marcus sense. As examples we consider α-stable Lévy perturbations of the Duffing equation and Pareto perturbations of a biochemical birhythmic system with two nested limit cycles.

A Smale Type Result and Applications to Contact Homology

In this note we will show that the injection of a suitable subspace of the space of Legendrian loops into the full loop space is an S 1 -equivariant homotopy equivalence. Moreover, since the smaller space is the space of variations of a given action functional, we will compute the relative Contact Homology of a family of tight contact forms on the three-dimensional torus.

Note on a general Palais–Smale condition

In this paper, we establish a qualitative deformation theorem for a locally Lipschitz function satisfying a general Palais–Smale type condition. Then we show that, with some assumptions, this compactness condition implies the coercivity of the function.

Smale's α-theory for inexact Newton methods under the γ-condition

The present paper is concerned with the convergence problem of inexact Newton methods. Assuming that the nonlinear operator satisfies the γ-condition, a convergence criterion for inexact Newton methods is established which includes Smale's type convergence criterion. The concept of an approximate zero for inexact Newton methods is proposed in this paper and the criterion for judging an initial point being an approximate zero is established. Consequently, Smale's α-theory is generalized to inexact Newton methods. Furthermore, a numerical example is presented to illustrate the applicability of our main results.

The Ekeland variational principle for Henig proper minimizers and super minimizers

In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the sense of Ioffe, Clarke and Mordukhovich. As applications we obtain sufficient conditions for F to have a Henig proper minimizer or a super minimizer under the Palais–Smale type conditions. The techniques are essentially based on the characterizations of Henig proper efficient points and super efficient points by mean of the Henig dilating cones and the Hiriart-Urruty signed distance function.

EXTENDED NEWTON’S METHOD FOR MAPPINGS ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE

Robinson's generalized Newton's method for nonlinear functions with values in a cone is extended to mappings on Riemannian manifolds with values in a cone. When ${\cal D}f$ satisfies the $L$-average Lipschitz condition, we use the majorizing function technique to establish the semi-local quadratic convergence of the sequences generated by the extended Newton's method. As applications, we also obtain Kantorovich's type theorem, Smale's type theorem under the $\gamma$-condition and an extension of the theory of Smale's approximate zeros.

EXTENDED NEWTON’S METHOD FOR MAPPINGS ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE

Robinson's generalized Newton's method for nonlinear functions with values in a cone is extended to mappings on Riemannian manifolds with values in a cone. When ${\cal D}f$ satisfies the $L$-average Lipschitz condition, we use the majorizing function technique to establish the semi-local quadratic convergence of the sequences generated by the extended Newton's method. As applications, we also obtain Kantorovich's type theorem, Smale's type theorem under the $\gamma$-condition and an extension of the theory of Smale's approximate zeros.

On finding sign-changing solutions

Some parameter-depending linking theorems are established, which allow to produce a bounded and sign-changing Palais–Smale sequence. For even functionals, a parameter-depending fountain theorem is obtained which provides infinitely many bounded and sign-changing Palais–Smale sequences. A variant mountain pass theorem is built in cones which yields bounded, positive and negative Palais–Smale sequences. The usual Palais–Smale type compactness condition and its variants are completely not necessary for these theories. More exact locations of the critical sequences can be determined. The abstract results are applied to the Schrödinger equation with (or without) critical Sobolev exponents: − Δ u + V ( x ) u = β | u | 2 * − 2 u + f ( x , u ) , x ∈ R N , β ⩾ 0 , where 2 * is the critical Sobolev exponent. The existence of (multiple) sign-changing solutions is obtained. The positive and negative solutions are also gained as by-products. We will also show that this Schrödinger problem with jumping nonlinearity is independent of the Fučík spectrum.

Variants of the Ekeland Variational Principle for a set-valued map involving the Clarke Normal Cone

In this paper we present two set-valued variants of the Ekeland variational principle involving the Clarke normal cone and establish sufficient conditions for a set-valued map to have a weak minimizer or a properly positive minimizer when it satisfies Palais–Smale type conditions.

On Functionals Bounded Below

In this paper, we prove that the infimum of a functional with a lower bound is a critical value. The novelty is that we do not need the Palais–Smale type condition.

Transverse Lusternik–Schnirelmann category of foliated manifolds

The purpose of this paper is to develop a transverse notion of Lusternik–Schnirelmann category in the field of foliations. Our transverse category, denoted by cat ⋔(M, F) , is an invariant of the foliated homotopy type which is finite on compact manifolds. It coincides with the classical notion when the foliation is by points. We prove that for any foliated manifold cat M⩽ cat L cat ⋔(M, F) , where L is a leaf of maximal category, thus generalizing a result of Varadarajan for fibrations. Also we prove that cat ⋔(M, F) is bounded below by the index of k ∗H b +(M) , the latter being the image in H DR ( M) of the algebra of basic cohomology in positive degrees. In the second part of the paper we prove that cat ⋔(M, F) is a lower bound for the number of critical leaves of any basic function provided that F is a foliation satisfying certain conditions of Palais–Smale type. As a consequence, we prove that the result is true for compact Hausdorff foliations and for foliations of codimension one. This generalizes the classical result of Lusternik and Schnirelmann about the number of critical points of a smooth function.

Smale's horseshoe map in a Hamiltonian system around a separatrix

A non-linear system with one degree of freedom is studied. It is assumed that the potential of the system is a double-well type and that a time-dependent external field is imposed on the system as a perturbation. The analytic solutions around a separatrix are obtained using the first order perturbation. These analytic solutions are in good agreement with the numerical integrations. It is further shown that these solutions draw a horseshoe map of the Smale type.

“Horseshoe chaos” in the space-independent double sine-Gordon system

For the driven, damped space-independent double sine-Gordon equation threshold curves for horseshoe chaos of the Smale type are derived by the Melnikov technique. Different qualitative behaviour of the solutions is found in different regions of parameter space.