Related papers: Infinitesimal Lyapunov functions for singular flow…
It is given notions of singular hyperbolicity and sectional Lyapunov exponents of orders beyond the classical ones, namely, other dimensions besides the dimension 2 and the full dimension of the central subbundle of the singular hyperbolic…
In this work, it is presented a characterization of star property for a $C^1$ vector field based on Lyapunov functions. It is also obtained conditions to strong homogeneity for singular sets by using the notion of infinitesimal Lyapunov…
We study the finiteness of uniform sinks for flow. Precisely, we prove that, for $\alpha>0$ $T>0$, if a vector field $X$ has only hyperbolic singularities or sectionally dissipative singularities, then $X$ can have only finitely many…
We consider some general classes of random dynamical systems and show that a priori very weak nonuniform hyperbolicity conditions actually imply uniform hyperbolicity.
In hyperbolic dynamics, a well-known result is: every hyperbolic Lyapunov stable set, is attracting; it's natural to wonder if this result is maintained in the sectional-hyperbolic dynamics. This question is still open, although some…
We study $C^1$-generic vector fields on closed manifolds without points accumulated by periodic orbits of different indices and prove that they exhibit finitely many sinks and sectional-hyperbolic transitive Lyapunov stable sets with…
From the analyticity properties of the equation governing infinitesimal perturbations, it is shown that all stability properties of spatially extended 1D systems can be derived from a single function that we call entropy potential since it…
We prove for a generic star vector field $X$ that, if for every chain recurrent class $C$ of $X$ all singularities in $C$ have the same index, then the chain recurrent set of $X$ is singular hyperbolic. We also prove that every Lyapunov…
We present new theorems characterizing robust Lyapunov functions and infinite horizon value functions in optimal control as unique viscosity solutions of partial differential equations. We use these results to further extend Zubov's method…
A new infinitesimal characterization of completely positive but not necessarily homomorphic Markov flows from a C^*-algebra to bounded operators on the boson Fock space over L^2(R) is given. Contrarily to previous characterizations, based…
We use Lyapunov type functions to find conditions of finite shadowing in a neighborhood of a nonhyperbolic fixed point of a one-dimensional or two-dimensional homeomorphism or diffeomorphism. A new concept of shadowing in which we control…
In this paper, a class of abstract dynamical systems is considered which encompasses a wide range of nonlinear finite- and infinite-dimensional systems. We show that the existence of a non-coercive Lyapunov function without any further…
In this article, we provide a general strategy based on Lyapunov functionals to analyse global asymptotic stability of linear infinite-dimensional systems subject to nonlinear dampings under the assumption that the origin of the system is…
Hyperbolicity plays an important role in the study of dynamical systems, and is a key concept in the iteration of rational functions of one complex variable. Hyperbolic systems have also been considered in the study of transcendental entire…
We study the amount of nonhyperbolicity within a broad class of (nonhyperbolic) partially hyperbolic diffeomorphisms with a one-dimensional center. For that, we focus on the center Lyapunov exponent and the entropy of its level sets. We…
In this paper we use a path-integral approach to represent the Lyapunov exponents of both deterministic and stochastic dynamical systems. In both cases the relevant correlation functions are obtained from a (one-dimensional) supersymmetric…
Singular and sectional hyperbolic sets are the objects of the extension of the classical Smale Hyperbolic Theory to flows having invariant sets with singularities accumulated by regular orbits within the set. It is by now well-known that…
We prove a converse Lyapunov theorem for boundedness of reachability sets for a general class of control systems whose flow is Lipschitz continuous on compact intervals with respect to trajectory-dominated inputs. We show that this…
The aim of this paper is to investigate the fractional combinatorial Calabi flow for hyperbolic bordered surfaces. By Lyapunov theory, it is proved that the flow exists for all time and converges exponentially to a conformal factor that…
It is a paper about models for isolated sets and the construction of special hyperbolic Lyapunov functions. We prove that after a suitable surgery every isolated set is the intersection of an attractor and a repeller. We give linear models…