Related papers: Characterizations of dominated splitting system an…
We present here the concept of Dominated Splitting and give an account of some important results on its dynamics.
A key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting. In this paper, we show that often partial derivatives of the splitting exist and the splitting…
The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms. One meets difficulties when one tries to extend these definitions to vector fields and Shantao…
We obtain sufficient conditions for an invariant splitting over a compact invariant subset of a $C^1$ flow $X_t$ to be dominated. In particular, we reduce the requirements to obtain sectional hyperbolicity and hyperbolicity.
We relate dominated splitting for a linear multiplicative cocyle with dominated splitting for the exterior powers of this cocycle. For a C1 vector field X on a 3-manifold, we can obtain singular-hyperbolicity using only the tangent map DX…
In this paper we obtain two criteria of stable ergodicity outside the partially hyperbolic scenario. In both criteria, we use a weak form of hyperbolicity called chain-hyperbolicity. It is obtained one criterion for diffeomorphisms with…
We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We…
A possible definition of strong/symmetric hyperbolicity for a second-order system of evolution equations is that it admits a reduction to first order which is strongly/symmetric hyperbolic. We investigate the general system that admits a…
We discuss the problems of modeling, control, and decision support in complex dynamic systems from a general system theoretic point of view. The main characteristics of complex systems and of system approach to complex system study are…
We define the notion of localizable property for a dynamical system. Then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems. These notions are the fast growth of the…
We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets $\Sigma$ in $GL(d,\mathbb{R})$ with the property that any cocycle with values…
Distinguishability and, by extension, observability are key properties of dynamical systems. Establishing these properties is challenging, especially when no analytical model is available and they are to be inferred directly from…
We consider the basic features of complex dynamic and control systems, including systems having hierarchical structure. Special attention is paid to the problems of design and synthesis of complex systems and control models, and to the…
This is a lightning introduction to some modern techniques used in the study of the statistical properties of hyperbolic dynamical systems. The emphasis is not in presenting a comprehensive theory but rather in fleshing out the main ideas…
This thesis attempts to contribute to the study of differentiable dynamics both from a semi-local and global point of view. The center of study is differentiable dynamics in manifolds of dimension 3 where we are interested in the…
After a short introduction to the characteristic geometry underlying weakly hyperbolic systems of partial differential equations we review the notion of symmetric hyperbolicity of first-order systems and that of regular hyperbolicity of…
Hyperbolicity is a graph parameter related to how much a graph resembles a tree with respect to distances. Its computation is challenging as the main approaches consist in scanning all quadruples of the graph or using fast matrix…
We propose and study a system whose dynamics are governed by predictions of its future states. General formalism and concrete examples are presented. We find that the dynamical characteristics depend on both how to shape predictions as well…
Several concepts on the measure of observability, reachability, and robustness are defined and illustrated for both linear and nonlinear control systems. Defined by using computational dynamic optimization, these concepts are applicable to…
This paper gives a summary of a body of work at the intersection of control theory and smooth nonlinear dynamics. The main idea is to transfer the concept of uniform hyperbolicity, central to the theory of smooth dynamical systems, to…