On chaotic extensions of dynamical systems

In this paper, inspired by some results in linear dynamics, we will show that every dynamical system (X,f), where f is a continuous self-map on a separable metric space X, can be extended to a chaotic (in the sense of Devaney) dynamical system in an isometric way.     

Parallel discrete dynamical systems on independent local functions

In this paper, we extend the manner of defining the evolution update of discrete dynamical systems on Boolean functions, without limiting the local functions to being dependent restrictions of a global one. Then, we analyze the cases concerned with parallel dynamical systems with the OR$OR$, AND$AND$, NAND$NAND$ and NOR$NOR$ functions as independent local functions over undirected and also directed dependency graphs. This extension of the update method widely generalizes the traditional one where only a global Boolean function is considered for establishing the evolution operator of the system. Besides, our analysis allows us to show a richer dynamics in these new kinds of parallel dynamical systems.     

Using machine learning to predict catastrophes in dynamical systems

Nonlinear dynamical systems, which include models of the Earth’s climate, financial markets and complex ecosystems, often undergo abrupt transitions that lead to radically different behavior. The ability to predict such qualitative and potentially disruptive changes is an important problem with far-reaching implications. Even with robust mathematical models, predicting such critical transitions prior to their occurrence is extremely difficult. In this work, we propose a machine learning method to study the parameter space of a complex system, where the dynamics is coarsely characterized using topological invariants. We show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner, we are able to predict with high accuracy the locations of critical transitions in parameter space.     

Sensitive dependence on initial conditions between dynamical systems and their induced hyperspace dynamical systems

The concepts of collective sensitivity and compact-type collective sensitivity are introduced as stronger conditions than the traditional sensitivity for dynamical systems and Hausdorff locally compact second countable (HLCSC) dynamical systems, respectively. It is proved that sensitivity of the induced hyperspace system defined on the space of non-empty compact subsets or non-empty finite subsets (Vietoris topology) is equivalent to the collective sensitivity of the original system; sensitivity of the induced hyperspace system defined on the space of all non-empty closed subsets (hit-or-miss topology) is equivalent to the compact-type collective sensitivity of the original HLCSC system. Moreover, relations between these two concepts and other dynamics concepts that describe chaos are investigated.     

Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems

Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in the t-direction. Hence, a numerical method would have to use infinitely many points.To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium for an autonomous system. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using radial basis functions.     

Stabilities in multi-valued dynamical systems

For the study of the various stabilities in multi-valued dynamical systems, we obtain some necessary and sufficient conditions of the notions of stability and Lyapunov stability, and investigate the connection between the concepts of attractions and the suitable versions of limit sets. Also we consider the notion of characteristic 0⁺ in multi-valued dynamical systems and obtain necessary and sufficient conditions for the concept of characteristic 0⁺.     

Positively expansive dynamical systems

We introduce the notions of weakly and strongly positively expansive (wPE and sPE, respectively) discrete dynamical systems. Both are topological generalizations of the well-known metric notion of positive expansiveness (PE). We prove that the three notions are identical on compact metrizable spaces, but not on noncompact spaces. We investigate properties of PE, wPE, and sPE dynamical systems and show how they are related. Finally, we show that the possible dynamics for wPE and sPE homeomorphisms are severely limited, and we classify sPE homeomorphisms.     

Sell’s conjecture for non-autonomous dynamical systems

This paper is dedicated to the study of the G. Sell’s conjecture for general non-autonomous dynamical systems. We give a positive answer for this conjecture and we apply this result to different classes of non-autonomous evolution equations: Ordinary Differential Equations, Functional Differential Equations and Semi-linear Parabolic Equations.     

Disguised toric dynamical systems

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric. They are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. By means of (real) algebraic geometry we show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms one can use to detect the disguised toric locus.     

Markov structures and decay of correlations for non-uniformly expanding dynamical systems

We consider non-uniformly expanding maps on compact Riemannian manifolds of arbitrary dimension, possibly having discontinuities and/or critical sets, and show that under some general conditions they admit an induced Markov tower structure. Moreover, the decay of the return time function can be controlled in terms of the time generic points need to achieve some uniform expanding behavior. As a consequence we obtain some rates for the decay of correlations of those maps and conditions for the validity of the Central Limit Theorem.     

Semicocycle extensions and the stroboscopic property

We indicate a large class of almost 1-1 extensions over minimal systems, which do not possess the stroboscopic property, as defined by Misiurewicz and studied by Jimenez and Snoha [Topology Appl. 129 (2003) 301-316]. Sturmian flows and all Toeplitz flows belong to this class. This generalizes a theorem of [Topology Appl. 129 (2003) 301-316] for Sturmian flows. Our result allows to easily construct minimal weakly mixing systems without the stroboscopic property, which answers in the negative a question posed in [Topology Appl. 129 (2003) 301-316]. Finally we prove that even the strong stroboscopic property does not imply the stroboscopic property for induced (first return time) systems.     

Topology of dynamical systems in finite groups and number theory

We study and develop a very new object introduced by V.I. Arnold: a monad is a triple consisting of a finite set, a map from that finite set to itself and the monad graph which is the directed graph whose vertices are the elements of the finite set and whose arrows lead each vertex to its image (by the map). We consider the case in which the finite set entering in the monad definition is a finite group G and the map is the Frobenius map, for some k∈Z. We study the Frobenius dynamical system defined by the iteration of the monad fk, and also study the combinatorics and topology (i.e., the discrete invariants) of the monad graph. Our study provides useful information about several structures on the group associated to the monad graph. So, for example, several properties of the quadratic residues of finite commutative groups can be obtained in terms of the graph of the Frobenius monad .     

Central sets and substitutive dynamical systems

In this paper we establish a new connection between central sets and the strong coincidence conjecture   for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of N$\mathbb{N}$ possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone–?ech compactification βN$\beta \mathbb{N}$. This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture  . The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone–?ech compactification of N$\mathbb{N}$.     

On minimal decomposition of p-adic polynomial dynamical systems

A polynomial of degree ?2 with coefficients in the ring of p-adic numbers Zp is studied as a dynamical system on Zp. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on Z₂, we exhibit all its minimal subsystems.     

Digitization of nonautonomous control systems

Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals.     

Digitization of nonautonomous control systems

Methods of the theory of nonautonomous differential equations are used to study the extent to which the properties of local null controllability and local feedback stabilizability are preserved when a control system with time-varying coefficients is digitized, e.g., approximated by piecewise autonomous systems on small time subintervals.     

Partial linearization of differentiable systems

For any given differentiable dynamical system with discrete time on a compact Riemannian manifold of finite dimension d, along its orbits, in this paper, we introduce a type of global linearization under natural moving orthonormal q-frames of the tangent space of the base manifold, 1 ≤ q ≤ d. As an application, we give a new proof of the Oseledec-Million??ikov Multiplicative Ergodic Theorem (MET) for a ergodic smooth systems (M, ν; φ).