Erratum to “Uniform convergence and chaotic behavior” [Nonlinear Anal. TMA 65 (4) (2006) 933–937]

We present counterexamples which show that the uniform limit of topologically transitive maps on a compact metric space is not necessarily topologically transitive. We also give stronger conditions that ensure the inheritance of topological transitivity for the case of uniform convergence and other situations.     

Entropy and ergodic probability for differentiable dynamical systems and their bundle extensions

We answer a problem of Liao [S.T. Liao, Standard systems of differential equations and obstruction sets—from linearity to perturbations, in: System Researches, Proceedings Dedicated to the 85th Anniversary of Qian Xue-Sen, Zhejiang Education Press, Hangzhou, China, 1996, pp. 279-290 (in Chinese)]: A C¹ vector field or a C¹ diffeomorphism on an n-dimensional manifold has equal entropy with that of its bundle extensions. We also prove that each ergodic probability with simple Lyapunov spectrum has at most n²n! covering probabilities on each bundle extension.     

Spatial complexity in multi-layer cellular neural networks

This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input.     

Topological entropy for set valued maps

Any continuous map T on a compact metric space X induces in a natural way a continuous map on the space K(X) of all non-empty compact subsets of X. Let T be a homeomorphism on the interval or on the circle. It is proved that the topological entropy of the induced set valued map is zero or infinity. Moreover, the topological entropy of is zero, where C(X) denotes the space of all non-empty compact and connected subsets of X. For general continuous maps on compact metric spaces these results are not valid.     

Polygonal approximation of flows

The analysis of the qualitative behavior of flows generated by ordinary differential equations often requires quantitative information beyond numerical simulation which can be difficult to obtain analytically. In this paper we present a computational scheme designed to capture qualitative information using ideas from the Conley index theory. Specifically we design an combinatorial multivalued approximation from a simplicial decomposition of the phase space, which can be used to extract isolating blocks for isolated invariant sets. These isolating blocks can be computed rigorously to provide computer-assisted proofs. We also obtain local conditions on the underlying simplicial approximation that guarantees that the chain recurrent set can be well-approximated.     

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.     

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.     

On the centre and the set of ω-limit points of continuous maps on dendrites

It is well known that for dynamical systems generated by continuous maps of a graph, the centre of the dynamical system is a subset of the set of ω-limit points.In this paper we provide an example of a continuous self-map f₁ of a dendrite such that ω(f₁) is a proper subset of C(f₁).The second example is a continuous self-map f₂ of a dendrite having a strictly increasing sequence of ω-limit sets which is not contained in any maximal one. Again, this is impossible for continuous maps on graphs.     

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.     

Choquet simplices as spaces of invariant probability measures on post-critical sets

A well-known consequence of the ergodic decomposition theorem is that the space of invariant probability measures of a topological dynamical system, endowed with the weak^∗ topology, is a non-empty metrizable Choquet simplex. We show that every non-empty metrizable Choquet simplex arises as the space of invariant probability measures on the post-critical set of a logistic map. Here, the post-critical set of a logistic map is the ω-limit set of its unique critical point. In fact we show the logistic map f can be taken in such a way that its post-critical set is a Cantor set where f   is minimal, and such that each invariant probability measure on this set has zero Lyapunov exponent, and is an equilibrium state for the potential −ln|f^′|$-\ln |f^{\prime }|$.     

Ergodic Optimization for Renewal Type Shifts

We consider a class of countable Markov shifts and a locally H?lder potential φ. We prove that the existence of φ-optimal measures is closely related to the behaviour of the pressure function t → P(tφ). Using a Theorem by Sarig it is possible to prove that there exists a critical value t _c ∈ (0, ∞] such that for t < t _c the pressure is analytic and for t > t _c is linear. We prove that if t _c is finite, then there are no φ-optimal measures, and if it is infinite, then φ-optimal measures do exist. The author was partially supported by FCT/POCTI/FEDER and the grant SFRH/BPD/21927/2005.     

Sharkovsky’s program for the classification of triangular maps is almost completed

For a continuous map of the interval, there are more than 50 conditions characterizing zero topological entropy. Some are applicable to the class of triangular maps (x,y)?(f(x),g_x(y)) of the square, but only a few of them are equivalent in this more general setting. In 1989, A.N. Sharkovsky posed the problem of proving or disproving all possible implications between them. During last 20 years, 32 conditions were considered, and most of the work was done. Only 45 relations out of 992 remained not clear. In this paper we give a survey of known results, provide two new examples disproving another 26 possible implications, and spell out the remaining 19 open problems; all but one concern distributional chaos.     

Quadratic Lyapunov functions and nonuniform exponential dichotomies

For nonautonomous linear equations x^′=A(t)x, we show how to characterize completely nonuniform exponential dichotomies using quadratic Lyapunov functions. The characterization can be expressed in terms of inequalities between matrices. In particular, we obtain converse theorems, by constructing explicitly quadratic Lyapunov functions for each nonuniform exponential dichotomy. We note that the nonuniform exponential dichotomies include as a very special case (uniform) exponential dichotomies. In particular, we recover in a very simple manner a complete characterization of uniform exponential dichotomies in terms of quadratic Lyapunov functions. We emphasize that our approach is new even in the uniform case.Furthermore, we show that the instability of a nonuniform exponential dichotomy persists under sufficiently small perturbations. The proof uses quadratic Lyapunov functions, and in particular avoids the use of invariant unstable manifolds which, to the best of our knowledge, are not known to exist in this general setting.     

Nonuniform exponential contractions and Lyapunov sequences

For a nonautonomous dynamics with discrete time obtained from the product of linear operators, we show that a nonuniform exponential contraction can be completely characterized in terms of what we call strict Lyapunov sequences. We note that nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics. We also obtain “inverse theorems” that give explicitly strict Lyapunov sequences for each nonuniform exponential contraction. Essentially, the Lyapunov sequences are obtained in terms of what are usually called Lyapunov norms, that is, norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform. We also show how the characterization of nonuniform exponential contractions in terms of quadratic Lyapunov sequences can be used to establish in a very simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations. Moreover, we describe an appropriate version of our results in the context of ergodic theory showing that the existence of an eventually strict Lyapunov function implies that all Lyapunov exponents are negative almost everywhere.