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Introduction to the modern theory of Dynamical Systems
... The purpose of this paper is to present a convergence analysis for Carleman linearization without the dissipative condition. One of our key contributions is the identification of another type of condition, called the no-resonance condition, motivated by the theory of dynamical systems [25]. Roughly speaking, this condition states that any one eigenvalue can not coincide with an integer combination of other eigenvalues. ...
... Therefore, for non-dissipative systems, ∆ in Eq. (3) plays a similar role to |σ| in the convergence rate in the dissipative regime Eq. (2). Resonance properties play a crucial role in the long-time behavior of dynamical systems, including phenomena such as bifurcation, chaos, and energy transport [10,25]. Important examples include Fermi-Pasta-Ulam (FPU) chains [34], the nonlinear Schrödinger equation [4], and the Korteweg-de Vries (KdV) equation [18]. ...
... In the theory of dynamical systems [25], the first part of the assumption comes from stability consideration, but we do not rule out non-hyperbolic equilibrium points. Meanwhile, the condition in Eq. (9) is known as the no-resonance condition. ...
In this paper, we explore the embedding of nonlinear dynamical systems into linear ordinary differential equations (ODEs) via the Carleman linearization method. Under dissipative conditions, numerous previous works have established rigorous error bounds and linear convergence for Carleman linearization, which have facilitated the identification of quantum advantages in simulating large-scale dynamical systems. Our analysis extends these findings by exploring error bounds beyond the traditional dissipative condition, thereby broadening the scope of quantum computational benefits to a new class of dynamical regimes. This novel regime is defined by a resonance condition, and we prove how this resonance condition leads to a linear convergence with respect to the truncation level $N$ in Carleman linearization. We support our theoretical advancements with numerical experiments on a variety of models, including the Burgers' equation, Fermi-Pasta-Ulam (FPU) chains, and the Korteweg-de Vries (KdV) equations, to validate our analysis and demonstrate the practical implications.
... The aim of this paper is the investigation of the topological dynamics of automorphisms on a 6-dimensional torus generated by an integer symplectic transformation of R 6 in the case of partial hyperbolicity. The hyperbolic case is rather well understood already [1,10,26,18]. ...
... Since the linear mapping L A : x → Ax maps the subgroup Z n onto itself, such a matrix generates a diffeomorphism f A of the torus T n called an automorphism of the torus [1,2,10]. Topological properties of such maps are the classical object of research (see, for example, [1,10,18,26]). Because this toral automorphism also preserves the standard volume element dx 1 ∧ · · · ∧ dx n on the torus carried over from R n , its ergodic properties have also been the subject of research [5,12,28]. ...
... [21,Ch. XIX], [19, ch. 8 & 9], [20], and [30]). ...
... So we have two different representations of trajectories, σ in (15) and α tr (σ) in (17), this second representation being closer to the time-evolution law of the theory of dynamical systems [20]. Notice that α tr (σ) is a function defined by parts on the timeline abstraction α tl (σ) of the trajectory σ so the time-evolution law α tr (σ) is not simpler that the trajectory σ to reason upon, in particular because the timeline information is abstracted away. ...
We formalize the semantics of hybrid systems as sets of hybrid trajectories, including those generated by an hybrid transition system. We study the abstraction of hybrid trajectory semantics for verification, static analysis, and refinement. We mainly consider abstractions of hybrid semantics which establish a correspondence between trajectories derived from a correspondence between states such as homomorphisms, simulations, bisimulations, and preservations with progress. We also consider abstractions that cannot be defined stepwise like discretization. All these abstractions are Galois connections between concrete and abstract hybrid trajectory or discrete trace semantics. In contrast to semantic based abstractions, we investigate the problematic trace-based composition of abstractions.
... The study of MAP systems dynamics is also related to the areas of Dynamical Systems (e.g. [20]) and Statistical Physics (e.g. [21,22,23]). ...
A large number of real and abstract systems involve the transformation of some basic resource into respective products under the action of multiple processing agents, giving rise to multiple-agent production systems (MAP). At each discrete time instant, for each agent, a fraction of the resources is assumed to be kept, forwarded to other agents, or converted into work/wasted. The present work describes a systematic study of nine basic MAP architectures subdivided into two main groups, namely parallel and sequential distribution of resources from a single respective source. Several types of interconnec-tions among the involved processing agents are also considered. The resulting MAP architectures are studied in terms of the total amount of work, the dispersion of the resources (states) among the agents, and the transition times from the start of operation until the respective steady state. Several interesting results are obtained and discussed, including the observation that some of the parallel designs have were able to yield maximum work and minimum states dispersion, achieved at the expense of the transition time and use of several interconnec-tions between the source and the agents. The performance obtained for the sequential designs indicated that relatively high performance can be obtained for some specific cases.
... Therefore, weak perturbations of the system from external sources may 40 cause a state of change due to the critical point, where the system will evolve into the 41 new stable or unstable state. The proposed mechanism of control ensures that the 42 system remains stable, in the Lyapunov sense [9], but with uniquely different stable 43 states. 44 Small interactions within the components of a SOC system are not significantly 45 large with respect to the scale of each element, but they contribute to the global 46 critical state due to nonlinear behaviour. ...
The control of extensive complex biological systems is considered to depend on feedback mechanisms. Reduced systems modelling has been effective to describe these mechanisms, but this approach does not sufficiently encompass the required complexity that is needed to understand how localised control in a biological system can provide global stable states. Self-Organised Criticality (SOC) is a characteristic property of locally interacting physical systems which readily emerges from changes to its dynamic state due to small nonlinear perturbations. Small changes in the local states, or in local interactions, can greatly affect the total system state of critical systems. It has long been conjectured that SOC is cardinal to biological systems that show similar critical dynamics and also may exhibit near power-law relations. Rate Control of Chaos (RCC) provides a suitable robust mechanism to generate SOC systems which operates at the edge of chaos. The bio-inspired RCC method requires only local instantaneous knowledge of some of the variables of the system, and is capable of adapting to local perturbations. Importantly, connected RCC controlled oscillators can maintain global multi-stable states, and domains with power-law relations may emerge. The network of oscillators deterministically stabilises into different orbits for different perturbations and the relation between the perturbation and amplitude can show exponential and power-law correlations. This is representative of a basic mechanism of protein production and control, that underlies complex processes such as homeostasis. Providing feedback from the global state, the total system dynamic behaviour can be boosted or reduced. Controlled SOC can provide much greater understanding of biological control mechanisms, that are based on distributed local producers, remote consumers of biological resources, with globally defined control.
Author summary
Using a nonlinear control method inspired by enzymatic control, which is capable of stabilising chaotic systems into periodic orbits or steady-states, it is shown that a controlled system can be created that is scale-free and in a critical state. This means that the system can easily move from one stable orbit to another using only a small local perturbation. Such a system is known as self-organised criticality, and is shown in this system to be deterministic. Using a known perturbation, it will result in a scale-free response of the system that can be in a power law relation. It has been conjectured that biosystems are in a self-organised critical state, and these models show that this is a suitable approach to allow local systems to control a global state, such as homeostatic control. The underlying principle is based on rate control of chaos, and can be used to understand how biosystems can use localised control to ensure stability at different dynamic scales without supervising mechanisms.
... Let T be an orientation-preserving circle homeomorphism. If the rotation number ρ of T is rational, then every orbit of T converges to a periodic orbit (see for instance [70,Proposition 11.2.2]). By Theorem 1 in [38], the oscillating sequence (c n ) n∈N is orthogonal to the system (S 1 , T ). ...
In this thesis, we solve Fuglede's conjecture on the field of p-adic numbers, and study some randomness and the oscillating properties of sequences related to Sarnak's conjecture. In the first part, we first prove Fuglede's conjecture for compact open sets in the field Q_p which states that a compact open set in Q_p is a spectral set if and only if it tiles Q_p by translation. It is also proved that a compact open set is a spectral set (or a tile) if and only if it is p-homogeneous. We characterize spectral sets in Z/p^n Z (p>1 prime, n>0 integer) by tiling property and also by homogeneity. Finally, we prove Fuglede's conjecture in Q_p without the assumption of compact open sets and also show that the spectral sets (or tiles) are the sets which differ by null sets from compact open sets. In the second part, we first give several equivalent definitions of oscillating sequences in terms of their disjointness from different dynamical systems on tori. Then we define Chowla property and Sarnak property for numerical sequences taking values 0 or complex numbers of modulus 1. We prove that Chowla property implies Sarnak property. It is also proved that for Lebesgue almost every b>1, the sequence (e^{2 pi b^n})_{n in N} shares Chowla property and consequently is orthogonal to all topological dynamical systems of zero entropy. We also discuss whether the samples of a given random sequence have almost surely Chowla property. Some dependent random sequences having almost surely Chowla property are constructed
Based on the Schweiger’s smooth fibered approach and the related Bernoulli shift transformation scheme, we prove the ergodicity of the two-dimensional Boole-type transformations. New multidimensional Boole-type transformations invariant with respect to the Lebesgue measure and their ergodicity properties are also discussed.
This work offers a comprehensive and fresh perspective on the bonding evolution theory (BET) framework, originally proposed by Silvi and collaborators [X. Krokidis, S. Noury and B. Silvi, Characterization of elementary chemical processes by catastrophe theory, J. Phys. Chem. A, 1997, 101, 7277-7282]. By underscoring Thom's foundational work, we identify the parametric function characterizing bonding events along a reaction pathway through a three-step sequence to establish such association rigorously, namely: (a) computing the determinant of the Hessian matrix at all potentially degenerate critical points, (b) computing the relative distance between these points, and (c) assigning the unfolding based on these computations and considering the maximum number of critical points for each unfolding. In-depth examination of the ammonia inversion and the dissociation of ethane and ammonia borane molecules yields a striking discovery: no elliptic umbilic flag is detected along the reactive coordinate for any of the systems, contradicting previous reports. Our findings indicate that the core mechanisms of these chemical reactions can be understood using only two folds, the simplest polynomial of Thom's theory, leading to considerable simplification. In contrast to previous reports, no signatures of the elliptic umbilic unfolding were detected in any of the systems examined. This finding dramatically simplifies the topological rationalization of electron rearrangements within the BET framework, opening new approaches for investigating complex reactions.
Amir et al. (CPM 2017) introduce the approximate string cover problem (ACP) motivated by applications including molecular biology, coding, automata theory, formal language theory and combinatorics. A cover of a string T is a string C for which every letter of T lies within some occurrence of C. The input of the ACP consists of a string T and the goal is to find a string C of length less than the length of T that covers a string \(T'\), which is as close to T as possible (under some predefined distance). Amir et al. study this problem for the Hamming distance and show that it is NP-hard.
In this paper we continue the work of Amir et al. and show the following results for the cover length relaxation of the ACP. After observing that the NP-hardness proof by Amir et al. (CPM 2017, TCS 2019) suffers from several lapses, we propose an amendment to the proof. We then introduce an approximation algorithm for a variant of the ACP, in which we aim to maximize the length of the input string minus the distance to the string covered by the approximate cover returned by the algorithm. This problem is naturally as hard as the ACP. We prove an asymptotic approximation ratio of \(\mathcal {O}(\sqrt{|T|})\), where |T| is the size of the input string. Finally, we present an FPT algorithm with respect to the alphabet size and the size of the cover based on a dynamic programming framework.
In this article, we consider a closed rank one \(C^\infty \) Riemannian manifold M without focal points and its universal cover X. Let \(b_t(x)\) be the Riemannian volume of the ball of radius \(t>0\) around \(x\in X,\) and h the topological entropy of the geodesic flow. We obtain the following Margulis-type asymptotic estimates $$\begin{aligned} \lim _{t\rightarrow \infty }b_t(x)/\frac{e^{ht}}{h}=c(x) \end{aligned}$$for some continuous function \(c: X\rightarrow {{{\mathbb {R}}}}.\) We prove that the Margulis function c(x) is in fact \(C^1.\) The result also holds for a class of manifolds without conjugate points, including all surfaces of genus at least 2 without conjugate points. If M is a rank one surface without focal points, we show that c(x) is constant if and only if M has constant negative curvature. We also obtain a rigidity result related to the flip invariance of the Patterson–Sullivan measure. These rigidity results are new even in the nonpositive curvature case.
Three classes of nonautonomous bifurcations are introduced for ordinary differential equations, namely attractor bifurcation, solution bifurcation and bifurcation of minimal sets. We review sufficient conditions for such bifurcations and present suitable examples. Rate-induced tipping is understood as a special case of solution bifurcation. The chapter closes with some remarks on nonautonomous Hopf bifurcations.KeywordsAttractor bifurcationSolution bifurcationBifurcation of minimal setsPullback attractorShovel bifurcationRate-induced tippingNonautonomous Hopf bifurcation
This work reveals an underlying correlation between the topology and energetic features of matter configurations/rearrangements by exploiting two topological concepts, namely, structural stability and persistency, leading thus to a model capable of predicting activation energies at 0 K. This finding provides some answers to the difficulties of applying Thom's functions for extracting energetic information of rate processes, which has been a limitation for exact, biological, and technological sciences. A linear relationship between the experimental barriers of 17 chemical reactions and both concepts was found by studying these systems' topography along the intrinsic reaction coordinate. Such a procedure led to the model , which accurately predicts the activation energy in reacting systems involving organic and organometallic compounds under different conditions, e.g., the gas-phase, solvent media, and temperature. This function was further recalibrated to enhance its predicting capabilities, generating the equation for this procedure, characterized by a squared Pearson correlation coefficient (r2 = 0.9774) 1.1 times higher. Surprisingly, no improvement was observed.
Extensive research has previously been carried out on the existence of strange attractors in 2D piecewise linear maps, including the renowned Lozi map. However, the rigorous analysis of strange attractors in 2D nonlinear maps remains an underdeveloped area of study. In this paper, we introduce a 2D map with a single piecewise-smooth nonlinear function that is suitable for analytic studies of its strange attractor. This 2D piecewise-smooth nonlinear map represents a broad range of chaotic maps, including a hybrid Lozi-Hénon map and the Belykh map. To prove the existence of a strange attractor in the 2D map, we (i) construct its trapping region that contains all limit sets and (ii) demonstrate that the invariant set's trajectories have negative and positive Lyapunov exponents. We develop an invariant cone approach to establish the latter property, which involves constructing expanding and contracting cones as bounds for the eigenvectors of the variational equations along the chaotic trajectories. We apply our approach to analyze the chaotic hybrid Lozi-Hénon map, the original Lozi map, and the Belykh map.
We introduce a family of stochastic models motivated by the study of nonequilibrium steady states of fluid equations. These models decompose the deterministic dynamics of interest into fundamental building blocks, i.e., minimal vector fields preserving some fundamental aspects of the original dynamics. Randomness is injected by sequentially following each vector field for a random amount of time. We show under general conditions that these random dynamics possess a unique, ergodic invariant measure and converge almost surely to the original, deterministic model in the small noise limit. We apply our construction to the Lorenz-96 equations, often used in studies of chaos and data assimilation, and Galerkin approximations of the 2D Euler and Navier–Stokes equations. An interesting feature of the models developed is that they apply directly to the conservative dynamics and not just those with excitation and dissipation.
A highly nonlinear lattice p-Laplacian equation driven by superlinear noise is considered. By using an appropriate stopping time technique and the dissipativeness of the nonlinear drift terms, we establish the global existence and uniqueness of the solutions in \(C\bigg ([\tau ,\infty ), L^2(\Omega , \ell ^2)\bigg )\cap L^p\bigg (\Omega , L_{\text {loc}}^p((\tau ,\infty ), \ell ^p )\bigg ) \) for any \(p>2\) when the coefficient of the noise has a superlinear growth order \(q\in [2,p)\). By the theory of mean random dynamical systems recently developed in Kloeden and Lorenz (J Differ Equ 253:1422–1438, 2012) and Wang (J Dyn Differ Equ 31:2177–2204, 2019), we prove that the nonautonomous system has a unique mean random attractor in the Bochner space \(L^2(\Omega , \ell ^2)\). When the drift and diffusion terms satisfy certain conditions, we show that the autonomous system has a unique, ergodic, mixing, and stable invariant probability measure in \(\ell ^2\). The idea of uniform tail-estimates is employed to establish the tightness of a family of distribution laws of the solutions in order to overcome the lack of compactness in infinite lattice as well as the infinite-dimensionalness of \(\ell ^2\). This work deepens and extends the results in Wang and Wang (Stoch Process Appl 130:7431–7462, 2020) where the coefficient of the noise grows linearly rather than superlinearly.
We formalize the semantics of hybrid systems as sets of hybrid trajectories, including those generated by an hybrid transition system. We study the abstraction of hybrid trajectory semantics for verification, static analysis, and refinement. We mainly consider abstractions of hybrid semantics which establish a correspondence between trajectories derived from a correspondence between states such as homomorphisms, simulations, bisimulations, and preservations with progress. We also consider abstractions that cannot be defined stepwise like discretization. All these abstractions are Galois connections between concrete and abstract hybrid trajectory or discrete trace semantics. In contrast to semantic based abstractions, we investigate the problematic trace-based composition of abstractions.KeywordsHybrid systemsSemanticsAbstractionHomomorphismSimulationsBisimulationsPreservationsProgressDiscretizationVerificationRefinementAbstract interpretationGalois connectionGalois relationLogical relation
In the 1970’s, Belinskii, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map’s dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright–Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright–Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows.
The approximation of trace(f(Ω)), where f is a function of a symmetric matrix Ω, can be challenging when Ω is exceedingly large. In such a case even the partial Lanczos decomposition of Ω is computationally demanding and the stochastic method investigated by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996) is preferred. Moreover, in the last years, a partial global Lanczos method has been shown to reduce CPU time with respect to partial Lanczos decomposition. In this paper we review these techniques, treating them under the unifying theory of measure theory and Gaussian integration. This allows generalizing the stochastic approach, proposing a block version that collects a set of random vectors in a rectangular matrix, in a similar fashion to the partial global Lanczos method. We show that the results of this technique converge quickly to the same approximation provided by Bai et al. (J. Comput. Appl. Math. 74:71–89, 1996), while the block approach can leverage the same computational advantages as the partial global Lanczos. Numerical results for the computation of the Von Neumann entropy of complex networks prove the robustness and efficiency of the proposed block stochastic method.
For a C1+α diffeomorphism f preserving a hyperbolic ergodic SRB measure μ, Katok's remarkable results assert that μ can be approximated by a sequence of hyperbolic sets {Λn}n≥1. In this paper, we prove that the Hausdorff dimension for Λn on the unstable manifold tends to the dimension of the unstable manifold. Furthermore, if the stable direction is one dimension, then the Hausdorff dimension of μ can be approximated by the Hausdorff dimension of Λn.
To establish these results, we utilize the u-Gibbs property of the conditional measure of the equilibrium measure of −ψs(⋅,fn) and the properties of the uniformly hyperbolic dynamical systems.
We study the Bowen topological entropy of generic and irregular points for certain dynamical systems. We define the topological entropy of noncompact sets for flows, analogous to Bowen's definition. We show that this entropy coincides with the Bowen topological entropy of the time-1 map on any set. We also show a Bowen's inequality for flows; namely, that the metric entropy with respect to every invariant measure for a continuous flow is an upper bound for the topological entropy of the set of generic points with respect to the same measure, and the equality is always true if the measure is ergodic. We propose a definition of almost specification property for flows and prove that a continuous flow has the almost specification property if the time-1 map satisfies this property. Using Bowen's inequality for flows, we show that every continuous flow with the almost specification property is saturated, extending a result of Meson and Vericat in [22]. Under the same hypotheses, we extend a result of Thompson on the entropy of irregular points in [29].
Relationships between a chaotic behaviour and closely related properties of topological transitivity, sensitivity to initial conditions, density of closed orbits of homeomorphism groups and their countable products are investigated. We provide a large number of new examples of chaotic groups of homeomorphisms of countable products of various metrizable topological spaces, including infinite-dimensional topological manifolds, whose factors can be as noncompact surfaces, so triangulable closed manifolds of an arbitrary dimension.
Let f : M → M be a C1+α local diffeomorphism/diffeomorphism of a compact Riemannian manifold M and μ be an expanding/hyperbolic ergodic f-invariant Borel probability measure on M. Assume f is average conformal expanding/hyperbolic on the support set W of μ and W is locally maximal. For any subset A ⊂ W with small entropy or dimension, we investigate the topological entropy and Hausdorff dimensions of the A-exceptional set and the limit A-exceptional set.
Given a C2− Anosov diffeomorphism f:M→M, we prove that the Jacobian condition Jfn(p)=1, for every point p such that fn(p)=p, implies transitivity. As application in the celebrated theory of Sinai-Ruelle-Bowen, this result allows us to state a classical theorem of Livsic-Sinai without directly assuming transitivity as a general hypothesis. A special consequence of our result is that every C2-Anosov diffeomorphism, for which every point is regular, is indeed transitive.
We prove that a partially hyperbolic attractor for a C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document} vector field with two dimensional center supports an SRB measure. In addition, we show that if the vector field is C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2$$\end{document}, and the center bundle admits the sectionally expanding condition w.r.t. Gibbs u-states, then the attractor can only support finitely many SRB/physical measures whose basins cover Lebesgue almost all points of the topological basin. The proof of these results has to deal with the difficulties which do not occur in the case of diffeomorphisms.
We prove a topological version of the classical shadowing theorem in differentiable dynamics (Encyclopedia of Mathematics and its Applications, vol. 54 (1995) (Cambridge: Cambridge University Press)). We use it to unify some stability results in literature as the GH-topological, Hartman and Walters stability theoremsDiscrete Contin. Dyn. Syst.37(7) (2017) 3531–3544; An. Acad. Brasil Ci.40 (1968) 263–266; Am. J. Math.91 (1969) 363–367; Proceedings of Conference, North Dakota State University, Fargo, N.D., 1977 (1978) (Berlin: Springer) pp. 231–244).
In this paper, we study the periods of interval exchange transformations. First, we characterize the periods of interval exchange transformations with one discontinuity. In particular, we prove that there is no forcing between periods of maps with two branches of continuity. This characterization is a partial solution to a problem by Misiurewicz. In particular, a periodic structure is not possible for a family of the maps with one point of discontinuity in which the monotonicity changes. Second, we study a similar problem for interval exchange transformations with two discontinuities. Here we classify these maps in several classes such that two maps in the same class have the same periods. Finally, we study the set of periods for two categories, obtaining partial results that prove that the characterization of the periods in each class is not an easy problem.
It is known that sectional-hyperbolic attracting sets, for a $C^2$ flow on a finite dimensional compact manifold, have at most finitely many ergodic physical invariant probability measures. We prove an upper bound for the number of distinct ergodic physical measures supported on a connected singular-hyperbolic attracting set for a $3$-flow. This bound depends only on the number of Lorenz-like equilibria contained in the attracting set. Examples of singular-hyperbolic attracting sets are provided showing that the bound is sharp.
According to the recurrent frequency, many levels of recurrent points are found such as periodic points, almost periodic points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points. In this paper, we consider symbolic dynamics and show the existence of six refined levels between Banach recurrence and general recurrence. Despite the fact that these refined levels are all null-measure under any invariant measure, we further show they carry strong topological complexity. Each refined level of recurrent points is dense in the whole space and contains an uncountable distributionally chaotic subset. For a wide range of dynamical systems such as expansive systems with the shadowing property, we also show the distributional chaos of the points that are recurrent but not Banach recurrent.
This paper analyses the stability of cycles within a heteroclinic network lying in a three-dimensional manifold formed by six cycles, for a one-parameter model developed in the context of game theory. We show the asymptotic stability of the network for a range of parameter values compatible with the existence of an interior equilibrium and we describe an asymptotic technique to decide which cycle (within the network) is visible in numerics. The technique consists of reducing the relevant dynamics to a suitable one-dimensional map, the so called \emph{projective map}. Stability of the fixed points of the projective map determines the stability of the associated cycles. The description of this new asymptotic approach is applicable to more general types of networks and is potentially useful in computational dynamics.
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