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  • Persistence and Expansivity through Pointwise Dynamics
    Dyn. Syst. (IF 0.986) Pub Date : 2020-10-12
    Abdul Gaffar Khan; Tarun Das

    Using the notion of topologically stable points it is proved that every equicontinuous pointwise topologically stable homeomorphism of a compact metric space is persistent. Also, using the notion of strong topologically stable points of a Borel probability measure, the above result is proved for homeomorphisms of perfect compact metric spaces. Further, it is shown that any homeomorphism of [ 0 , 1

  • Equilibrium states of intermediate entropies
    Dyn. Syst. (IF 0.986) Pub Date : 2020-09-21
    Peng Sun

    We explore an approach to the conjecture of Katok on intermediate entropies based on uniqueness of equilibrium states, provided the entropy function is upper semi-continuous. As an application, we prove Katok's conjecture for Mañé diffeomorphisms.

  • Amorphic complexity can take any nonnegative value in general metric spaces
    Dyn. Syst. (IF 0.986) Pub Date : 2020-08-10
    Marcin Kulczycki

    This note proves that for every real nonnegative number there exists a noncomplete metric space and a map on it that has amorphic complexity equal to this number.

  • Family of chaotic maps from game theory
    Dyn. Syst. (IF 0.986) Pub Date : 2020-07-20
    Thiparat Chotibut; Fryderyk Falniowski; Michał Misiurewicz; Georgios Piliouras

    From a two-agent, two-strategy congestion game where both agents apply the multiplicative weights update algorithm, we obtain a two-parameter family of maps of the unit square to itself. Interesting dynamics arise on the invariant diagonal, on which a two-parameter family of bimodal interval maps exhibits periodic orbits and chaos. While the fixed point b corresponding to a Nash equilibrium of such

  • On the topological entropy of induced transformations for free semigroup action
    Dyn. Syst. (IF 0.986) Pub Date : 2020-07-20
    Yong Ji; Yunping Wang

    The aim of this paper is to reveal the relationship between the topological entropy of a free semigroup action and that of its induced transformations. More precisely, we prove that the topological entropy of a free semigroup action ( X , F ) vanishes if and only if the topological entropy of its induced system ( M ( X ) , F ) is zero; if the topological entropy of ( X , F ) is positive, then that

  • Relative Equilibria, Stability and Bifurcations in Hamiltonian Galactic-Tidal Models
    Dyn. Syst. (IF 0.986) Pub Date : 2020-07-10
    J. L. Zapata; F. Crespo; S. Ferrer

    We study special solutions of Galactic-Tidal models and the existence, stability and bifurcations of relative equilibria. Precisely, we found four equilibria and rectilinear, as well as circular solutions, previous to any manipulation of the original system. Moreover, by averaging of the Keplerian energy and making use of Reeb's theorem, we find four periodic orbits. The stability of these relative

  • Asymptotic behavior of stochastic heat equations in materials with memory on thin domains
    Dyn. Syst. (IF 0.986) Pub Date : 2020-07-09
    Ji Shu; Hui Li; Xin Huang; Jian Zhang

    This paper deals with the dynamical behavior of a stochastic integro-differential equation driven by additive noise defined on thin domains. We prove the existence and uniqueness of random attractors for the equation in an ( n + 1 ) -dimensional narrow domain. Due to the fact that the memory term includes the whole past history of the phenomenon, we are not able to prove compactness of the generated

  • On the regularity and approximation of invariant densities for random continued fractions
    Dyn. Syst. (IF 0.986) Pub Date : 2020-06-30
    Toby Taylor-Crush

    We study perturbations of random dynamical systems whose associated transfer operators admit a uniform spectral gap. We provide a k t h -order approximation for the invariant density of the associated random dynamical system. We apply our result to random continued fractions.

  • A Central limit theorem for the Birkhoff sum of the Riemann zeta-function over a Boolean type transformation
    Dyn. Syst. (IF 0.986) Pub Date : 2020-06-12
    Tanja I. Schindler

    We prove a central limit theorem for the real and imaginary part and the absolute value of the Riemann zeta-function sampled along a vertical line in the critical strip with respect to an ergodic transformation similar to the Boolean transformation. This result complements a result by Steuding who has proven a strong law of large numbers for the same system. As a side result we state a general central

  • Global dynamics of the Maxwell-Bloch system with invariant algebraic surfaces
    Dyn. Syst. (IF 0.986) Pub Date : 2020-06-01
    F.S. Dias; Claudia Valls

    In this paper by using the Poincaré compactification in R3 we make a global analysis of the Maxwell-Bloch system x˙=y,y˙=cy+dx−xzz˙=bz+xy with (x,y,z)∈R3, b,c and d∈R. We give the complete description of its dynamics on the sphere at infinity. For some values of the the parameters, this system has first integrals and invariant algebraic surfaces. For these sets we provide the global phase portraits

  • Decomposition of stochastic flow and an averaging principle for slow perturbations
    Dyn. Syst. (IF 0.986) Pub Date : 2020-05-18
    Diego Sebastian Ledesma; Fabiano Borges da Silva

    We use decomposition of stochastic flow to obtain components that represent the slow and fast motion of a given perturbed stochastic differential equation. Then, through a special metric which works well with stochastic calculation tools, Lipschitz condition for vector fields and an averaging principle we obtain an approximation for the slow motion. Finally, we get an estimate for the approximation

  • Orbit Growth of Dyck and Motzkin Shifts via Artin-Mazur Zeta Function
    Dyn. Syst. (IF 0.986) Pub Date : 2020-05-18
    Azmeer Nordin; Mohd Salmi MdNoorani; Syahida Che Dzul-Kifli

    For a discrete dynamical system, the prime orbit and Mertens' orbit counting functions indicate the growth of the closed orbits in the system in a certain way. In this paper, we prove the asymptotic behaviours of the counting functions for Dyck and Motzkin shifts. The proof relies on the analyticity and non-vanishing property of their Artin-Mazur zeta functions.

  • Geometry of symplectic partially hyperbolic automorphisms on 4-torus
    Dyn. Syst. (IF 0.986) Pub Date : 2020-05-04
    L.M. Lerman; K.N. Trifonov

    We study topological properties of automorphisms of 4-dimensional torus generated by integer symplectic matrices. The main classifying element is the structure of the topology of a foliation generated by unstable leaves of the automorphism. There are two different cases, transitive and decomposable ones. For both cases the classification is given.

  • Equicontinuity and Li-Yorke pairs of dendrite maps
    Dyn. Syst. (IF 0.986) Pub Date : 2020-04-22
    Ghassen Askri

    In this paper, relationships between equicontinuity of a dendrite map f on Λ(f), absence of Li-Yorke pairs, collection of minimal sets and regularly recurrent points are investigated.

  • Minimality of 5-adic polynomial dynamics
    Dyn. Syst. (IF 0.986) Pub Date : 2020-03-30
    Donggyun Kim; Youngwoo Kwon; Kyunghwan Song

    We characterize the dynamical systems consisting of the set of 5-adic integers and polynomial maps which consist of one minimal component.

  • On inverse shadowing
    Dyn. Syst. (IF 0.986) Pub Date : 2020-03-18
    Chris Good; Joel Mitchell; Joe Thomas

    We give a reformulation of the inverse shadowing property with respect to the class of all pseudo-orbits. This reformulation bears witness to the fact that the property is far stronger than might initially seem. We give some implications of this reformulation, in particular showing that systems with inverse shadowing are not sensitive. Finally we show that, on compact spaces, inverse shadowing is equivalent

  • Optimal quantization via dynamics
    Dyn. Syst. (IF 0.986) Pub Date : 2020-03-14
    Joseph Rosenblatt; Mrinal Kanti Roychowdhury

    Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using stationary processes arising in dynamical systems, followed by a discussion of the special cases of stationary processes: random processes and Diophantine processes

  • Dimensions in infinite iterated function systems consisting of bi-Lipschitz mappings
    Dyn. Syst. (IF 0.986) Pub Date : 2020-03-10
    Chih-Yung Chu; Sze-Man Ngai

    We study infinite iterated function systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focussing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set in terms of certain topological pressure. By

  • Weak Gibbs measures and equilibrium states
    Dyn. Syst. (IF 0.986) Pub Date : 2020-02-18
    Maria Carvalho; Sebastián A. Pérez

    We present a simple proof of the fairly general fact that an invariant right-weak Gibbs probability measure is an equilibrium state, using a generalization of the formula of Brin and Katok for the local metric entropy. This new formula also allows us to clarify the role of the topological pressure on the standard definition of the Gibbs property.

  • Simultaneous occurrence of sliding and crossing limit cycles in piecewise linear planar vector fields
    Dyn. Syst. (IF 0.986) Pub Date : 2020-02-10
    João L. Cardoso; Jaume Llibre; Douglas D. Novaes; Durval J. Tonon

    In the present study, we consider planar piecewise linear vector fields with two zones separated by the straight line x = 0. Our goal is to study the existence of simultaneous crossing and sliding limit cycles for such a class of vector fields. First, we provide a canonical form for these systems assuming that each linear system has centre, a real one for y<0 and a virtual one for y>0, and such that

  • A groupoid approach to C*-algebras associated with λ-graph systems and continuous orbit equivalence of subshifts
    Dyn. Syst. (IF 0.986) Pub Date : 2020-02-05
    Kengo Matsumoto

    A λ-graph system L is a labelled Bratteli diagram with shift operation. It is a generalized notion of finite labelled graph and presents a subshift. We will study continuous orbit equivalence of one-sided subshifts and topological conjugacy of two-sided subshifts from the view points of groupoids and C ∗ -algebras constructed from λ-graph systems.

  • Limsup is needed in the definitions of topological entropy via spanning or separation numbers
    Dyn. Syst. (IF 0.986) Pub Date : 2020-02-04
    Winfried Just; Ying Xin

    The notion of topological entropy can be conceptualized in terms of the number of forward trajectories that are distinguishable at resolution ϵ within T time units. It can then be formally defined as a limit of a limit superior that involves either covering numbers, or separation numbers, or spanning numbers. If covering numbers are used, the limit superior reduces to a limit. While it has been generally

  • Weighted upper metric mean dimension for amenable group actions
    Dyn. Syst. (IF 0.986) Pub Date : 2020-01-07
    Dingxuan Tang; Haiyan Wu; Zhiming Li

    In this paper, we formulate the notions of weighted upper metric mean dimensions and weighted upper measure-theoretic mean dimensions for amenable group actions. In particular, a variational principle for amenable group actions is presented. We also define weighted upper metric mean dimensions with respect to pseudo-orbits and establish their relation to weighted upper metric mean dimensions.

  • The measures of shadowable pseudo-orbits
    Dyn. Syst. (IF 0.986) Pub Date : 2019-12-28
    Kazuhiro Sakai; Naoya Sumi

    Let f be a continuous map of a compact metric space. In Moriyasu et al. [Diffeomorphisms with shadowable measures, Axioms 7(4) (2018), p. 93. Available at https://doi.org/10.3390/axioms7040093], the notion of shadowable measures for f is introduced, and the property of dynamical systems admitting shadowable measures is studied. In this paper, we introduce a probability measure on the space of pseudo-orbits

  • On the dynamics of the Euler equations on so(4)
    Dyn. Syst. (IF 0.986) Pub Date : 2019-12-22
    Claudio A. Buzzi; Jaume Llibre; Rubens Pazim

    This paper deals with the Euler equations on the Lie Algebra so(4). These equations are given by a polynomial differential system in R 6 . We prove that this differential system has four 3-dimensional invariant manifolds and we give a complete description of its dynamics on these invariant manifolds. In particular, each of these invariant manifolds are fulfilled by periodic orbits except in a zero

  • Equivalent notions of hyperbolicity
    Dyn. Syst. (IF 0.986) Pub Date : 2019-12-18
    Luis Barreira; Claudia Valls

    For a linear nonautonomous dynamics, we give a characterization of its hyperbolicity in terms of a certain cocycle, both for discrete and continuous time. The base of the cocycle is obtained taking the closure of the translations of the dynamics with respect to the topology of uniform convergence on compact sets, which in the case of discrete time corresponds to pointwise convergence.

  • Suspensions of homeomorphisms with the two-sided limit shadowing property
    Dyn. Syst. (IF 0.986) Pub Date : 2019-11-25
    Jesús Aponte; Bernardo Carvalho; Welington Cordeiro

    In this paper, we discuss the two-sided limit shadowing property for continuous flows defined in compact metric spaces. We analyse some of the results known for the case of homeomorphisms in the case of continuous flows and observe that some differences appear in this scenario. We prove that the suspension flow of a homeomorphism satisfying the two-sided limit shadowing property also satisfies it.

  • Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type
    Dyn. Syst. (IF 0.986) Pub Date : 2019-10-06
    Marc Kesseböhmer; Tanja I. Schindler

    We prove strong laws of large numbers under intermediate trimming for Birkhoff sums over subshifts of finite type. This gives another application of a previous trimming result only proven for interval maps. To prove these statements we introduce the space of quasi-Hölder continuous functions for subshifts of finite type. Additionally, we prove a trimmed strong law for St. Petersburg type distribution

  • Leafwise shadowing property for partially hyperbolic diffeomorphisms
    Dyn. Syst. (IF 0.986) Pub Date : 2019-09-26
    A. Castro; F. Rodrigues; P. Varandas

    In this note, we characterize hyperbolicity of invariant laminations for partially hyperbolic diffeomorphisms in terms of a leafwise shadowing property. We prove that a C 1 -diffeomorphism with partially hyperbolic non-wandering set is Axiom A if and only if the leafwise shadowing property holds along the central lamination for all C 1 -close diffeomorphisms. Similar results hold for open classes of

  • On the computation of extended rescaled Poincaré maps for singular vector fields
    Dyn. Syst. (IF 0.986) Pub Date : 2019-09-23
    Qianying Xiao; Yiwei Zhang

    In this paper, we aim to compute extended rescaled Poincaré maps for singular vector fields. For explicit 2-dimensional linear vector fields, we are able to compute the extended rescaled Poincaré maps upto second order derivatives. For singular vector fields, we show that the extended rescaled Poincaré maps over the non-degenerate singularity are equal to the extended rescaled Poincaré maps of the

  • Almost-periodic bifurcations for one-dimensional degenerate vector fields
    Dyn. Syst. (IF 0.986) Pub Date : 2019-09-23
    Wen Si; Xiaodan Xu; Jianguo Si

    Quasi-periodic high order degenerate bifurcation theories have been well established, but works which are related to almost-periodic bifurcations seem to be very few. In this paper, we consider the almost-periodic time-dependent perturbations of one-dimensional degenerate vector field x ˙ = x l . With the KAM theory and singularity theory, we show that the universal unfolding of the vector field can

  • Gibbs states and Gibbsian specifications on the space ℝℕ
    Dyn. Syst. (IF 0.986) Pub Date : 2019-09-18
    Artur O. Lopes; Victor Vargas

    We are interested in the study of Gibbs and equilibrium probabilities on the space state R N . We consider the unilateral full-shift σ defined on the non-compact set R N , that is σ ( x 1 , x 2 , . . , x n , . . ) = ( x 2 , x 3 , . . , x n , . . ) , and a Hölder continuous potential A : R N → R . From a suitable class of a priori probability measures ν we define the Ruelle operator associated to A

  • On the equivariance properties of self-adjoint matrices
    Dyn. Syst. (IF 0.986) Pub Date : 2019-09-18
    Michael Dellnitz; Bennet Gebken; Raphael Gerlach; Stefan Klus

    We investigate self-adjoint matrices A ∈ R n , n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ 2 ( A ) ⊂ O ( n ) which is isomorphic to ⊗ k = 1 n Z 2 . If the self-adjoint matrix possesses multiple eigenvalues – this may, for instance, be induced by symmetry properties of an

  • Volume of the hyperbolic cantor sets
    Dyn. Syst. (IF 0.986) Pub Date : 2019-09-16
    Habibulla Akhadkulov; Yunping Jiang

    In this paper, we study hyperbolic Cantor sets on the line. We prove that the Lebesgue measure of a hyperbolic Cantor set generated by a degree two expanding map satisfying a certain Zygmund condition is zero. Moreover, we show that for any ρ ∈ ( 0 , 1 ) there exists a C 1 -smooth degree two expanding map f such that the Lebesgue measure of the hyperbolic Cantor set generated by f is ρ.

  • On the transitivity and sensitivity of group actions
    Dyn. Syst. (IF 0.986) Pub Date : 2019-06-07
    Xiaoxiao Nie, Jiandong Yin

    Let (T,X) be a flow, which means that X is a compact metric space and T is a group action on X and let K(X), M(X) and F(X) be the sets of all nonempty compact subsets of X, all Borel probability measures on X and all upper semi-continuous fuzzy sets on X, respectively. Then K(X), M(X) and F(X) are metric spaces under the Hausdorff metric, the prohorov metric and the level-wise metric, respectively

  • On the connection between a skew product IFS and the ergodic optimization for a finite family of potentials
    Dyn. Syst. (IF 0.986) Pub Date : 2019-05-04
    Elismar R. Oliveira

    We study a skew product IFS on the cylinder defined by Baker-like maps associated to a finite family of potential functions and the doubling map. We show that there exist a compact invariant set with attractive behaviour and a random SRB measure whose support is in that set. We also study the IFS ergodic optimization problem for that finite family of potential functions and characterize the maximizing

  • Geometric limits of Julia sets and connectedness locus of the family of polynomials Pc(z) = zn + czk
    Dyn. Syst. (IF 0.986) Pub Date : 2019-04-06
    Alexandre Miranda Alves

    Let n and k be positive integers. For n>k≥2, we consider the connectedness locus Mn,k of the family of polynomials Pc(z)=zn+czk, where c is a complex parameter. We show that the geometric limit of the connectedness locus sets Mn,k, when n tends to infinity, exists and is the closed unit disk. In addition, we give an upper bound for the geometric size of Mn,k.When parameter c belongs to the open unit

  • Attractors for the nonclassical diffusion equations of Kirchhoff type with critical nonlinearity on unbounded domain† † All authors contributed equally to each part of this work. All authors read and approved the final manuscript.View all notes
    Dyn. Syst. (IF 0.986) Pub Date : 2019-03-28
    Fanghong Zhang, Lihong Bai

    In this paper, we investigate the long-time behavior of the solutions for the following nonclassical diffusion equations of Kirchhoff type ut−Δut−M(∥∇u∥2)Δu+f(x,u)=g(x). First, we prove the well-posedness of solution for the nonclassical diffusion equations of Kirchhoff type with critical nonlinearity on RN, then the existence of global attractor A is established in the natural energy space H1(RN)

  • On the number of zeros of Abelian integral for a class of cubic Hamiltonian systems
    Dyn. Syst. (IF 0.986) Pub Date : 2019-03-01
    Jihua Yang

    We prove that the maximal number of zeros of Abelian integral for system x˙=2y(b+cx2+2y2)+εf(x,y),y˙=−2x(a−2x2+cy2)+εg(x,y) is 90n+24 (taking into account the multiplicity), where a,b,c∈R such that the unperturbed system has at least one center, f(x,y) and g(x,y) are arbitrary polynomials in x and y of degree n.

  • Slow–fast systems and sliding on codimension 2 switching manifolds
    Dyn. Syst. (IF 0.986) Pub Date : 2019-03-01
    Paulo Ricardo da Silva, Willian Pereira Nunes

    In this work, we consider piecewise smooth vector fields X defined in Rn∖Σ, where Σ is a self-intersecting switching manifold. A double regularization of X is a 2-parameter family of smooth vector fields Xε.η, ε,η>0, satisfying that Xε,η converges uniformly to X in each compact subset of Rn∖Σ when ε,η→0. We define the sliding region on the non-regular part of Σ as a limit of invariant manifolds of

  • Duck traps: two-dimensional critical manifolds in planar systems
    Dyn. Syst. (IF 0.986) Pub Date : 2019-02-27
    Christian Kuehn, Christian Münch

    In this work we consider two-dimensional critical manifolds in planar fast-slow systems near fold and so-called canard ( =‘duck’) points. These higher-dimension, and lower-codimension, situation is directly motivated by the case of hysteresis operators limiting onto fast-slow systems as well as by systems with constraints. We use geometric desingularization via blow-up to investigate two situations

  • Explicit bounds for separation between Oseledets subspaces
    Dyn. Syst. (IF 0.986) Pub Date : 2019-02-22
    Anthony Quas, Philippe Thieullen, Mohamed Zarrabi

    We consider a two-sided sequence of bounded operators in a Banach space which are not necessarily injective and satisfy two properties (SVG) and (FI). The singular value gap (SVG) property says that two successive singular values of the cocycle at some index d admit a uniform exponential gap; the fast invertibility (FI) property says that the cocycle is uniformly invertible on the fastest d-dimensional

  • Chaos near a reversible homoclinic bifocus
    Dyn. Syst. (IF 0.986) Pub Date : 2019-02-14
    Pablo G. Barrientos, Artem Raibekas, Alexandre A. P. Rodrigues

    We show that any neighbourhood of a non-degenerate reversible bifocal homoclinic orbit contains chaotic suspended invariant sets on N-symbols for all N≥2. This will be achieved by showing switching associated with networks of secondary homoclinic orbits. We also prove the existence of super-homoclinic orbits (trajectories homoclinic to a network of homoclinic orbits), whose presence leads to a particularly

  • The asymptotically additive topological pressure: variational principle for non-compact and intersection of irregular sets
    Dyn. Syst. (IF 0.986) Pub Date : 2019-01-30
    G. Ferreira

    Let (X,d,f) be a dynamical system, where (X,d) is a compact metric space and f:X→X is a continuous map. Using the concepts of g-almost product property and uniform separation property introduced by Pfister and Sullivan in Pfister and Sullivan [On the topological entropy of saturated sets, Ergodic Theory Dyn. Syst. 27 (2007), pp. 929–956], we give a variational principle for certain non-compact with

  • Structural stability and a characterization of Anosov families
    Dyn. Syst. (IF 0.986) Pub Date : 2018-12-30
    Jeovanny de Jesus Muentes Acevedo

    Anosov families are non-stationary dynamical systems with hyperbolic behaviour. Non-trivial examples of Anosov families will be given in this paper. We show the existence of invariant manifolds, the structrural stability and a characterization for a certain class of Anosov families.

  • Saturation of generalized partially hyperbolic attractors
    Dyn. Syst. (IF 0.986) Pub Date : 2018-12-27
    Abbas Fakhari, Mohammad Soufi

    We prove the saturation of a generalized partially hyperbolic attractor of a C2 map. As a consequence, we show that any generalized partially hyperbolic horseshoe-like attractor of a C1-generic diffeomorphism has zero volume. In contrast, by modification of the Poincaré cross section of Lorenz geometric model, we build a C1-diffeomorphism with a partially hyperbolic horseshoe-like attractor of positive

  • Geometric method for global stability and repulsion in Kolmogorov systems
    Dyn. Syst. (IF 0.986) Pub Date : 2018-12-14
    Zhanyuan Hou

    A class of autonomous Kolmogorov systems that are dissipative and competitive with the origin as a repellor are considered when each nullcline surface is either concave or convex. Geometric method is developed by using the relative positions of the upper and lower planes of the nullcline surfaces for global asymptotic stability of an interior or a boundary equilibrium point. Criteria are also established

  • Coherent measures and the unstable manifold of isolated unstable attractors
    Dyn. Syst. (IF 0.986) Pub Date : 2018-11-29
    Konstantin Athanassopoulos

    In this note we give a statistical approximation of the unstable manifold of a connected isolated unstable attractor of a smooth flow using coherent measures relative to it. In the main result we show that almost all orbits in the support of a coherent measure relative to an isolated unstable attractor are contained in its unstable manifold.

  • Invariant cone and synchronization state stability of the mean field models
    Dyn. Syst. (IF 0.986) Pub Date : 2018-11-28
    W. Oukil, Ph. Thieullen, A. Kessi

    In this article we prove the stability of some mean field systems similar to the Winfree model in the synchronized state. The model is governed by the coupling strength parameter κ and the natural frequency of each oscillator. The stability is proved independently of the number of oscillators and the distribution of the natural frequencies. The main result is proved using the positive invariant cone

  • Reducibility of a class of 2k-dimensional Hamiltonian systems with quasi-periodic coefficients
    Dyn. Syst. (IF 0.986) Pub Date : 2018-11-03
    Jia Li, Youhui Su, Yanling Shi

    In this paper, we consider the following real analytic Hamiltonian system x˙=(A+εQ(t,ε))x,x∈R2k, where A is a constant Hamiltonian matrix with the different eigenvalues ±w1−1, ±λ2,…,±λk, where w1∈R, ±λi≠0 for 2≤i≤k are real, and Q(t,ε) is quasi-periodic with frequencies w1,w2,…wr. Without any non-degeneracy condition with respect to ϵ, we prove that by a quasi-periodic symplectic mapping, then for

  • On the effective reducibility of a class of Quasi-periodic nonlinear systems near the equilibrium
    Dyn. Syst. (IF 0.986) Pub Date : 2018-11-03
    Jia Li, Chunpeng Zhu

    In this paper, we consider the effective reducibility of the following quasi-periodic nonlinear system x˙=(A+εQ(t,ε))x+εg(t,ε)+h(x,t,ε),|ε|≤ε0, where A is a constant matrix with the different and nonzero eigenvalues, h=O(x2)(x→0), and h(x,t), Q(t) and g(t) are analytic quasi-periodic on Dρ with respect to t. Under non-resonance conditions, without any non-degeneracy condition, by a quasi-periodic transformation

  • The chaotic behaviour of piecewise smooth differential equations on two-dimensional torus and sphere
    Dyn. Syst. (IF 0.986) Pub Date : 2018-11-02
    Ricardo M. Martins, Durval J. Tonon

    This paper studies the global dynamics of piecewise smooth differential equations defined in the two-dimensional torus and sphere in the case when the switching manifold breaks the manifold into two connected components. Over the switching manifold, we consider the Filippov's convention for discontinuous differential equations. The study of piecewise smooth dynamical systems over torus and sphere is

  • Centre manifolds for infinite dimensional random dynamical systems
    Dyn. Syst. (IF 0.986) Pub Date : 2018-10-25
    Xiaopeng Chen, Anthony J. Roberts, Jinqiao Duan

    Stochastic centre manifolds theory are crucial in modelling the dynamical behaviour of complex systems under stochastic influences. The existence of stochastic centre manifolds for infinite dimensional random dynamical systems is shown under the assumption of exponential trichotomy. The theory provides a support for the discretisations of nonlinear stochastic partial differential equations with space–time

  • Phase portraits of Abel quadratic differential systems of second kind with symmetries
    Dyn. Syst. (IF 0.986) Pub Date : 2018-10-16
    Antoni Ferragut, Johanna D. García-Saldaña, Claudia Valls

    We provide normal forms and the global phase portraits on the Poincaré disk of the Abel quadratic differential equations of the second kind having a symmetry with respect to an axis or to the origin. Moreover, we also provide the bifurcation diagrams for these global phase portraits.

  • Fractal dimension of random attractors for non-autonomous fractional stochastic Ginzburg–Landau equations with multiplicative noise
    Dyn. Syst. (IF 0.986) Pub Date : 2018-09-26
    Yun Lan, Ji Shu

    This paper deals with the asymptotic behaviour of solutions for non-autonomous stochastic fractional Ginzburg–Landau equations driven by multiplicative noise with α∈(0,1). We first present some conditions for bounding the fractal dimension of a random invariant set of non-autonomous random dynamical system. Then we derive uniform estimates of solutions and establish the existence and uniqueness of

  • Measure-theoretic pressure and topological pressure in mean metrics
    Dyn. Syst. (IF 0.986) Pub Date : 2018-09-20
    Ping Huang, Chenwei Wang

    In this paper, using spanning sets in mean metrics we construct a new definition of measure-theoretic pressure of ergodic measures over a topological dynamical system. And we establish a pressure version of Katok's entropy formula in the case of mean metrics. Furthermore, we also introduce a new definition of topological pressure by replacing the Bowen metrics with the corresponding mean metrics, and

  • Block conjugacy of irreducible toral automorphisms
    Dyn. Syst. (IF 0.986) Pub Date : 2018-08-28
    Lennard F. Bakker, Pedro Martins Rodrigues

    We establish a matrix characterization (called block conjugacy) of the notion of weak equivalence of ideals. This gives the existence of a one-to-one correspondence between equivalence classes of block conjugate toral automorphisms (of a given irreducible characteristic polynomial) and equivalence classes of the weakly equivalent associated ideals. We show that there exist nonconjugate irreducible

  • Independence and Alpern multitowers
    Dyn. Syst. (IF 0.986) Pub Date : 2018-08-20
    James T. Campbell, Randall McCutcheon, Alistair Windsor

    Let T be any invertible, ergodic, aperiodic measure-preserving transformation of a Lebesgue probability space (X,B,μ), and P any finite measurable partition of X. We show that a (finite) Alpern multitower may always be constructed whose base is independent of P.

  • Erratum
    Dyn. Syst. (IF 0.986) Pub Date : 2018-08-13

    (2019). Erratum. Dynamical Systems: Vol. 34, No. 1, pp. i-i.

  • A new universal real flow of the Hilbert-cubical type
    Dyn. Syst. (IF 0.986) Pub Date : 2018-08-13
    Lei Jin, Siming Tu

    We provide a new universal real flow of the Hilbert-cubical type. We prove that any real flow can be equivariantly embedded in the translation on L(R)N, where L(R) denotes the space of 1-Lipschitz functions f:R→[0,1]. Furthermore, all those functions in L(R)N that are images of such embeddings can be chosen as C1-functions.

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