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We show that the mode-locking region of the family of quasi-periodically forced Arnold circle maps with a topologically generic forcing function is dense. This gives a rigorous verification of certain numerical observations in [M. Ding, C. Grebogi and E. Ott. Evolution of attractors in quasiperiodically forced systems: from quasiperiodic to strange nonchaotic to chaotic. Phys. Rev. A 39 (5) (1989), 2593–2598] for such forcing functions. More generally, under some general conditions on the base map, we show the density of the mode-locking property among dynamically forced maps (defined in [Z. Zhang. On topological genericity of the mode-locking phenomenon. Math. Ann. 376 (2020), 707–72]) equipped with a topology that is much stronger than the $C^0$ topology, compatible with smooth fiber maps. For quasi-periodic base maps, our result generalizes the main results in [A. Avila, J. Bochi and D. Damanik. Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. Duke Math. J. 146 (2009), 253–280], [J. Wang, Q. Zhou and T. Jäger. Genericity of mode-locking for quasiperiodically forced circle maps. Adv. Math. 348 (2019), 353–377] and Zhang (2020). [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new concept of expansiveness, similar to the separating property. Specifically, we consider a compact Riemannian manifold M without boundary and a C1 vector field X on M, which generates a flow φt on M. We say that X is rescaling separating on a compact invariant set Λ of X if there is a constant δ > 0 such that, for any x, y ∈ Λ, if d(φt(x), φt(y)) ≤ δ∥X (φt(x))∥ for all t ∈ ℝ, then y ∈ Orb(x). We prove that if X is rescaling separating on Λ and every singularity of X in Λ is hyperbolic, then any C1 vector field Y, whose flow commutes with φt on Λ, must be collinear to X on Λ. As applications of this result, we show that the centralizer of a rescaling separating C1 vector field without nonhyperbolic singularity is quasi-trivial. We also proved that there is an open and dense set U ⊂ X 1 (M) such that for any star vector field X ∈ U , the centralizer of X is collinear to X on the chain recurrent set of X. [ABSTRACT FROM AUTHOR]

We consider the space of C 1 -diffeomorphims of a three dimensional closed manifold equipped with the C 1 -topology. It is known that there are open sets in which C 1 -generic diffeomorphisms display uncountably many chain recurrence classes, while only countably many of them may contain periodic orbits. The classes without periodic orbits, called aperiodic classes, are the main subject of this paper. The aim of the paper is to show that aperiodic classes of C 1 -generic diffeomorphisms can exhibit a variety of topological properties. More specifically, there are C 1 -generic diffeomorphisms with (1) minimal expansive aperiodic classes, (2) minimal but non-uniquely ergodic aperiodic classes, (3) transitive but non-minimal aperiodic classes, (4) non-transitive, uniquely ergodic aperiodic classes. [ABSTRACT FROM AUTHOR]

We mainly study the existence, structural stability and robustness of fuzzy slightly altruistic equilibria for a class of generalized multiobjective fuzzy games which are expressed as ϖ . Firstly, we introduce the concept of fuzzy slightly altruistic equilibrium and prove the existence of equilibrium for the ϖ by Fan–Glicksberg fixed point theorem. Secondly, the connections between ϖ and bounded rationality are discussed by an abstract rationality functions. Moreover, we construct the problem space of ϖ which is represented by Λ and show that most ϖ ∈ Λ are structurally stable and robust to ε -equilibrium on the meaning of Baire category. These results are new and extend some existing results in recent literature. [ABSTRACT FROM AUTHOR]

We consider a generic symplectic partially hyperbolic diffeomorphism close to direct/skew products of symplectic Anosov diffeomorphisms with area-preserving diffeomorphisms and prove that every hyperbolic periodic point has transverse homoclinic points. [ABSTRACT FROM AUTHOR]

In this paper, we extend the classical theory of normal forms for continuous and difference dynamical systems to dynamic equations on discrete time scales. As consequences of the well known results from the theory of analytic differential equations, we obtain some versions of the Poincaré and Siegel theorems for dynamic equations on discrete time scales. Using these results and known results on the stability of dynamic equations on time scales, we obtain some stability results for the nonlinear dynamic equations. We also prove some results on generic properties of bifurcation curves and the saddle-node bifurcation for one-parameter families of dynamic equations on arbitrary time scales. [ABSTRACT FROM AUTHOR]

In this text we study billiards on symmetric ovals and investigate some consequences of the symmetry of the boundary on the dynamics. As it simplifies some calculations, the symmetry helps to obtain the results. We focus on periodic orbits with the same symmetry of the boundary which always exist and prove that typically half of them are elliptic and Moser stable and the other half are hyperbolic with homo(hetero)clinic intersections. [ABSTRACT FROM AUTHOR]

Perfect adaptation in a dynamical system is the phenomenon that one or more variables have an initial transient response to a persistent change in an external stimulus but revert to their original value as the system converges to equilibrium. With the help of the causal ordering algorithm, one can construct graphical representations of dynamical systems that represent the causal relations between the variables and the conditional independences in the equilibrium distribution. We apply these tools to formulate sufficient graphical conditions for identifying perfect adaptation from a set of first-order differential equations. Furthermore, we give sufficient conditions to test for the presence of perfect adaptation in experimental equilibrium data. We apply this method to a simple model for a protein signalling pathway and test its predictions in both simulations and using real-world protein expression data. We demonstrate that perfect adaptation can lead to misleading orientation of edges in the output of causal discovery algorithms.

In this paper, we show that smooth TA-endomorphisms of compact manifolds with c-expansivity (that is, expansive in the inverse limit) and C1-stable shadowing are Axiom A. [ABSTRACT FROM AUTHOR]

The aim of this note is to give an alternative proof for the following result originally proved by Bonatti, Díaz and Kwietniak. For every n ≥ 3 there exists a compact manifold without boundary M of dimension n and a non-empty open set U ⊂ Diff (M) such that for every f ∈ U there exists a non-hyperbolic measure μ invariant for f with positive entropy and full support. We also investigate the connection between the Feldman-Katok convergence of measures and the Kuratowski convergence of their supports. [ABSTRACT FROM AUTHOR]

In this paper, we will assume MM to be a compact smooth manifold and f:M→Mf:M\to M to be a diffeomorphism. We herein demonstrate that a C1{C}^{1} generic diffeomorphism ff is Axiom A and has no cycles if ff is asymptotic measure expansive. Additionally, for a C1{C}^{1} generic diffeomorphism ff, if a homoclinic class H(p,f)H\left(\hspace{0.08em}p,f) that contains a hyperbolic periodic point pp of ff is asymptotic measure-expansive, then H(p,f)H\left(\hspace{0.08em}p,f) is hyperbolic of ff.

The aim of this paper is to give a complete description of the input–output methods for uniform exponential stability of discrete dynamical systems. We present a new study from four perspectives, in each case providing a deep analysis of the input–output criteria and of the axiomatic structure of the admissible pairs. The first stage is devoted to global conditions of uniform type for stability of discrete variational systems, in the most general case. Next, the results are applied to characterize the stability of discrete nonautonomous systems, without any assumptions on their coefficients. In both cases, the optimality of the methods is motivated by examples, showing that the hypotheses regarding the input and output spaces cannot be removed. The third method is focused on ergodic theory approaches, providing criteria of nonuniform type for stability of discrete variational systems. We characterize the uniform exponential stability by means of nonuniform input–output stabilities relative to ergodic measures, using some very general classes of sequence spaces. Thus, we extend our recent stability results obtained in Dragičević et al. (J Differ Equ 268:4786–4829, 2020). At the fourth stage, we prove even more, that in certain conditions, in the variational case, the exponential stability can be characterized in terms of stabilities along periodic orbits. [ABSTRACT FROM AUTHOR]

In this article we study some statistical aspects of surface diffeomorphisms. We first show that for a C 1 generic diffeomorphism, a Dirac invariant measure whose statistical basin of attraction is dense in some open set and has positive Lebesgue measure, must be supported in the orbit of a sink. We then construct an example of a C 1 -diffeomorphism having a Dirac invariant measure, supported on a saddle-type hyperbolic fixed point, whose statistical basin of attraction is a nowhere dense set with positive Lebesgue measure. Our technique can be applied also to construct a C 1 diffeomorphism whose set of points with historic behaviour has positive measure and is nowhere dense. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of nontrivial solutions for an anisotropic discrete nonlinear problem with a p(k)-Laplacian operator. The proof of our main result is based on a local minimum theorem for differentiable functionals due to Ricceri. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new notion of integrability for billiard tables, namely, integrability away from the boundary. One key feature of our notion is that the integrable region could be disjoint from the boundary with a positive distance. We prove that if a strictly convex billiard table, whose boundary is a small perturbation of an ellipse with small eccentricity, is integrable in this sense, then its boundary must be itself an ellipse. [ABSTRACT FROM AUTHOR]

Analytical study of ball vibration absorber behavior is presented in the paper. The dynamics of trajectories of a heavy ball moving without slipping inside a spherical cavity are analyzed. Following our previous work, where a similar system was investigated through various numerical simulations, research of the dynamic properties of a sphere moving in a spherical cavity was carried out by methods of analytical dynamics. The strategy of analytical investigation enabled definition of a set of special and limit cases which designate individual domains of regular trajectories. In order to avoid any mutual interaction between the domains along a particular trajectory movement, energy dissipation at the contact of the ball and the cavity has been ignored, as has any kinematic excitation due to cavity movement. A governing system was derived using the Lagrangian formalism and complemented by appropriate non-holonomic constraints of the Pfaff type. The three first integrals are defined, enabling the evaluation of trajectory types with respect to system parameters, the initial amount of total energy, the angular momentum of the ball and its initial spin velocity. The neighborhoods of the limit trajectories and their dynamic stability are assessed. Limit and transition special cases are investigated along with their individual elements. The analytical means of investigation enabled the performance of broad parametric studies. Good agreement was found when comparing the results achieved by the analytical procedures in this paper with those obtained by means of numerical simulations, as they followed from the Lagrangian approach and the Appell–Gibbs function presented in previous papers. [ABSTRACT FROM AUTHOR]

In this paper, we show that for generic C1, if a flow Xt has the shadowing property on a bi-Lyapunov stable homoclinic class, then it does not contain any singularity and it is hyperbolic.

Let M be a closed smooth Riemannian manifold and let f : M → M be a diffeomorphism. We show that if f has the C1 robustly asymptotic orbital shadowing property then it is an Anosov diffeomorphism. Moreover, for a C1 generic diffeomorphism f, if f has the asymptotic orbital shadowing property then it is a transitive Anosov diffeomorphism. In particular, we apply our results to volume-preserving diffeomorphisms.

In this paper, we present a criterion for pitchfork bifurcations of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale℉s result [15] significantly. Based on our criterion, we present a class of families of non-symmetric vector fields undergoing a pitchfork bifurcation. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for C 1 -generic diffeomorphisms, if a homoclinic class contains periodic orbits of indices i and j with j > i + 1 , and the homoclinic class has no-domination of index l for any l ∈ { i + 1 , ... , j - 1 } , then there exists a non-hyperbolic ergodic measure with more than one vanishing Lyapunov exponents and whose support is the whole homoclinic class. Some other results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper we consider $C^{\infty }$ -generic families of area-preserving diffeomorphisms of the torus homotopic to the identity and their rotation sets. Let $f_{t}:\text{T}^{2}\rightarrow \text{T}^{2}$ be such a family, $\widetilde{f}_{t}:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ be a fixed family of lifts and $\unicode[STIX]{x1D70C}(\widetilde{f}_{t})$ be their rotation sets, which we assume to have interior for $t$ in a certain open interval  $I$. We also assume that some rational point $(p/q,l/q)\in \unicode[STIX]{x2202}\unicode[STIX]{x1D70C}(\widetilde{f}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$ , and we want to understand the consequences of the following hypothesis: for all $t>\overline{t}$ , $t\in I$ , $(p/q,l/q)\in \text{int}(\unicode[STIX]{x1D70C}(\widetilde{f}_{t}))$. Under these very natural assumptions, we prove that there exists a $f_{\overline{t}}^{q}$ -fixed hyperbolic saddle $P_{\overline{t}}$ such that its rotation vector is $(p/q,l/q)$. We also prove that there exists a sequence $t_{i}>\overline{t}$ , $t_{i}\rightarrow \overline{t}$ , such that if $P_{t}$ is the continuation of $P_{\overline{t}}$ with the parameter, then $W^{u}(\widetilde{P}_{t_{i}})$ (the unstable manifold) has quadratic tangencies with $W^{s}(\widetilde{P}_{t_{i}})+(c,d)$ (the stable manifold translated by $(c,d)$), where $\widetilde{P}_{t_{i}}$ is any lift of $P_{t_{i}}$ to the plane. In other words, $\widetilde{P}_{t_{i}}$ is a fixed point for $(\widetilde{f}_{t_{i}})^{q}-(p,l)$ , and $(c,d)\neq (0,0)$ are certain integer vectors such that $W^{u}(\widetilde{P}_{\overline{t}})$ do not intersect $W^{s}(\widetilde{P}_{\overline{t}})+(c,d)$ , and these tangencies become transverse as $t$ increases. We also prove that, for $t>\overline{t}$ , $W^{u}(\widetilde{P}_{t})$ has transverse intersections with $W^{s}(\widetilde{P}_{t})+(a,b)$ , for all integer vectors $(a,b)$ , and thus one may consider that the tangencies above are associated to the birth of the heteroclinic intersections in the plane that do not exist for $t\leq \overline{t}$. [ABSTRACT FROM AUTHOR]

In the paper we use a special geometric structure of selected one-dimensional continua to prove that some stronger versions of the shadowing property are generic (or at least dense) for continuous maps acting on these spaces. Specifically, we prove that (i) the periodic T S -bi-shadowing property, where T S means some class of continuous methods, is generic as well as the s-limit shadowing property is dense in the space of all continuous maps (and all continuous surjective maps) of any topological graph; (ii) the T S -bi-shadowing property is generic as well as the s-limit shadowing property is dense in the space of all continuous maps of any dendrite; (iii) the T S -bi-shadowing property is generic in the space of all continuous maps of chainable continuum that can by approximated by arcs from the inside. The results of the paper extend ones obtained over the last few decades by various authors (see, e.g., Kościelniak in J Math Anal Appl 310:188–196, 2005; Kościelniak and Mazur in J Differ Equ Appl 16:667–674, 2010; Kościelniak et al. in Discret Contin Dyn Syst 34:3591–3609, 2014; Mazur and Oprocha in J Math Anal Appl 408:465–475, 2013; Mizera in Generic Properties of One-Dimensional Dynamical Systems, Ergodic Theory and Related Topics, III, Springer, Berlin, 1992; Odani in Proc Am Math Soc 110:281–284, 1990; Pilyugin and Plamenevskaya in Topol Appl 97:253–266, 1999; and Yano in J Fac Sci Univ Tokyo Sect IA Math 34:51–55, 1987) for both homeomorphisms and continuous maps of compact manifolds, including (in particular) an interval and a circle, which are the simplest examples of one-dimensional continua. Moreover, from a technical point of view our considerations are a continuation of those carried out in the earlier work by Mazur and Oprocha in J. Math. Anal. Appl. 408:465–475, 2013. [ABSTRACT FROM AUTHOR]

It is well known that iterated function systems generated by orientation preserving homeomorphisms of the unit interval with positive Lyapunov exponents at its ends admit a unique invariant measure on (0, 1) provided their action is minimal. With the additional requirement of continuous differentiability of maps on a fixed neighbourhood of { 0 , 1 } , we present a metric in the space of such systems which renders it complete. Using then a classical argument (and an alternative uniqueness proof), we show that almost singular invariant measures are admitted by systems lying densely in the space. This allows us to construct a residual set of systems with unique singular stationary distribution. Dichotomy between singular and absolutely continuous unique measures is assured by taking a subspace of systems with absolutely continuous maps; the closure of this subspace is where the residual set is found. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide a non-local approach to study aspects of structural stability of 3D Filippov systems. We introduce a notion of semi-local structural stability which detects when a piecewise smooth vector field is robust around the entire switching manifold, as well as, provides a complete characterization of such systems. In particular, we present some methods in the qualitative theory of piecewise smooth vector fields, which make use of geometrical analysis of the foliations generated by their orbits. Such approach displays surprisingly rich dynamical behavior which is studied in detail in this work. It is worth mentioning that this subject has not been treated in dimensions higher than two from a non-local point of view, and we hope that the approach adopted herein contributes to the understanding of structural stability for piecewise-smooth vector fields in its most global sense. [ABSTRACT FROM AUTHOR]

The aim of the study is to demonstrate a couple of special states which can be encountered at the system of a ball moving in a spherical cavity working as a passive tuned mass damper (TMD) of slender engineering structures. The system includes six degrees of freedom with three non-holonomic constraints being under horizontal additive kinematic excitation. The Appell–Gibbs approach is used to deduce the governing differential system. Uniaxial and biaxial types of kinematic excitation are considered. Among biaxial, a special attention is paid to circular setting. Influence of the rolling and spinning damping in contact of the ball with cavity is discussed. Under uniaxial excitation is the system auto-parametric and posses multiple solutions. The individual response branches can be identified when the excitation frequency is swept up or down with respect to setting up of initial conditions. Among stable branches reveal those with very low and sometimes zero approaching stability level. Although the accessibility of relevant trajectories is often very subtle due to effect of the dynamic stability, these post-critical phenomena accumulate a lot of energy. Hence, they can be very dangerous for TMD and other important engineering systems. Some general recommendations for practice are formulated. [ABSTRACT FROM AUTHOR]

We consider a noncooperative Stackelberg game, where the two players choose their strategies within domains X ⊆ R m and Y ⊆ R n . Assuming that the cost functions F, G for the two players are sufficiently smooth, we study the structure of the best reply map for the follower and the optimal strategy for the leader. Two main cases are considered: either X = Y = [ 0 , 1 ] , or X = R , Y = R n with n ≥ 1 . Using techniques from differential geometry, including a multi-jet version of Thom's transversality theorem, we prove that, for an open dense set of cost functions F ∈ C 2 and G ∈ C 3 , the Stackelberg equilibrium is unique and is stable w.r.t. small perturbations of the two cost functions. [ABSTRACT FROM AUTHOR]