Attractor-repeller Pairs Research Articles

Critical transitions for scalar nonautonomous systems with concave nonlinearities: some rigorous estimates

The global dynamics of a nonautonomous Carathéodory scalar ordinary differential equation x′=f(t,x) , given by a function f which is concave in x, is determined by the existence or absence of an attractor-repeller pair of hyperbolic solutions. This property, here extended to a very general setting, is the key point to classify the dynamics of an equation which is a transition between two nonautonomous asymptotic limiting equations, both with an attractor-repeller pair. The main focus of the paper is to get rigorous criteria guaranteeing tracking (i.e. connection between the attractors of the past and the future) or tipping (absence of connection) for the particular case of equations x′=f(t,x−Γ(t)) , where Γ is asymptotically constant. Some computer simulations show the accuracy of the obtained estimates, which provide a powerful way to determine the occurrence of critical transitions without relying on a numerical approximation of the (always existing) locally pullback attractor.

Criterion for the existence of a connected characteristic space of orbits in a gradient-like diffeomorphism of a surface

The classical approach to the study of dynamical systems consists in representing the dynamics of the system in the "source-sink" form, that is, by singling out a dual attractor-repeller pair, consisting of the attracting and repelling sets for all other trajectories of the system. A choice of an attractor-repeller dual pair so that the space of orbits in their complement (the characteristic space of orbits) is connected paves the way for finding complete topological invariants of the dynamical system. In this way, in particular, several classification results for Morse-Smale systems were obtained. Thus, a complete topological classification of Morse-Smale 3-diffeomorphisms is essentially based on the existence of a connected characteristic space of orbits associated with the choice of a one-dimensional dual attractor-repeller pair. For Morse-Smale diffeomorphisms with heteroclinic points on surfaces, there are examples in which the characteristic spaces of orbits are disconnected in all cases. In this paper, we prove a criterion for the existence of a connected characteristic space of orbits for gradient-like (without heteroclinic points) diffeomorphisms on surfaces. This result implies, in particular, that any orientation-preserving diffeomorphism admits a connected characteristic space. For an orientable surface of any kind, we also construct an orientation-changing gradient-like diffeomorphism that does not have a connected characteristic space. On any non-orientable surface of any kind, we also construct a gradient-like diffeomorphism which does not admit a connected characteristic space.

Critical Transitions in Piecewise Uniformly Continuous Concave Quadratic Ordinary Differential Equations

A critical transition for a system modelled by a concave quadratic scalar ordinary differential equation occurs when a small variation of the coefficients changes dramatically the dynamics, from the existence of an attractor–repeller pair of hyperbolic solutions to the lack of bounded solutions. In this paper, a tool to analyze this phenomenon for asymptotically nonautonomous ODEs with bounded uniformly continuous or bounded piecewise uniformly continuous coefficients is described, and used to determine the occurrence of critical transitions for certain parametric equations. Some numerical experiments contribute to clarify the applicability of this tool.

The Morse Equation in the Conley Index Theory for Discrete Multivalued Dynamical Systems

A recent generalization of the Conley index to discrete multivalued dynamical systems without a continuous selector is motivated by applications to data–driven dynamics. In the present paper we continue the program by studying attractor–repeller pairs and Morse decompositions in this setting. In particular, we prove Morse equation and Morse inequalities.

Rate-Induced Tipping and Saddle-Node Bifurcation for Quadratic Differential Equations with Nonautonomous Asymptotic Dynamics

An in-depth analysis of nonautonomous bifurcations of saddle-node type for scalar differential equations $x'=-x^2+q(t)\,x+p(t)$, where $q\colon\mathbb{R}\to\mathbb{R}$ and $p\colon\mathbb{R}\to\mathbb{R}$ are bounded and uniformly continuous, is fundamental to explain the absence or occurrence of rate-induced tipping for the differential equation $y' =(y-(2/\pi)\arctan(ct))^2+p(t)$ as the rate $c$ varies on $[0,\infty)$. A classical attractor-repeller pair, whose existence for $c=0$ is assumed, may persist for any $c>0$, or disappear for a certain critical rate $c=c_0$, giving rise to rate-induced tipping. A suitable example demonstrates that one can have more than one critical rate, and the existence of the classical attractor-repeller pair may return when $c$ increases.

Mathematical Modeling of the Flight of the Starlings. A Particular Case of an Attractor Repeller Pairs

It presents a mathematical modeling of a biological system, this is a particular case of two fractal sets, an attractor repeller pair, according to the definition given by the Morse-Smalle theory; these are complementary fractal sets, since the complementary of the fractal repeller set is the attractor set, as we can see in the demonstration. The existence of this model in Nature is found in the flight of starlings. This dynamic system works like a beat of a heart, where the movement of the systole is equivalent to the contraction (attractor set) of the flight trajectory of birds and the diastole movement to its expansion (repeller set).The modeling is presented with differential equations written in Matlab code, and several images of the fractal sets associated with the flight pattern of the starlings are generated, so that we can see this model more clearly. We will begin with a definition and its subsequent mathematical demonstration to study this complex system of complementary fractals.

Topological dynamics on finite directed graphs

We establish that finite directed graphs give rise to semiflows on the power set of their nodes. We analyze the topological dynamics for semiflows on finite directed graphs by characterizing Morse decompositions, recurrence behavior and attractor–repeller pairs under weaker assumptions. As is expected, the discrete metric plays an important role in our constructions and their consequences.

The Selgrade Decomposition for Linear Semiflows on Banach Spaces

We extend Selgrade's Theorem, Morse spectrum, and related concepts to the setting of linear skew product semiflows on a separable Banach bundle. We recover a characterization, well-known in the finite-dimensional setting, of exponentially separated subbundles as attractor-repeller pairs for the associated semiflow on the projective bundle.

Attractor-repeller pair of topological zero modes in a nonlinear quantum walk

The quantum-mechanical counterpart of a classical random walk offers a rich dynamics that has recently been shown to include topologically protected bound states (zero-modes) at boundaries or domain walls. Here we show that a topological zero-mode may acquire a dynamical role in the presence of nonlinearities. We consider a one-dimensional discrete-time quantum walk that combines zero-modes with a particle-conserving nonlinear relaxation mechanism. The presence of both particle-hole and chiral symmetry converts two zero-modes of opposite chirality into an attractor-repeller pair of the nonlinear dynamics. This makes it possible to steer the walker towards a domain wall and trap it there.

Simulation of a Tornadoes as a Particular Case of an Attractor-repeller Pairs

The attractor-repeller pairs are binary systems which are stable systems in termo-dynamic equilibrium, and one of the leaders we have in Nature to understand Tornadoes. A mathematical model of a tornado is presented, within a model of chaos theory, where two complementary fractals are combined to understand this natural phenomenon. It is a thermodynamic state where the wind formed by warm air rises (the repeller) while the swirling cold (attractor) wind descends joining together and creating a tornado, which is an equilibrium system. The mathematical modeling we present here is based on algorithms and it has been performed with Matlab code.

Developments in fractal geometry

Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Mobius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor–repeller pairs and the Conley “landscape picture” for an IFS.

An estimate on the fractal dimension of attractors of gradient-like dynamical systems

This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets, Lipschitz properties of the semigroup and the rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A,A∗) is an attractor–repeller pair for the attractor A of a semigroup {T(t):t≥0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of A∗, the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. As we said previously, we generalize this result for some evolution processes using the same basic ideas.

Breakdown of symmetry in reversible systems

We review results on local bifurcations of codimension 1 in reversible systems (flows and diffeomorphisms) which lead to the birth of attractor-repeller pairs from symmetric equilibria (for flows) or periodic points (for diffeomorphisms).

Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations

Recently, the existence of Morse decompositions for nonautonomous dynamical systems was shown for three different time domains: the past, the future and—in the linear case—the entire time. In this article, notions of exponential dichotomy are discussed with respect to the three time domains. It is shown that an exponential dichotomy gives rise to an attractor–repeller pair in the projective space, which is a building block of a Morse decomposition. Moreover, based on the notions of exponential dichotomy, dichotomy spectra are introduced, and it is proved that the corresponding spectral manifolds lead to Morse decompositions in the projective space.

The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations

The author proposes a general mechanism by which strange non-chaotic attractors (SNA) are created during the collision of invariant curves in quasiperiodically forced systems. This mechanism, and its implementation in different models, is first discussed on an heuristic level and by means of simulations. In the considered examples, a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant curve there exists a strange non-chaotic attractor-repeller pair at the bifurcation point. This process is accompanied by a very characteristic behaviour of the invariant curves prior to their collision, which the author calls 'exponential evolution of peaks'.

ATTRACTOR–REPELLER PAIR, MORSE DECOMPOSITION AND LYAPUNOV FUNCTION FOR RANDOM DYNAMICAL SYSTEMS

In the stability theory of dynamical systems, Lyapunov functions play a fundamental role. In this paper, we study the attractor–repeller pair decomposition and Morse decomposition for compact metric space in the random setting. In contrast to [7,17], by introducing slightly stronger definitions of random attractor and repeller, we characterize attractor–repeller pair decompositions and Morse decompositions for random dynamical systems through the existence of Lyapunov functions. These characterizations, we think, deserve to be known widely.

Conley indices on convex sets, attractor-repeller pairs and applications to multiple solutions of nonlinear elliptic equations

We discuss the calculation of homotopy indices on closed convex sets in a Banach space and then use these ideas to prove the existence of multiple solutions of nonlinear elliptic equations. We obtain more information on the solutions than in earlier work. A key idea is to calculate the homotopy index of the stable and mountain pass solutions and the connections joining them.

The boundary map and the connecting orbits

The purpose of this article is to show that the image of the homological boundary map attached to a filtration for an attractor–repeller pair of a smooth flow on a compact manifold is a submodule of the Alexander cohomology of certain order of the connecting set (some restrictions have to be imposed in order to have a valid argument). In particular, this gives an affirmative answer to a conjecture in Conley index theory which states that if the boundary map is not zero in two dimensions, the connecting set cannot be contractible.

A Computational Approach to Conley’s Decomposition Theorem

Background. The discrete dynamics generated by a continuous map can be represented combinatorially by an appropriate multivalued map on a discretization of the phase space such as a cubical grid or triangulation. Method of approach. We describe explicit algorithms for computing dynamical structures for the combinatorial multivalued maps. Results. We provide computational complexity bounds and numerical examples. Specifically we focus on the computation attractor-repeller pairs and Lyapunov functions for Morse decompositions. Conclusions. The computed discrete Lyapunov functions are weak Lyapunov functions and well-approximate a continuous Lyapunov function for the underlying map.

Homology index braids in infinite-dimensional Conley index theory

We extend the notion of a categorial Conley-Morse index, as defined in [K. P. rybakowski, The Morse index, repeller-attractor pairs and the connection index for semiflows on noncompact spaces , J. Differential Equations 47 (1987), 66–98], to the case based on a more general concept of an index pair introduced in [R. D. Franzosa and K. Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces , J. Differential Equations 71 (1988), 270–287]. We also establish a naturality result of the long exact sequence of attractor-repeller pairs with respect to the choice of index triples. In particular, these results immediately give a complete and rigorous existence result for homology index braids in infinite dimensional Conley index theory. Finally, we describe some general regular and singular continuation results for homology index braids obtained in our recent papers [M. C. Carbinatto and K. P. Rybakowski, Nested sequences of index filtrations and continuation of the connection matrix , J. Differential Equations 207 (2004), 458–488] and [M. C. Carbinatto and K. P. Rybakowski, Continuation of the connection matrix in singular perturbation problems ].