Minimal Attractor Research Articles

A converse sum of squares lyapunov function for outer approximation of minimal attractor sets of nonlinear systems

<p style='text-indent:20px;'>Many dynamical systems described by nonlinear ODEs are unstable. Their associated solutions do not converge towards an equilibrium point, but rather converge towards some invariant subset of the state space called an attractor set. For a given ODE, in general, the existence, shape and structure of the attractor sets of the ODE are unknown. Fortunately, the sublevel sets of Lyapunov functions can provide bounds on the attractor sets of ODEs. In this paper we propose a new Lyapunov characterization of attractor sets that is well suited to the problem of finding the minimal attractor set. We show our Lyapunov characterization is non-conservative even when restricted to Sum-of-Squares (SOS) Lyapunov functions. Given these results, we propose a SOS programming problem based on determinant maximization that yields an SOS Lyapunov function whose <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-sublevel set has minimal volume, is an attractor set itself, and provides an optimal outer approximation of the minimal attractor set of the ODE. Several numerical examples are presented including the Lorenz attractor and Van-der-Pol oscillator.</p>

Attractors of Direct Products

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows \(\varphi \) on \(S^2\) for which the attractor of the Cartesian square of \(\varphi \) is smaller than the Cartesian square of the attractor of \(\varphi \). In the example for the minimal attractors, the flow \(\varphi \) also has a global physical measure such that its square does not coincide with the global physical measure of the square of \(\varphi \).

Vertical scaling optimization algorithm for a fractal interpolation model of time offsets prediction in navigation systems

System time offsets between Global Navigation Satellite Systems is a key issue in multi-system positioning. A high-precision time offsets prediction model is required in actual navigation applications. In this study, the fractal characteristics of time offsets in American and Russian navigation systems are directly confirmed through fractal dimension calculations. Then, the influence of the vertical scaling factor on the prediction accuracy of the fractal interpolation prediction model is confirmed. It is found that perturbation on the estimated vertical scaling factor causes a minimal attractor bias, which is positively correlated with the prediction error. After proving the existence of the estimated vertical scaling factor of the minimal attractor biases, an optimization method of the fractal interpolation prediction model is proposed, and the related algorithms are given. The numerical results show that the time offsets prediction errors using the new model are less than 1 ns for the American navigation system and close to 2 ns for the Russian navigation system. The prediction accuracy of this model has an improvement of at least 60% over the commonly used quadratic and grey models, and an improvement of more than 13% over the standard fractal interpolation prediction model.

Small and minimal attractors of an IFS

Lasota and Myjak demonstrated that one can study attractors to an iterated function system (IFS) possibly containing discontinuous functions. We continue this line of thought by working with IFS with a lower semicontinuous Hutchinson-Barnsley operator and two new (but not significantly different) types of attractors, the small attractor (the smallest closed nonempty invariant set) and minimal attractors (a minimal closed nonempty invariant set). We characterize exactly when an IFS possesses a small attractor and provide several practically verifiable sufficient conditions for this. To study minimal attractors we create the notion of the weak basin; we show a minimal attractor behaves much like a small attractor on its weak basin and that the weak basin is the largest set in which a minimal attractor behaves like this. We then give a characterization of when the weak basin is open. Further, we show that, when the iterates of the Hutchinson-Barnsley operator of an IFS form an equicontinuous set of multifunctions, both the weak basin and the point wise basin (of a point wise attractor) is closed.

What can Koopmanism do for attractors in dynamical systems?

We characterize the longterm behavior of a semiflow on a compact space K by asymptotic properties of the corresponding Koopman semigroup. In particular, we compare different concepts of attractors, such as asymptotically stable attractors, Milnor attractors and centers of attraction. Furthermore, we give a characterization for the minimal attractor for each mentioned property. The main aspect is that we only need techniques and results for linear operator semigroups, since the Koopman semigroup permits a global linearization for a possibly non-linear semiflow.

Locally Topologically Generic Diffeomorphisms with Lyapunov Unstable Milnor Attractors

We prove that for every smooth compact manifold $M$ and any $r \ge 1$, whenever there is an open domain in $\mathrm{Diff}^r(M)$ exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic saddle, topologically generic diffeomorphisms in this domain have Lyapunov unstable Milnor attractors. This implies, in particular, that the instability of Milnor attractors is locally topologically generic in $C^1$ if $\mathrm{dim}\,M \ge 3$ and in $C^2$ if $\mathrm{dim}\,M = 2$. Moreover, it follows from the results of C. Bonatti, L. J. D\'iaz and E. R. Pujals that, for a $C^1$ topologically generic diffeomorphism of a closed manifold, either any homoclinic class admits some dominated splitting, or this diffeomorphism has an unstable Milnor attractor, or the inverse diffeomorphism has an unstable Milnor attractor. The same results hold for statistical and minimal attractors.

A global attractor for one semilinear hyperbolic equation with memory operator

The mixed problem for a semilinear hyperbolic equation with a memory operator is considered. The existence of a minimal global attractor for this problem is proved.

Robust Control of Linear Systems With Disturbances Bounded in a State Dependent Set

This technical note examines attractiveness and minimality of invariant sets for linear systems subject to additive disturbances confined in a state-dependent set. Existence of a minimal attractor is proved under the assumption that the state-dependent set, in which the disturbance is confined, is upper-semi-continuous. In many practical applications, the disturbance may evolve in a compact set while being generated by a dynamic process with a given model and bounded input. Our results for systems subject to general disturbances confined in state-dependent sets are applied to this case to prove the existence of a minimal robust invariant attractor.

Minimal attractors in digraph system models of neuronal networks

We study a class of discrete dynamical systems models of neuronal networks. In these models, each neuron is represented by a finite number of states and there are rules for how a neuron transitions from one state to another. In particular, the rules determine when a neuron fires and how this affects the state of other neurons. In an earlier paper [D. Terman, S. Ahn, X. Wang, W. Just, Reducing neuronal networks to discrete dynamics, Physica D 237 (2008) 324–338], we demonstrate that a general class of excitatory–inhibitory networks can, in fact, be rigorously reduced to the discrete model. In the present paper, we analyze how the connectivity of the network influences the dynamics of the discrete model. For randomly connected networks, we find two major phase transitions. If the connection probability is above the second but below the first phase transition, then starting in a generic initial state, most but not all cells will fire at all times along the trajectory as soon as they reach the end of their refractory period. Above the first phase transition, this will be true for all cells in a typical initial state; thus most states will belong to a minimal attractor of oscillatory behavior (in a sense that is defined precisely in the paper). The exact positions of the phase transitions depend on intrinsic properties of the cells including the lengths of the cells’ refractory periods and the thresholds for firing. Existence of these phase transitions is both rigorously proved for sufficiently large networks and corroborated by numerical experiments on networks of moderate size.

Reducing neuronal networks to discrete dynamics

We consider a general class of purely inhibitory and excitatory–inhibitory neuronal networks, with a general class of network architectures, and characterize the complex firing patterns that emerge. Our strategy for studying these networks is to first reduce them to a discrete model. In the discrete model, each neuron is represented as a finite number of states and there are rules for how a neuron transitions from one state to another. In this paper, we rigorously demonstrate that the continuous neuronal model can be reduced to the discrete model if the intrinsic and synaptic properties of the cells are chosen appropriately. In a companion paper [W. Just, S. Ahn, D. Terman. Minimal attractors in digraph system models of neuronal networks (preprint)], we analyse the discrete model.

Trajectory and Global Attractors of the Boundary Value Problem for Autonomous Motion Equations of Viscoelastic Medium

We introduce the concept of minimal trajectory attractor generalizing the known concept of trajectory attractor of an abstract evolution equation. We obtain several results on existence and properties of minimal trajectory and global attractors without assumptions of any invariance of the trajectory space of an equation. With the help of these results we prove existence of minimal trajectory and global attractors for weak solutions of the boundary value problem for autonomous motion equations of an incompressible viscoelastic medium with the Jeffreys constitutive law.

An example of non-coincidence of minimal and statistical attractors

In this work we construct an example of a smooth dynamical system for which the minimal and statistical attractors do not coincide. In the language of SRB measures, this is an example for which there exists a unique natural measure, but there is no observable one. Also, as the statistical attractor describes the mean behaviour of individual points, and the minimal one the mean behaviour of the Lebesgue measure, this example provides us with a dynamical realization of the Riesz example.

On delta-modulated control: A simple system with complex dynamics

Abstract In this paper, we investigate some interesting properties of a scalar system controlled by Δ -modulated feedback. We show that there are three different cases. In the first case, there is a minimal global attractor which consists of only two points. The two points form either one 2-periodic orbit or two 1-periodic orbits (fixed points). We also characterize the attracting region for each of these two points. In the second case, the maximal stabilizable region is bounded, and there is a minimal local attractor inside this stabilizable region. In the third case, the maximal stabilizable set is a Cantor set, which is a repeller of the system, and the system is chaotic on the Cantor set.

A $C^{\infty}$ diffeomorphism with infinitely many intermingled basins

Let M be the four-dimensional compact manifold $M=T^2\times S^2$ and let $k\ge2$. We construct a $C^\infty$ diffeomorphism $F:M\to M$ with precisely k intermingled minimal attractors $A_1,\dotsc, A_k$. Moreover the union of the basins is a set of full Lebesgue measure. This means that Lebesgue almost every point in M lies in the basin of attraction of Aj for some j, but every non-empty open set in M has a positive measure intersection with each basin. We also construct $F:M\to M$ with a countable infinity of intermingled minimal attractors.

Attractors for General Operators

In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua.

Minimal attractors and bifurcations of random dynamical systems

We consider attractors for certain types of random dynamical systems. These are skew–product systems whose base transformations preserve an ergodic invariant measure. We discuss definitions of invariant sets, attractors and invariant measures for deterministic and random dynamical systems. Under assumptions that include, for example, iterated function systems, but that exclude stochastic differential equations, we demonstrate how random attractors can be seen as examples of Milnor attractors for a skew–product system. We discuss the minimality of these attractors and invariant measures supported by them. As a further connection between random dynamical systems and deterministic dynamical systems, we show how dynamical or D–bifurcations of random attractors with multiplicative noise can be seen as blowout bifurcations, and we relate the issue of branching at such D–bifurcations to branching at blowout bifurcations.

Vertex-reinforced random walks and a conjecture of Pemantle

We discuss and disprove a conjecture of Pemantle concerning vertex-reinforced random walks. The setting is a general theory of non-Markovian discrete-time random 4 processes on a finite space $E = {1, \dots, d}$, for which the transition probabilities at each step are influenced by the proportion of times each state has been visited. It is shown that, under mild conditions, the asymptotic behavior of the empirical occupation measure of the process is precisely related to the asymptotic behavior of some deterministic dynamical system induced by a vector field on the $d - 1$ unit simplex. In particular, any minimal attractor of this vector field has a positive probability to be the limit set of the sequence of empirical occupation measures. These properties are used to disprove a conjecture and to extend some results due to Pemantle. Some applications to edge-reinforced random walks are also considered.

On the Existence of the Compact Global Attractor for Semilinear Reaction Diffusion Systems on [formula omitted

We show that a class of reaction diffusion systems on RNgenerates an asymptotically compact semiflow on the Banach space of bounded uniformly continuous functions. If such a semiflow is dissipative, then a unique, non-empty, compact minimal attractor is known to exist. We apply this abstract result to obtain the existence of the compact minimal attractor for reaction diffusion systems on RNthat contain appropriate weight functions. We also state conditions, which guarantee that the attractor has finite Hausdorff-dimension.

MINIMAL AND STRANGE ATTRACTORS

A general concept going back to Kolmogorov claims that if a dynamical system has a complicated attracting set then its behavior has not a deterministic, but rather probabilistic character. This concept was not formalized up to now. Even the definition of attractor has a lot of different versions. This paper presents an attempt to give some definitions and results formalizing this heuristic ideas. It contains a definition of a minimal attractor, modifying the one given in Ilyashenko [1991]. The actual minimality of the attractor is discussed. The principal result is the Triple Choice Theorem. It claims that the existence of a strange minimal attractor implies some mild form of chaos for the map itself or for a nearby one. The program of further investigation is proposed as a chain of problems at the end of the paper.

On the minimal global attractor of a system of phase field equations

The unique global solvability of the initial-boundary value problem (1)–(3) is proved for the system of phase field equations (1), (2). It is shown that the problem (1)–(3) generates a continuous compact semigroup Vt, t>0, for which there exists a minimal global B-attractor.