Smale Systems Research Articles

Non-chattering finite-time and fixed-time flocking of Cucker–Smale systems

In this paper, the non-chattering finite/fixed-time flocking problems of Cucker–Smale (C–S) systems are investigated. To address the chattering issue, two novel chattering-free protocols are proposed to achieve the flocking of C–S systems within a finite/fixed time. Using the stability theory of differential equations, sufficient conditions for achieving the flocking are established, and the upper bounds of convergence time are given. The connection between the chattering-free flocking protocols and the classic finite/fixed-time control schemes involving the signum function is shown for the first time. Moreover, we discover a correlation between the settling time and the group size. Finally, the effectiveness of the presented protocols is validated via several simulations.

On limit measures and their supports for stochastic ordinary differential equations

This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin–Wentzell or Dembo–Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse–Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincaré–Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. Four nontrivial examples admitting a unique limit measure are provided, two of which possess infinite equivalent classes and their supports are Liapunov stable periodic orbits, one is supported on saddle-nodes.

Emergent behaviors of thermodynamic Cucker–Smale ensemble with a unit-speed constraint

We propose the thermodynamic Cucker–Smale system with unit-speed constraints (TCSUS) and study its emergent dynamics with rigorous proofs. For this, three dissipative differential inequalities are derived in terms of $ L^\infty $-diameters of the position–velocity–temperature to verify mono-cluster flocking in the TCSUS model under suitable frameworks using two approaches. Motivated by the first approach, we provide a sufficient framework for bi-cluster flocking to the TCSUS model. Unlike to the mono-cluster flocking behavior of TCSUS, the decay rate of velocity–temperature for each cluster group is not exponential and is dependent on the decay rate of the corresponding communication weight, respectively.

Community integration algorithms (CIAs) for dynamical systems on networks

Dynamics of large-scale network processes underlies crucial phenomena ranging across all sciences. Forward simulation of large network models is often computationally prohibitive. Yet, most networks have intrinsic community structure. We exploit these communities and propose a fast simulation algorithm for network dynamics. In particular, aggregating the inputs a node receives constitutes the limiting factor in numerically simulating large-scale network dynamics. We develop community integration algorithms (CIAs) significantly reducing function-evaluations. We obtain a substantial reduction from polynomial to linear computational complexity. We illustrate our results in multiple applications including classical and higher-order Kuramoto-type systems for synchronisation and Cucker–Smale systems exhibiting flocking behaviour on synthetic as well as real-world networks. Numerical comparison and theoretical analysis confirm the robustness and efficiency of CIAs.

Controlled pattern formation of stochastic Cucker–Smale systems with network structures

We present a new stochastic particle system on networks which exhibits flocking behavior and pattern formation. More precisely, we consider Cucker–Smale particles with decentralized formation control and multiplicative noises on symmetric connected networks. Under suitable assumptions on the initial configurations and the network structure, we establish time-asymptotic stochastic flocking behavior and pattern formation of solutions for the proposed stochastic particle system. Our approach is based on the Lyapunov functional energy estimates and does not require any spectral information of the graph associated with the network structure.

Hydrodynamic limit of a coupled Cucker–Smale system with strong and weak internal variable relaxation

In this paper, we present the hydrodynamic limit of a multiscale system describing the dynamics of two populations of agents with alignment interactions and the effect of an internal variable. It consists of a kinetic equation coupled with an Euler-type equation inspired by the thermomechanical Cucker–Smale (TCS) model. We propose a novel drag force for the fluid-particle interaction reminiscent of Stokes’ law. While the macroscopic species is regarded as a self-organized background fluid that affects the kinetic species, the latter is assumed sparse and does not affect the macroscopic dynamics. We propose two hyperbolic scalings, in terms of a strong and weak relaxation regime of the internal variable towards the background population. Under each regime, we prove the rigorous hydrodynamic limit towards a coupled system composed of two Euler-type equations. Inertial effects of momentum and internal variable in the kinetic species disappear for strong relaxation, whereas a nontrivial dynamics for the internal variable appears for weak relaxation. Our analysis covers both the case of Lipschitz and weakly singular influence functions.

Collision avoiding finite-time and fixed-time flocking of Cucker–Smale systems with pinning control

Dealing with the flocking problems of Cucker–Smale systems is a challenging issue. In this paper, the finite-time and fixed-time flocking problems of the Cucker–Smale system are investigated. By designing new continuous non-Lipschitz pinning controllers and using the theory of differential equations, sufficient conditions for the flocking of Cucker–Smale systems are obtained. Besides, sufficient conditions are given to ensure that there is no collision during the process of flocking. The effect of control parameters on the convergence speed is also investigated. The results show that the convergence time is related to control parameters and the density of pinned nodes. Finally, some numerical simulations are used to verify the theoretical results.

Fixed-time flocking problem of a Cucker–Smale type self-propelled particle model

In this paper, we consider the fixed-time flocking problem of the Cucker–Smale system. We present a variation of the standard Cucker–Smale system, and a simple fixed-time controller is designed. Based on the fixed-time stability theory, sufficient conditions for the system to achieve flocking within fixed time are obtained. Both our theoretical and numerical results demonstrate that the convergence time depends on the group size, and the group with high density can transit to ordered collective motion more rapidly.

Entropy Hierarchies for Equations of Compressible Fluids and Self-Organized Dynamics

We develop a method of obtaining a hierarchy of new higher-order entropies in the context of compressible models with local and nonlocal diffusion and isentropic pressure. The local viscosity is allowed to degenerate as the density approaches vacuum. The method provides a tool to propagate initial regularity of classical solutions provided no vacuum has formed and serves as an alternative to the classical energy method. We obtain a series of global well-posedness results for state laws in previously uncovered cases, including $p(\rho) = c_p \rho$. As an application we prove global well-posedness of collective behavior models with pressure arising from an agent-based Cucker--Smale system.

Effects of Measurement Noise on Flocking Dynamics of Cucker–Smale Systems

Understanding the emergence of coordination behavior is of paramount importance in science and engineering. Environmental noise may have important impact on the coordination. In this brief, we consider a variation of the standard Cucker-Smale system in which individuals can not measure the states of others exactly. We present sufficient conditions on group size, noise intensity and communication rate to guarantee the almost surely exponential convergence toward flocking. We challenge the prevailing paradigm that noise suppressing the emergence of collective motion and report that noise may facilitate the emergence of ordered flocking behavior. We also find that group with high density can reach the ordered state more rapidly.

Consensus and flocking under communication failures for a class of Cucker–Smale systems

In this paper, we study sufficient conditions for the emergence of asymptotic consensus and flocking in a certain class of non-linear generalised Cucker-Smale systems subject to multiplicative communication failures. Our approach is based on the combination of strict Lyapunov design together with the formulation of a suitable persistence condition for multi-agent systems. The latter can be interpreted as a lower bound on the algebraic connectivity of the time-average of the interaction graph generated by the communication weights, and provides quantitative decay estimates for the variance functional along the solutions of the system.

Morse–Smale Systems and Topological Structure of Carrier Manifolds

We review the results describing the connection between the global dynamics of Morse–Smale systems on closed manifolds and the topology of carrier manifolds. Also we consider the results related to topological classification of Morse–Smale systems.

Classification of Morse–Smale systems and topological structure of the underlying manifolds

Morse–Smale systems arise naturally in applications for mathematical modelling of processes with regular dynamics (for example, in chains of coupled maps describing diffusion reactions, or in the study of the topology of magnetic fields in a conducting medium, in particular, in the study of the question of existence of separators in magnetic fields of highly conducting media). Since mathematical models in the form of Morse–Smale systems appear in the description of processes of various nature, the first step in the study of such models is to distinguish properties independent of the physical context but determining a partition of the phase space into trajectories. The relation preserving the partition into trajectories up to a homeomorphism is called topological equivalence, and the relation preserving also the time of motion along trajectories (continuous in the case of flows, and discrete in the case of cascades) is called topological conjugacy. The problem of topological classification of dynamical systems consists in finding invariants that uniquely determine the equivalence class or the conjugacy class for a given system. The present survey is devoted to a description of results on topological classification of Morse–Smale systems on closed manifolds, including results recently obtained by the authors. Also presented are recent results of the authors concerning the interconnections between the global dynamics of such systems and the topological structure of the underlying manifolds. Bibliography: 112 titles.

On the structure of the ambient manifold for Morse–Smale systems without heteroclinic intersections

It is shown that if a closed smooth orientable manifold M n , n ≥ 3, admits a Morse–Smale system without heteroclinic intersections (the absence of periodic trajectories is additionally required in the case of a Morse–Smale flow), then this manifold is homeomorphic to the connected sum of manifolds whose structure is interconnected with the type and number of points that belong to the non-wandering set of the Morse–Smale system.

Certified computation of planar Morse–Smale complexes

The Morse–Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse–Smale complexes have been presented in case of piecewise linear manifolds (Edelsbrunner et al., 2003a). However, previous research in this field is incomplete in the case of smooth functions. In the current paper we address the following question: Given an arbitrarily complex Morse–Smale system on a planar domain, is it possible to compute its certified (topologically correct) Morse–Smale complex? Towards this, we develop an algorithm using interval arithmetic to compute certified critical points and separatrices forming the Morse–Smale complexes of smooth functions on bounded planar domain. Our algorithm can also compute geometrically close Morse–Smale complexes.

Non-autonomous dynamical systems

This review paper treats the dynamics of non-autonomous dynamical systems. To study the forwards asymptotic behaviour of a non-autonomous differential equation we need to analyse the asymptotic configurations of the non-autonomous terms present in the equations. This fact leads to the definition of concepts such as skew-products and cocycles and their associated global, uniform, and cocycle attractors. All of them are closely related to the study of the pullback asymptotic limits of the dynamical system, from which naturally emerges the concept of a pullback attractor. In the first part of this paper we want to clarify these different dynamical scenarios and the relations between their corresponding attractors.  &nbspIf the global attractor of an autonomous dynamical system is given as the union of a finite number of unstable manifolds of equilibria, a detailed understanding of the continuity of the local dynamics under perturbation leads to important results on the lower-semicontinuity and topological structural stability for the pullback attractors of evolution processes that arise from small non-autonomous perturbations, with respect to the limit regime. Finally, continuity with respect to global dynamics under non-autonomous perturbation is also studied, for which appropriate concepts for Morse decomposition of attractors and non-autonomous Morse--Smale systems are introduced. All of these results will also be considered for uniform attractors. As a consequence, this paper also makes connections between different approaches to the qualitative theory of non-autonomous differential equations, which are often treated independently.

Emergent Dynamics for the Hydrodynamic Cucker--Smale System in a Moving Domain

We study the emergent dynamics for the hydrodynamic Cucker--Smale system arising in the modeling of flocking dynamics in interacting many-body systems. Specifically, the initial value problem with a moving domain is considered to investigate the global existence and time-asymptotic behavior of classical solutions, provided that the initial mass density has bounded support and the initial data are in an appropriate Sobolev space. In order to show the emergent behavior of flocking, we make use of an appropriate Lyapunov functional that measures the total fluctuation in the velocity relative to the mean velocity. In our analysis, we present the local well-posedness of the smooth solutions via Lagrangian coordinates, and we extend to the global-in-time solutions by establishing the uniform flocking estimates.

Bruhat Order in Full Symmetric Toda System

In this paper we discuss some geometrical and topological properties of the full symmetric Toda system. We show by a direct inspection that the phase transition diagram for the full symmetric Toda system in dimensions n = 3, 4 coincides with the Hasse diagram of the Bruhat order of symmetric groups S 3 and S 4. The method we use is based on the existence of a vast collection of invariant subvarieties of the Toda flow in orthogonal groups. We show how one can extend it to the case of general n. The resulting theorem identifies the set of singular points of dim = n Toda flow with the elements of the permutation group S n , so that points will be connected by a trajectory, if and only if the corresponding elements are Bruhat comparable. We also show that the dimension of the submanifolds, spanned by the trajectories connecting two singular points, is equal to the length of the corresponding segment in the Hasse diagram. This is equivalent to the fact that the full symmetric Toda system is in fact a Morse–Smale system.

Morse–Smale systems with few non-wandering points

Let MSflow(Mn,k) and MSdiff(Mn,k) be Morse–Smale flows and diffeomorphisms respectively the non-wandering set of those consists of k fixed points on a closed n-manifold Mn. For k=3, we show that the only values of n possible are n∈{2,4,8,16}, and M2 is the projective plane. For n⩾4, Mn is simply connected and orientable. We prove that the closure of any separatrix of ft∈MSflow(Mn,3) is a locally flat n2-sphere while there is ft∈MSflow(Mn,4) such that the closure of separatrix of ft is a wildly embedded codimension two sphere. This allows us to classify flows from MSflow(M4,3). For n⩾6, one proves that the closure of any separatrix of f∈MSdiff(Mn,3) is a locally flat n2-sphere while there is f∈MSdiff(M4,3) such that the closure of any separatrix is a wildly embedded 2-sphere.

Vector Fields in the Interior of Kupka–Smale Systems Satisfy Axiom A

In 1994, S. Hu conjectured in his paper (1994, Trans. Amer. Math. Soc.342, 753–772) that Axiom A systems are dense in the interior of Kupka–Smale systems. We will show that if X is in the interior of Kupka–Smale systems, then X satisfies Axiom A.