Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x_n+1=g(x_n) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.     

Construction of Lyapunov functions for nonlinear planar systems by linear programming

Recently the authors proved the existence of piecewise affine Lyapunov functions for dynamical systems with an exponentially stable equilibrium in two dimensions (Giesl and Hafstein, 2010 [7]). Here, we extend these results by designing an algorithm to explicitly construct such a Lyapunov function. We do this by modifying and extending an algorithm to construct Lyapunov functions first presented in Marinosson (2002) [17] and further improved in Hafstein (2007) [10]. The algorithm constructs a linear programming problem for the system at hand, and any feasible solution to this problem parameterizes a Lyapunov function for the system. We prove that the algorithm always succeeds in constructing a Lyapunov function if the system possesses an exponentially stable equilibrium. The size of the region of the Lyapunov function is only limited by the region of attraction of the equilibrium and it includes the equilibrium.     

The estimate of the time of motion of certain dynamical systems

An estimate of the time necessary for the phase points of a dynamical system to reach a specified finite domain, containing an asymptotically stable solution, from any initial position belonging to the specified domain is obtained with the sole assumption that the derivative of the Lyapunov function for autonomous second-order systems and for certain higher-order systems has a negative sign.     

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium?s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C²$C^2$ and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.     

An alternative proof of the Barker, Berman, Plemmons (BBP) result on diagonal stability and extensions

We revisit the theorem of Barker, Berman and Plemmons on the existence of a diagonal quadratic Lyapunov function for a stable linear time-invariant (LTI) dynamical system [G.P. Barker, A. Berman, R.J. Plemmons, Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra 5(3) (1978) 249-256]. We use recently derived results to provide an alternative proof of this result and to derive extensions.     

Global stability of a generalized epidemic model

The competitive exclusion principle is one of the most interesting and important phenomena in both theoretical epidemiology and biology. We show that the equilibrium in which only the strain with the maximum basic reproductive number exists is globally asymptotically stable by using an average Lyapunov function theorem and some dynamical system theory. This result is anticipated by H.J. Bremermann and H.R. Thieme (1989) [6] where they showed that the equilibrium is locally stable — the global result has not been established previously.     

Existence of piecewise affine Lyapunov functions in two dimensions

In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.     

Structure and properties of the attractor of a marine dynamical system

We investigate the properties of a marine dynamical system by means of time series of the sea-level height at four locations in the Saronicos Gulf in the Aegean Sea, Greece. In order to characterize the dynamics, we estimate the dimension of the underlying system attractor, and we compute its Lyapunov exponents. Dimension estimates indicate that the dynamics can be explained by a low-dimensional deterministic dynamical system. Lyapunov exponent estimates further substantiate the above conclusion, while at the same time, indicate that the dynamical system is a rather nonuniform chaotic one.     

Semistability of switched dynamical systems,Part II: Non-linear system theory

In a companion paper [Q. Hui, W.M. Haddad, Semistability of switched dynamical systems. Part I: Linear system theory, Non-linear Anal. Hybrid Syst. 3 (3) (2009) 343–353] semistability and uniform semistability results for switched linear systems were developed. In this paper we develop semistability analysis results for non-linear switched systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. The main results of the paper involve sufficient conditions for semistability using multiple Lyapunov functions and integral-type inequalities.     

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.     

Finite-time stabilization of nonlinear impulsive dynamical systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. For continuous-time dynamical systems finite-time convergence implies nonuniqueness of system solutions in reverse time, and hence, such systems possess non-Lipschitzian dynamics. For impulsive dynamical systems, however, it may be possible to reset the system states to an equilibrium state achieving finite-time convergence without requiring non-Lipschitzian system dynamics. In this paper, we develop sufficient conditions for finite-time stability of impulsive dynamical systems using both scalar and vector Lyapunov functions. Furthermore, we design hybrid finite-time stabilizing controllers for impulsive dynamical systems that are robust against full modelling uncertainty. Finally, we present a numerical example for finite-time stabilization of large-scale impulsive dynamical systems.     

On invariant closed curves for one-step methods

Summary We show that a one-step method as applied to a dynamical system with a hyperbolic periodic orbit, exhibits an invariant closed curve for sufficiently small step size. This invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit. The dynamics of the one-step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression. Our results complement those of [2, 3] where one-step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a Hopf bifurcation.     

Time-reversal symmetry,Poincaré recurrence,irreversibility, and the entropic arrow of time: From mechanics to system thermodynamics

In this paper, we use a large-scale dynamical systems perspective to provide a system-theoretic foundation for thermodynamics. Specifically, using a state space formulation, we develop a nonlinear compartmental dynamical system model characterized by energy conservation laws that is consistent with basic thermodynamic principles. In addition, we establish the existence of a unique, continuously differentiable global entropy function for our large-scale dynamical system, and using Lyapunov stability theory we show that the proposed thermodynamic model has convergent trajectories to Lyapunov stable equilibria determined by the system initial energies. Finally, using the system entropy, we establish the absence of Poincaré recurrence for our thermodynamic model and develop a clear connection between irreversibility, the second law of thermodynamics, and the entropic arrow of time.     

Control and adaptive modified function projective synchronization of Liu chaotic dynamical system

In this work, the feedback control method is proposed to control the behaviour of Liu chaotic dynamical system. The controlled system is stable under some conditions on the parameters of the system determined by Routh-Hurwitz criterion. This paper also presents the adaptive modified function projective synchronization (AMFPS) between two identical Liu chaotic dynamical systems. Based on the Lyapunov stability theorem, adaptive control laws are designed to achieving the AMFPS. Finally, some numerical simulations are obtained to validate the proposed methods.     

Semistability of switched dynamical systems,Part I: Linear system theory

This paper develops semistability and uniform semistability analysis results for switched linear systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system’s initial conditions. Since solutions to switched systems are a function of the system’s initial conditions as well as the switching signals, uniformity here refers to the convergence rate of the multiple solutions as the switching signal evolves over a given switching set. The main results of the paper involve sufficient conditions for semistability and uniform semistability using multiple Lyapunov functions and sufficient regularity assumptions on the class of switching signals considered.     

Aut0(Λ<Superscript>2</Superscript>)-symmetrical 4-extensions of the 2-dimensional grid Λ<Superscript>2</Superscript>

There are different non-equivalent definitions of attractors in the theory of dynamical systems. The most common are two definitions: the maximal attractor and the Milnor attractor. The maximal attractor is by definition Lyapunov stable, but it is often in some ways excessive. The definition of Milnor attractor is more realistic from the physical point of view. The Milnor attractor can be Lyapunov unstable though. One of the central problems in the theory of dynamical systems is the question of how typical such a phenomenon is. This article is motivated by this question and contains new examples of so-called relatively unstable Milnor attractors. Recently I. Shilin has proved that these attractors are Lyapunov stable in the case of one-dimensional fiber under some additional assumptions. However, the question of their stability in the case of multidimensional fiber is still an open problem.     

Feedback Stabilization Methods for the Solution of Nonlinear Programming Problems

In this work, we show that, given a nonlinear programming problem, it is possible to construct a family of dynamical systems, defined on the feasible set of the given problem, so that: (a) the equilibrium points are the unknown critical points of the problem, which are asymptotically stable, (b) each dynamical system admits the objective function of the problem as a Lyapunov function, and (c) explicit formulas are available without involving the unknown critical points of the problem. The construction of the family of dynamical systems is based on the Control Lyapunov Function methodology, which is used in mathematical control theory for the construction of stabilizing feedback. The knowledge of a dynamical system with the previously mentioned properties allows the construction of algorithms, which guarantee the global convergence to the set of the critical points.     

Control vector Lyapunov functions for large-scale impulsive dynamical systems

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.     

Smooth Lyapunov Functions for Discontinuous Stable Systems

It has been proved that a differential system d x / d t = f(t, x) with a discontinuous right-hand side admits some continuous weak Lyapunov function if and only if it is robustly stable. This paper focuses on the smoothness of such a Lyapunov function. An example of an (asymptotically) stable system for which there does not exist any (even weak) Lyapunov functions of class C ¹ is given. In the more general context of differential inclusions, the existence of a weak Lyapunov function of class C ¹ (or C ) is shown to be equivalent to the robust stability of some perturbed system obtained in introducing measurement error with respect to x and t. This condition is proved to be satisfied by most of the robustly stable systems encountered in the literature. Analogous results are given for the Lagrange stability. As an application to the study of the links between internal and external stability for control systems, an extension of a result by Bacciotti and Beccari is obtained by means of a smooth Lyapunov function associated with a robustly Lagrange stable system.