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Discrete & Continuous Dynamical Systems - B

2015 , Volume 20 , Issue 8

Special Section Papers: 2291-2526; Regular Papers: 2527-2763

Special Section Papers are related to the Special Section on computational methods for Lyapunov functions
    Guest Editors: Peter Giesl and Sigurdur Hafstein

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Computational methods for Lyapunov functions
Peter Giesl and Sigurdur Hafstein
2015, 20(8): i-ii doi: 10.3934/dcdsb.2015.20.8i +[Abstract](699) +[PDF](95.7KB)
Abstract:
Lyapunov functions, introduced by Lyapunov more than 100 years ago, are to this day one of the most important tools in the stability analysis of dynamical systems. They are functions which decrease along solution trajectories of systems, and they can be used to show stability of an invariant set, such as an equilibrium, as well as to determine its basin of attraction. Lyapunov functions have been considered for a variety of dynamical systems, such as continuous-times, discrete-time, linear, non-linear, non-smooth, switched, etc. Lyapunov functions are used and studied in different communities, such as Mathematics, Informatics and Engineering, often using different notations and methods.

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Review on computational methods for Lyapunov functions
Peter Giesl and Sigurdur Hafstein
2015, 20(8): 2291-2331 doi: 10.3934/dcdsb.2015.20.2291 +[Abstract](1015) +[PDF](948.3KB)
Abstract:
Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them.
    Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ different methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function.
Classical converse theorems in Lyapunov's second method
Christopher M. Kellett
2015, 20(8): 2333-2360 doi: 10.3934/dcdsb.2015.20.2333 +[Abstract](711) +[PDF](606.5KB)
Abstract:
Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.
Advances in computational Lyapunov analysis using sum-of-squares programming
James Anderson and Antonis Papachristodoulou
2015, 20(8): 2361-2381 doi: 10.3934/dcdsb.2015.20.2361 +[Abstract](576) +[PDF](592.9KB)
Abstract:
The stability of an equilibrium point of a nonlinear dynamical system is typically determined using Lyapunov theory. This requires the construction of an energy-like function, termed a Lyapunov function, which satisfies certain positivity conditions. Unlike linear dynamical systems, there is no algorithmic method for constructing Lyapunov functions for general nonlinear systems. However, if the systems of interest evolve according to polynomial vector fields and the Lyapunov functions are constrained to be sum-of-squares polynomials then stability verification can be cast as a semidefinite (convex) optimization programme. In this paper we describe recent advances in sum-of-squares programming that facilitate advanced stability analysis and control design.
Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares
Reza Kamyar and Matthew M. Peet
2015, 20(8): 2383-2417 doi: 10.3934/dcdsb.2015.20.2383 +[Abstract](807) +[PDF](1592.3KB)
Abstract:
In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of $H_\infty$ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.
Efficient computation of Lyapunov functions for Morse decompositions
Arnaud Goullet, Shaun Harker, Konstantin Mischaikow, William D. Kalies and Dinesh Kasti
2015, 20(8): 2419-2451 doi: 10.3934/dcdsb.2015.20.2419 +[Abstract](613) +[PDF](3150.6KB)
Abstract:
We present an efficient algorithm for constructing piecewise constant Lyapunov functions for dynamics generated by a continuous nonlinear map defined on a compact metric space. We provide a memory efficient data structure for storing nonuniform grids on which the Lyapunov function is defined and give bounds on the complexity of the algorithm for both time and memory. We prove that if the diameters of the grid elements go to zero, then the sequence of piecewise constant Lyapunov functions generated by our algorithm converge to a continuous Lyapunov function for the dynamics generated the nonlinear map. We conclude by applying these techniques to two problems from population biology.
Grid refinement in the construction of Lyapunov functions using radial basis functions
Najla Mohammed and Peter Giesl
2015, 20(8): 2453-2476 doi: 10.3934/dcdsb.2015.20.2453 +[Abstract](521) +[PDF](1393.5KB)
Abstract:
Lyapunov functions are a main tool to determine the domain of attraction of equilibria in dynamical systems. Recently, several methods have been presented to construct a Lyapunov function for a given system. In this paper, we improve the construction method for Lyapunov functions using Radial Basis Functions. We combine this method with a new grid refinement algorithm based on Voronoi diagrams. Starting with a coarse grid and applying the refinement algorithm, we thus manage to reduce the number of data points needed to construct Lyapunov functions. Finally, we give numerical examples to illustrate our algorithms.
Computation of local ISS Lyapunov functions with low gains via linear programming
Huijuan Li, Robert Baier, Lars Grüne, Sigurdur F. Hafstein and Fabian R. Wirth
2015, 20(8): 2477-2495 doi: 10.3934/dcdsb.2015.20.2477 +[Abstract](611) +[PDF](665.7KB)
Abstract:
In this paper, we present a numerical algorithm for computing ISS Lyapunov functions for continuous-time systems which are input-to-state stable (ISS) on compact subsets of the state space. The algorithm relies on a linear programming problem and computes a continuous piecewise affine ISS Lyapunov function on a simplicial grid covering the given compact set excluding a small neighborhood of the origin. The objective of the linear programming problem is to minimize the gain. We show that for every ISS system with a locally Lipschitz right-hand side our algorithm is in principle able to deliver an ISS Lyapunov function. For $C^2$ right-hand sides a more efficient algorithm is proposed.
Separable Lyapunov functions for monotone systems: Constructions and limitations
Gunther Dirr, Hiroshi Ito, Anders Rantzer and Björn S. Rüffer
2015, 20(8): 2497-2526 doi: 10.3934/dcdsb.2015.20.2497 +[Abstract](570) +[PDF](707.0KB)
Abstract:
For monotone systems evolving on the positive orthant of $\mathbb{R}^n_+$ two types of Lyapunov functions are considered: Sum- and max-separable Lyapunov functions. One can be written as a sum, the other as a maximum of functions of scalar arguments. Several constructive existence results for both types are given. Notably, one construction provides a max-separable Lyapunov function that is defined at least on an arbitrarily large compact set, based on little more than the knowledge about one trajectory. Another construction for a class of planar systems yields a global sum-separable Lyapunov function, provided the right hand side satisfies a small-gain type condition. A number of examples demonstrate these methods and shed light on the relation between the shape of sublevel sets and the right hand side of the system equation. Negative examples show that there are indeed globally asymptotically stable systems that do not admit either type of Lyapunov function.
On the homogenization of multicomponent transport
Grégoire Allaire and Harsha Hutridurga
2015, 20(8): 2527-2551 doi: 10.3934/dcdsb.2015.20.2527 +[Abstract](480) +[PDF](531.4KB)
Abstract:
This paper is devoted to the homogenization of weakly coupled cooperative parabolic systems in strong convection regime with purely periodic coefficients. Our approach is to factor out oscillations from the solution via principal eigenfunctions of an associated spectral problem and to cancel any exponential decay in time of the solution using the principal eigenvalue of the same spectral problem. We employ the notion of two-scale convergence with drift in the asymptotic analysis of the factorized model as the lengthscale of the oscillations tends to zero. This combination of the factorization method and the method of two-scale convergence is applied to upscale an adsorption model for multicomponent flow in an heterogeneous porous medium.
Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3,1/2]$
María J. Garrido–Atienza, Kening Lu and Björn Schmalfuss
2015, 20(8): 2553-2581 doi: 10.3934/dcdsb.2015.20.2553 +[Abstract](690) +[PDF](556.2KB)
Abstract:
In this article we are concerned with the study of the existence and uniqueness of pathwise mild solutions to evolutions equations driven by a Hölder continuous function with Hölder exponent in $(1/3,1/2)$. Our stochastic integral is a generalization of the well-known Young integral. To be more precise, the integral is defined by using a fractional integration by parts formula and it involves a tensor for which we need to formulate a new equation. From this it turns out that we have to solve a system consisting of a path and an area equations. In this paper we prove the existence of a unique local solution of the system of equations. The results can be applied to stochastic evolution equations with a non-linear diffusion coefficient driven by a fractional Brownian motion with Hurst parameter in $(1/3,1/2]$, which in particular includes white noise.
Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one
Yingwen Guo and Yinnian He
2015, 20(8): 2583-2609 doi: 10.3934/dcdsb.2015.20.2583 +[Abstract](734) +[PDF](507.4KB)
Abstract:
In this paper, we study a fully discrete finite element method with second order accuracy in time for the equations of motion arising in the Oldroyd model of viscoelastic fluids. This method is based on a finite element approximation for the space discretization and the Crank-Nicolson/Adams-Bashforth scheme for the time discretization. The integral term is discretized by the trapezoidal rule to match with the second order accuracy in time. It leads to a linear system with a constant matrix and thus greatly increases the computational efficiency. Taking the nonnegativity of the quadrature rule and the technique of variable substitution for the trapezoidal rule approximation, we prove that this fully discrete finite element method is almost unconditionally stable and convergent. Furthermore, by the negative norm technique, we derive the $H^1$ and $L^2$-optimal error estimates of the velocity and the pressure.
Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system
Sirui Li, Wei Wang and Pingwen Zhang
2015, 20(8): 2611-2655 doi: 10.3934/dcdsb.2015.20.2611 +[Abstract](608) +[PDF](702.5KB)
Abstract:
In this paper, we consider a hydrodynamic $Q$-tensor system for nematic liquid crystal flow, which is derived from Doi-Onsager molecular theory by the Bingham closure. We first prove the existence and uniqueness of local strong solution. Furthermore, by taking Deborah number goes to zero and using the Hilbert expansion method, we present a rigorous derivation from the molecule-based $Q$-tensor theory to the Ericksen-Leslie theory.
Analytic integrability of a class of planar polynomial differential systems
Jaume Llibre and Claudia Valls
2015, 20(8): 2657-2661 doi: 10.3934/dcdsb.2015.20.2657 +[Abstract](512) +[PDF](278.4KB)
Abstract:
In this paper we find necessary and sufficient conditions in order that the differential systems of the form $\dot x = x f(y)$, $\dot y =g(y)$, with $f$ and $g$ polynomials, have a first integral which is analytic in the variable $x$ and meromorphic in the variable $y$. We also characterize their analytic first integrals in both variables $x$ and $y$.
    These polynomial differential systems are important because after a convenient change of variables they contain all quasi--homogeneous polynomial differential systems in $\mathbb{R}^2$.
Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays
Francisco Montes de Oca and Liliana Pérez
2015, 20(8): 2663-2690 doi: 10.3934/dcdsb.2015.20.2663 +[Abstract](565) +[PDF](518.4KB)
Abstract:
The qualitative properties of certain type of nonautonomous competitive Lotka-Volterra systems with infinite delay are considered.
    By constructing suitable Lyapunov-type functional, we establish a series of easily verifiable algebraic conditions on the coefficients and the kernels, which are sufficient to ensure the extinction and survival of a determined number of species. The surviving part stabilizes around any solution of a subsystem of the systems in study. These conditions also guarantee the persistence, extreme stability and asymptotic behavior of the systems.
Coexistence solutions of a competition model with two species in a water column
Hua Nie, Sze-Bi Hsu and Jianhua Wu
2015, 20(8): 2691-2714 doi: 10.3934/dcdsb.2015.20.2691 +[Abstract](475) +[PDF](534.6KB)
Abstract:
Competition between species for resources is a fundamental ecological process, which can be modeled by the mathematical models in the chemostat culture or in the water column. The chemostat-type models for resource competition have been extensively analyzed. However, the study on the competition for resources in the water column has been relatively neglected as a result of some technical difficulties. We consider a resource competition model with two species in the water column. Firstly, the global existence and $L^\infty$ boundedness of solutions to the model are established by inequality estimates. Secondly, the uniqueness of positive steady state solutions and some dynamical behavior of the single population model are attained by degree theory and uniform persistence theory. Finally, the structure of the coexistence solutions of the two-species system is investigated by the global bifurcation theory.
Long time dynamics of a multidimensional nonlinear lattice with memory
Jáuber Cavalcante Oliveira, Jardel Morais Pereira and Gustavo Perla Menzala
2015, 20(8): 2715-2732 doi: 10.3934/dcdsb.2015.20.2715 +[Abstract](541) +[PDF](452.3KB)
Abstract:
This work is devoted to study the nature of vibrations arising in a multidimensional nonlinear periodic lattice structure with memory. We prove the existence of a global attractor. In the homogeneous case under a restriction on the nonlinear term we obtain decay rates of the total energy. These rates could be exponential, polynomial or several other intermediate types.
Exponential-stability and super-stability of a thermoelastic system of type II with boundary damping
Lei Wang, Zhong-Jie Han and Gen-Qi Xu
2015, 20(8): 2733-2750 doi: 10.3934/dcdsb.2015.20.2733 +[Abstract](676) +[PDF](357.3KB)
Abstract:
In this paper, the stability of a one-dimensional thermoelastic system with boundary damping is considered. The theory of thermoelasticity under consideration is developed by Green and Naghdi, which is named as ``thermoelasticity of type II''. This system consists of two strongly coupled wave equations. By the frequency domain method, we prove that the energy of this system generally decays to zero exponentially. Furthermore, by showing the spectrum of the system is empty under certain condition and estimating the norm of the resolvent operator, we give a sufficient condition on the super-stability of this thermoelastic system. Under this condition, the solution to the system is identical to zero after finite time. Moreover, we also estimate the maximum existence time of the nonzero part of the solution. Finally, we give some numerical simulations.
Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system
Qingshan Zhang and Yuxiang Li
2015, 20(8): 2751-2759 doi: 10.3934/dcdsb.2015.20.2751 +[Abstract](636) +[PDF](374.1KB)
Abstract:
We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equation \begin{eqnarray*} \left\{\begin{array}{lll} n_t + u \cdot \nabla n = \Delta n - \chi\nabla\cdot(n \nabla c),&{} x\in\Omega,\ t>0,\\ c_t + u \cdot \nabla c = \Delta c - nc, &{} x \in \Omega,\ t>0,\\ u_t + \kappa(u\cdot\nabla)u = \Delta u + \nabla P + n\nabla\phi ,&{} x\in\Omega,\ t>0,\\ \nabla\cdot u=0, &{}x\in\Omega,\ t>0, \end{array}\right. \end{eqnarray*} in a bounded domain $\Omega\subset\mathbb{R}^2$. It is known that if $\chi>0$, $\kappa\in\mathbb{R}$ and $\phi\in C^2(\bar{\Omega})$, for sufficiently smooth initial data, the model possesses a unique global classical solution which satisfies $(n, c, u)\rightarrow(\bar{n}_0, 0, 0)$ as $t\rightarrow\infty$ uniformly with respect to $x\in\Omega$, where $\bar{n}_0:=\frac{1}{|\Omega|}\int_{\Omega}n(x, 0)dx$. In the present paper, we prove this solution converges to $(\bar{n}_0, 0, 0)$ exponentially in time.
A note on 'Spin-polarized transport: Existence of weak solutions'
Carlos J. García-Cervera and Xiao-Ping Wang
2015, 20(8): 2761-2763 doi: 10.3934/dcdsb.2015.20.2761 +[Abstract](487) +[PDF](250.3KB)
Abstract:
The authors presented a proof of existence of weak solutions to a model for spin-polarized transport in ferromagnetic multilayers in [1]. The proof of the previous result is valid only in the case when the external current in parallel to the boundary of the domain. We present here an extension of that result, which applies to more general currents.

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