Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems

Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in t-direction. Hence, a numerical method would have to use infinitely many points.\ud \ud To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using Radial Basis Functions

Converse theorem on a global contraction metric for a periodic orbit

Contraction analysis uses a local criterion to prove the long-term behaviour of a dynamical system. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions contracts. If adjacent solutions in all directions perpendicular to the flow are contracted, then there exists a unique periodic orbit, which is exponentially stable and we obtain an upper bound on the rate of exponential attraction. In this paper we study the converse question and show that, given an exponentially stable periodic orbit, a contraction metric exists on its basin of attraction and we can recover the upper bound on the rate of exponential attraction

Converse theorems on contraction metrics for an equilibrium

The stability and basin of attraction of an equilibrium can be determined by a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent trajectories decreases. The advantage of a contraction metric over, e.g., a Lyapunov function is that the contraction condition is robust under perturbations of the system. While the sufficiency of a contraction metric for the existence, stability and basin of attraction of an equilibrium has been extensively studied, in this paper we will prove converse theorems, showing the existence of several different contraction metrics. This will be useful to develop algorithms for the construction of contraction metrics

Construction of Finsler-Lyapunov functions with meshless collocation

We study the stability of invariant sets such as equilibria or periodic orbits of a Dynamical System given by a general autonomous nonlinear ordinary differential equation (ODE). A classical tool to analyse the stability are Lyapunov functions, i.e. scalar-valued functions, which decrease along solutions of the ODE. An alternative to Lyapunov functions is contraction analysis. Here, stability (or incremental stability) is a consequence of the contraction property between two adjacent solutions, formulated as the local property of a Finsler-Lyapunov function. This has the advantage that the invariant set plays no special role and does not need to be known a priori. In this paper, we propose a method to numerically construct a Finsler-Lyapunov function by solving a first-order partial differential equation using meshless collocation. Depending on the expected attractor, the contraction only takes place in certain directions, which can easily be implemented within the method. In the basin of attraction of an exponentially stable equilibrium or periodic orbit, we show that the PDE problem has a solution, which provides error estimates for the numerical method. Furthermore, we show how the method can also be applied outside the basin of attraction and can detect the stability as well as the stable/unstable directions of equilibria. The method is illustrated with several examples

Computation and verification of Lyapunov functions

Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding. In this paper, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a potential Lyapunov function. Then we interpolate this function by a CPA function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction. We show that this combined method always succeeds in computing and verifying a Lyapunov function, as well as in determining arbitrary compact subsets of the basin of attraction. The method is applied to two examples

Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation

We consider a non-autonomous ordinary differential equation over a finite time interval [T1; T2]. The area of exponential attraction consists of solutions such that the distance to adjacent solutions exponentially contracts from T1 to T2. One can use a contraction metric to determine an area of exponential attraction and to provide a bound on the rate of attraction. In this paper, we will give the first method to algorithmically construct a contraction metric for finite-time systems in one spatial dimension. We will show the existence of a contraction metric, given by a function which satisfies a second-order partial differential equation with boundary conditions. We then use meshless collocation to approximately solve this equation, and show that the resulting approximation itself defines a contraction metric, if the collocation points are sufficiently dense. We give error estimates and apply the method to an example

Kernel-based discretisation for solving matrix-valued PDEs

In this paper, we discuss the numerical solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these pproximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical system