Abstract:Ordinary differential equations arise in a variety of applications, including e.g. climate systems, and can exhibit complicated dynamical behaviour. Complete Lyapunov functions can capture this behaviour by dividing the phase space into the chain-recurrent set, determining the long-time behaviour, and the transient part, where solutions pass through. In this paper, we present an algorithm to construct complete Lyapunov functions. It is based on mesh-free numerical approximation and uses the failure of convergence in certain areas to determine the chain-recurrent set. The algorithm is applied to three examples and is able to determine attractors and repellers, including periodic orbits and homoclinic orbits.

The minimum mode following method for finding first order saddle points on an energy surface is used, for example, in simulations of long time scale evolution of materials and surfaces of solids. Such simulations are increasingly being carried out in combination with computationally demanding electronic structure calculations of atomic interactions, so it is essential to reduce as much as possible the number of function evaluations needed to find the relevant saddle points. Several improvements to the method are presented here and tested on a benchmark system involving rearrangements of a heptamer island on a close packed crystal surface. Instead of using a uniform or Gaussian random initial displacement of the atoms, as has typically been done previously, the starting points are arranged evenly on the surface of a hypersphere and its radius is adjusted during the sampling of the saddle points. This increases the diversity of saddle points found and reduces the chances of reconverging on previously located saddle points. The minimum mode is estimated using the Davidson method, and it is shown that significant savings in the number of function evaluations can be obtained by assuming the minimum mode is unchanged until the atomic displacement exceeds a threshold value. The number of function evaluations needed for a recently published benchmark (S. T. Chill et al. J. Chem. Theory Comput. 2014, 10, 5476) is reduced to less than a third with the improved method as compared with the best previously reported results.

Abstract. We establish existence of infinitely many distinct solutions to the multi-configurative Hartree-Fock type equations for N -electron Coulomb systems with quasi-relativistic kinetic energy −α −2 ∆ xn + α −4 −α −2 for the n th electron. Finitely many of the solutions are interpreted as excited states of the molecule. Moreover, we prove existence of a ground state. The results are valid under the hypotheses that the total charge Z tot of K nuclei is greater than N − 1 and that Z tot is smaller than a critical charge Z c . The proofs are based on a new application of the Lions-FangGhoussoub critical point approach to nonminimal solutions on a complete analytic Hilbert-Riemann manifold.

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Ordinary differential equations arise in a variety of applications, including climate modeling, electronics, predator-prey modeling, etc., and they can exhibit highly complicated dynamical behaviour. Complete Lyapunov functions capture this behaviour by dividing the phase space into two disjoint sets: the chain-recurrent part and the transient part. If a complete Lyapunov function is known for a dynamical system the qualitative behaviour of the system's solutions is transparent to a large degree. The computation of a complete Lyapunov function for a given system is, however, a very hard task. We present significant improvements of an algorithm recently suggested by the authors to compute complete Lyapunov functions. Previously this methodology was incapable to fully detect chain-recurrent sets in dynamical systems with high differences in speed. In the new approach we replace the system under consideration with another one having the same solution trajectories but such that they are traversed at a more uniform speed. The qualitative properties of the new system such as attractors and repellers are the same as for the original one. This approach gives a better approximation to the chain-recurrent set of the system under study.

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A complete Lyapunov function characterizes the behaviour of a general discrete-time dynamical system. In particular, it divides the state space into the chain-recurrent set where the complete Lyapunov function is constant along trajectories and the part where the flow is gradient-like and the complete Lyapunov function is strictly decreasing along solutions. Moreover, the level sets of a complete Lyapunov function provide information about attractors, repellers, and basins of attraction. We propose two novel classes of methods to compute complete Lyapunov functions for a general discrete-time dynamical system given by an iteration. The first class of methods computes a complete Lyapunov function by approximating the solution of an ill-posed equation for its discrete orbital derivative using meshfree collocation. The second class of methods computes a complete Lyapunov function as solution of a minimization problem in a reproducing kernel Hilbert space. We apply both classes of methods to several examples.