Lyapunov Functions For Dynamical Systems Research Articles

Behaviour of Lyapunov’s function for different strategies of circuit optimisation

The process of analogue circuit optimisation is mathematically defined as a controllable dynamic system. In this context the minimisation of the processor time of designing can be formulated as a problem of time minimisation for transitional process of dynamic system. In order to analyse the properties of such a system, it is proposed to use the concept of Lyapunov function of dynamic system. Using this function and its time derivative, the special functions have been built that allow us to predict the total processor time for circuit optimisation by analysing the initial interval of the optimisation process. Numerical results indicate the possibility of predicting the processor time of different strategies for circuit optimisation.

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.

GEO-MATHEMATICAL MODELLING OF SPATIAL-ECOLOGICAL COMPLEX SYSTEMS: AN EVALUATION

Assessing the complexity of landscapes is one of the top research priorities for Physical Geography and Ecology. This paper aims at a methodological evaluation of the discrete and analytical mathematical models hitherto available for quantitative assessments of spatial ecological complex systems. These models are derived from cellular automata and nonlinear dynamics. They describe complex features and processes in landscapes, such as spatial ecological nonlinear interactions, unpredictability and chaos, self-organization and pattern formation. Beginning with a distinction between two basic types of spatial ecological complexity (structural, functional), and after reviewing the quantitative methods so far available to assess it, the areas where the major challenges (and hence, difficulties) for future research arise are identified. These are: a) to develop measures of structural spatial-ecological complexity, b) to find Lyapunov functions for dynamical systems describing spatial interactions on the landscape (and related attractors), and c) to combine discrete time and continuous spatial data and models.

Meshless Collocation: Error Estimates with Application to Dynamical Systems

In this paper, we derive error estimates for generalized interpolation, in particular collocation, in Sobolev spaces. We employ our estimates in collocation problems using radial basis functions and extend and improve previously known results for elliptic problems. Finally, we use meshless collocation to approximate Lyapunov functions for dynamical systems.