Sufficiency Conditions for Finite Escape Times in Systems of Quadratic Differential Equations

Abstract

Sufficient conditions are derived for solutions of a general autonomous system of quadratic ordinary vector differential equations to exhibit finite escape times. These results are extended t o a matrix Riccati-type differential equation using Kronecker matrix products and a stacking operator. Finally, these results are applied to a class of models used extensively in population dynamics. GENERAL procedures are well-known for analysing the stability properties of systems of non-linear ordinary differential equations. Important among these procedures are: (i) the various Lyapunov-type theorems (Hahn, 1967; Cesari, 1963, chapter 3) and (ii) the analysis of the stability properties of the corresponding variational system, viz. the non-linear system linearized about one of its stationary points (Willems, 1970, chapter 5). Stability results pertaining to specific systems are known for many different types of non-linear differential equations. Often these results are derived by finding a suitable Lyapunov function for the system concerned and accordingly the applicability of the results is limited to that system. However, few explicit results exist for the stability properties of quadratic differential equations, especially since suitable Lyapunov functions for general quadratic systems are difficult to find. The primary reason for this is that the time derivative of a scalar function o f even order is odd and it is therefore difficult t o comment upon its sign without knowing more about the structure of the solutions of the quadratic differential equations. In this paper we derive sufficiency conditions for solutions of autonomous systems of quadratic differential equations to have a finite escape time. An upper bound for the escape time is also given. Using the Kronecker matrix product and an operator which stacks columns of a matrix into an extended vector, we show that the matrix Riccati-equation that arises in the linear-quadratic regulator problem of optimal control theory, can be rewritten into a format such that our results complement known sufficient conditions of the Kalman-type (Jacobson, 1970). Finally, we demonstrate that the results derived in this paper can be applied to a general class of systems of autonomous quadratic differential equations used to model interacting populations. The Lotka-Volterra model of predator-prey dynamics is a special case of this class. 25 377