Positive limit set in novel hybrid systems with interactions between differential dynamics and evolutionary games

Abstract

This paper proposes a novel class of hybrid systems and investigates its positive limit set theory. The new model is motivated by hybrid characters of evolutionary games, sparse research concerning interactions between games and dynamics, and practical backgrounds that varying environmental factors lead to various impacts. The model consists of continuous dynamics, transient impulse, logical evolution of game strategies, and their interactions where evolutionary games determine the type of impulse. Such model has generality and practicality, and is rarely studied. Firstly, via semi-tensor product, proposed hybrid systems are converted into the complete algebraic representation. Next, we investigate the basic properties to lay the foundation for discussions about the positive limit set, including left-continuity, semi-group, quasi-continuous dependence, and boundedness. Then we discuss non-emptiness, closeness, compactness, and invariance of the positive limit set. Finally a numerical example is given to show the validity of main results.