Nonlinear Dynamics in Game Theory as a New Mathematical Approach to Analysing Strategic Behaviour

This research presents a novel mathematical framework integrating nonlinear dynamics with game theory to analyse strategic behaviour in complex multi-agent systems. Traditional game-theoretic approaches often assume equilibrium convergence and rational decision-making, yet empirical observations reveal persistent oscillations, chaotic behaviour, and multi-stability in strategic interactions. We develop a unified theory incorporating bifurcation analysis, strange attractors, and Lyapunov stability to characterise the full spectrum of dynamical behaviours in strategic settings. Our framework introduces the concept of strategic bifurcations—qualitative changes in equilibrium structure induced by parameter variations in payoff functions or behavioural rules. We establish conditions for Hopf bifurcations in replicator dynamics, derive analytical expressions for limit cycle amplitudes, and characterise routes to chaos through period-doubling cascades. The theory extends to n-player games with heterogeneous learning rates, revealing that chaos becomes increasingly prevalent as system complexity grows. We prove that the basin of attraction for stable Nash equilibria shrinks exponentially withthe number of players, whilst the measure of chaotic regimes expands. Applications to evolutionary biology, financial markets, and social dynamics demonstratethe framework’s predictive power. Our results challenge the primacy of equilibrium analysis in game theory and establish nonlinear dynamics as fundamental to understanding strategic behaviour in complex systems.