From Local Payoffs to Global Instabilities: A Spectral Cartography of Spatiotemporal Chaos in Canonical 2x2 Evolutionary Games

Abstract

We develop a motif-based framework for spatiotemporal chaos in spatial evolutionary games and use it to map the dynamical phase diagram in the payoff plane. Using Boolean linearization of the imitate-the-best rule, we derive analytical instability thresholds for local motifs including invaders, cooperative pairs, stripe interfaces, and cooperative cores. These thresholds are obtained from payoff balance at contested motif interfaces and recover classical invasion thresholds of spatial evolutionary games, which emerge here as boundaries of the chaotic phase. Combining the Derrida slope with the asymptotic Hamming distance, we obtain a four-region cartography: ordered, transient-chaotic, sustained-chaotic, and subcritical-chaotic dynamics. The phase diagram is organized by density-dependent motif selection: different initial cooperator densities activate different instability mechanisms, yet a small set of motif-instability lines consistently bounds the sustained-chaos region across densities. This cartography reveals a subcritical chaotic phase (Derrida slope $s<1$ but asymptotic Hamming distance $d_\infty>0$), where infinitesimal perturbations decay while finite-amplitude perturbations sustain chaos. The motif-based framework is anchored by an exact benchmark: for homogeneous backgrounds, the Boolean Jacobian yields an exact correspondence between the Derrida slope and spectral radius, linking damage spreading to deterministic instability.