Global bifurcational approach to the onset of spatio-temporal chaos in reaction diffusion systems

Abstract

Abstract. A new criterion for the onset of spatio-temporal chaos arising in the Gray-Scott model is presented. This is based on the interrelationship of global bifurcating branches of stationary or periodic solutions with respect to the removal rate contained in the model, especially their locations of saddle-node points and the Hopf bifurcation point of a constant state play a key role. At the onset point there exists a generalized heteroclinic cycle on the whole line and spatio-temporal chaos emerges by unfolding this cycle.

The existence and stability of spike equilibria in the one-dimensional Gray–Scott model: The pulse-splitting regime

“…An important remark is that in our analysis we require that εA/ √ D 1. Therefore, the pulse-splitting that occurs in our regime is distinctly different from the pulse-splitting that occurs in the parameter regime of [10,33,35,36,43], where D = O(ε 2 ). When D = O(ε 2 ), the inter-spike interactions become exponentially weak and there is a new core problem in the vicinity of each spike that involves a full balance of the terms in (1.1).…”

“…In this regime, called the semi-strong spike interaction regime, spike patterns are such that v is localized near the spikes, while u varies more globally across the domain. This regime is to be contrasted with the weak interaction regime studied in [33,35,36,43], where D = O(ε 2 ) and ε 1. In the semi-strong interaction limit there are three regimes for A where different behaviors occur; the low feedrate regime A = O(ε 1/2 ), the intermediate feed-rate regime O(ε 1/2 ) A O(1), and the high feed-rate, or pulsesplitting, regime, A = O(1).…”

“…For various ranges of these parameters, (1.1) and its two-dimensional counterpart, are known to posses a rich solution structure involving oscillating standing pulses, the propagation of traveling waves, pulse-replication behavior, and chaotic phenomena (cf. [5][6][7][8][9]19,24,25,[28][29][30]33,[35][36][37][38][39][40]43]). Self-replication of spots for the GS model was first observed numerically in [37].…”

The Existence and Stability of Spike Equilibria in the One‐Dimensional Gray–Scott Model: The Low Feed‐Rate Regime

“…For various ranges of these parameters, (1a) and (1b) and its two-dimensional counterpart are known to possess a rich solution structure including the existence of stable standing pulses, the propagation of traveling waves, pulse-replication behavior, and spatio-temporal chaos (cf. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). We will analyze (1) in the singularly perturbed limit, where D v /D u is asymptotically small.…”