A comprehensive stability classification and dynamical analysis of a brain-inspired coupled Rössler-type system

To gain systematic insight into the role of parameter heterogeneity and the uncertainty of initial-conditions in determining stability, sensitivity, and extreme responses, this work conducts a systematic parametric and ensemble-based study of a three-dimensional coupled Rössler-type system with brain-inspired interaction characteristics. The analysis is carried out on the parameter space a 1 , b 1 , c 1 ∈ [ 0 , 6 ] , where three coupling strengths k i j = 0 % , 10 % , 20 % , with several representative initial conditions ( x 0 , y 0 , z 0 ) = ( − 0.2 , 1.0 , 0.05 ) , ( 0.2 , 0.07 , − 0.15 ) , and ( 0.1 , 0.05 , 0.08 ) , are considered. The primary study is carried out using an 8 × 8 × 8 parameter discretization, while supplementary studies employing 10 × 10 × 10 and 15 × 15 × 15 grids are conducted to ensure that the results obtained from the primary study are robust with respect to parameter space resolution. The eigenvalue analysis of the Jacobian matrix is used to define the local stability properties of the solution, and the analysis of trajectory statistics determines whether the response is divergent, non-divergent, or periodic. The extreme responses are measured by large samples of randomized realizations of parameters and perturbed initial conditions using the maximum radial amplitude measure. An empirical statistical characterization of system responses is presented using cumulative distribution functions and kernel density estimates to demonstrate convergence and decreased variability at stronger coupling. Changes in sensitivity indices imply that there is a change in the predominant initial condition, which depends on the coupling with the system response dominated by y 0 at weaker coupling and by z 0 at stronger coupling. The abrupt changes in behavior are revealed using heat-map and phase-space visualizations, and the parameter c 1 is one of the most important factors in stability loss and amplified response.