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The temporal period dt and the spatial period di were defined as x (i+ di, t+ dt) =x (i,t). A two-dimensional autocorrelation analysis for the patterns in Figs. c,d,e yields a temporal period of dt =712.0 and di =0.0 for (c), dt =702.2 and di =-7.4 for (d), and dt =969.8 and di =-24.5 for (e). In comparison the period of the limit cycle for the uncoupled WS system is dt =257.6.
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The temporal period dt and the spatial period di were defined as x (i+ di, t+ dt) =x (i,t). A two-dimensional autocorrelation analysis for the patterns in Figs.c,d,e yields a temporal period of dt =712.0 and di =0.0 for (c), dt =702.2 and di =-7.4 for (d), and dt =969.8 and di =-24.5 for (e). In comparison the period of the limit cycle for the uncoupled WS system is dt =257.6.
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This method determines the lifetime with sufficient accuracy, assuming that the maximum Lyapunov exponent is positive on average during the chaotic transient dynamics and zero on average during the nonchaotic dynamics, which is the case when the attractor is not a fixed point. All simulations for the Wacker-Schöll system fulfilled this criterion.
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This method determines the lifetime with sufficient accuracy, assuming that the maximum Lyapunov exponent is positive on average during the chaotic transient dynamics and zero on average during the nonchaotic dynamics, which is the case when the attractor is not a fixed point. All simulations for the Wacker-Schöll system fulfilled this criterion.
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In the Gray-Scott system the probability for local extinction could be associated with the probability P of a unit to be conducive to collapse. Test simulations show that no-flux boundary conditions as well as shortcuts in the network change locally the probability for local extinctions.
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In the Gray-Scott system the probability for local extinction could be associated with the probability P of a unit to be conducive to collapse. Test simulations show that no-flux boundary conditions as well as shortcuts in the network change locally the probability for local extinctions.
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Transient spatiotemporal chaos can become asymptotic for a constant boundary condition that provides a superthreshold perturbation to the excitable reaction-diffusion network.
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Transient spatiotemporal chaos can become asymptotic for a constant boundary condition that provides a superthreshold perturbation to the excitable reaction-diffusion network.
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Computation of Lyapunov exponents can be sensitive to integration time step and time between renormalizations. One representative system for each of the three models (Gray-Scott, Bär-Eiswirth, and Wacker-Schöll) was run with an integration time step five times smaller and then with a renormalization interval five times shorter. In all cases the difference in either Lyapunov dimension or sum of positive Lyapunov exponents between the trial and the baseline cases was less than 1.8 times the estimated convergence error.
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Computation of Lyapunov exponents can be sensitive to integration time step and time between renormalizations. One representative system for each of the three models (Gray-Scott, Bär-Eiswirth, and Wacker-Schöll) was run with an integration time step five times smaller and then with a renormalization interval five times shorter. In all cases the difference in either Lyapunov dimension or sum of positive Lyapunov exponents between the trial and the baseline cases was less than 1.8 times the estimated convergence error.
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Runs that survived for a time less than 5× 106 Δt were discarded under the assumption that not enough data would be available to get a reliable value for the Lyapunov exponents. Additionally, each system is run to time 10000Δt before perturbation vectors are followed to minimize the significance of the initial period before the chaotic saddle has been reached, and the perturbation vector data gathered in the first 4500 time units were discarded in order to allow the vectors to align in a natural way.
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Runs that survived for a time less than 5× 106 Δt were discarded under the assumption that not enough data would be available to get a reliable value for the Lyapunov exponents. Additionally, each system is run to time 10000Δt before perturbation vectors are followed to minimize the significance of the initial period before the chaotic saddle has been reached, and the perturbation vector data gathered in the first 4500 time units were discarded in order to allow the vectors to align in a natural way.
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