Figure 3 | Scientific Reports

Figure 3

From: Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications

Figure 3

Long-range dependence of nonlinear dynamics and control mechanisms. SACF(70) denotes the partial sum of the first 70 terms of the simulated autocorrelation function sequence as a measure of dependence. A large SACF(70) reflects high level of dependence. (a,b) The traces of SACF(70) versus the stability coefficient γ for the NLARI’ stochastic stable fixed point, period cycles, and chaos when the slope indicator |η1| is increased by increasing the disturbance mean |ω| with the restorative delays of κ2 = 1 and κ2 = 4, respectively. (c,d) The traces of SACF(70) of the deterministic stable fixed point, period cycles, and chaos for the NLARI in (a,b) without noise for κ2 = 1 and κ2 = 4, respectively. (e,f) The traces of SACF(70) versus the stability coefficient γ for the NLARI’ stochastic stable fixed point, period cycles, and chaos when the slope indicator |η1| is increased by decreasing the resistance coefficient α for κ2 = 1 and κ2 = 4, respectively. (g,h) The traces of SACF(70) of the deterministic stable fixed point, period cycles, and chaos for the NLARI in (e and f) without noise for κ2 = 1 and κ2 = 4, respectively. In (ad), α = 1.1 and the restoration coefficient \(\beta \in (0.036,5.760)\). In (eh), \(\alpha \in (0.001,1.7)\) and \(\beta \in (0.012,12.79)\). In (a,b,e,f), the value of the amplitude indicator η2 decreases as the value of the stability coefficient γ increases.

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