Fig. 8: Positive/negative dual feedback can induce chaotic behavior when the difference in delay times is significant.
From: Nonlinear delay differential equations and their application to modeling biological network motifs
a The double feedback motif, in which two regulation arms feed back directly, each with its own explicit delay. Here we show one arm activating and one repressing; for double repressive feedback, see Supplementary Fig. 5. b Time trace of chaotic dynamics after initial transients. c Trace of dynamics in phase space, with the derivative on the vertical axis. While a simple oscillator would trace a loop (possibly with multiple sub-loops if the waveform is complicated), the chaotic dynamics appear to trace out a fractal attractor. Consistent with chaos, a reconstructed phase space with coordinates (X(T), X(T − 10), X(T − 20)) traces out a fractal attractor, with box dimension 1.81 with 95% confidence interval (1.76, 1.87) and a positive dominant Lyapunov exponent (0.0040 ± 0.00055 bits), see “Methods” for details. d Fourier transform of chaotic dynamics show many peaks, indicating that there is no simple set of frequencies underlying the dynamics. e Bifurcation diagram for double positive/negative feedback, with local maxima plotted. Simple oscillations intersperse regimes with complex dynamics, where local maxima with a range of values are found. η1 = 15, η2 = 1, n1 = 11, n2 = −3, and γ1 = 1. For (b–d), γ2 = 11.
