Analytical criterion for amplitude death in nonautonomous systems with piecewise nonlinear coupling

We investigate the amplitude death phenomenon in a nonautonomous chained network with complicated piecewise nonlinear coupling functions. An analytical criterion for the boundary of the amplitude death region is derived by using the average method. The mechanism of the amplitude death in the nonautonomous networks is very different from that of autonomous systems and rapid dynamic transitions could halt the amplitude death. Numerical verifications are carried out to check jump transitions among different solution branches and further confirm the correctness of the theoretical results.

Figure 1

(a) Sketch for the rubber-cable connector; (b)–(d) $G$ function value against relative displacement $x_j-x_i$ and $z_j-z_i$.

Figure 2

Variation of dynamic response of the nonautonomous chain network in Eq. (2); (a) the oscillation amplitudes, (b) the bifurcation diagram of the state variable $x_4$ of Module 4, as the strength ${\varepsilon }_2$ of the piecewise nonlinear coupling function sweeps forwards for the parameter setting of $\eta =0.5,N=5$.

Figure 3

Resonance curve versus the coupling strength ${\varepsilon }_2$ for $\eta =0.5,N=3$.

Figure 4

The folded sheet represents the response amplitude, the dashed and solid curves are the projection of the boundaries of the bottom surface (deep blue) on parametric space (${\varepsilon }_2,\eta$); amplitude death occurs when the parameter setting falls in the area surrounded by the curves, for $N=3$.

Figure 5

Resonance curves for (a) the second oscillator and (b) the third oscillator, when the parameters are set at $\eta =0.5,N=5$.

Figure 6

Solution diagram in parameter space $({\varepsilon }_2-\eta )$ for $N=5$. AD: amplitude death; P1: period-1 motion; P2: period-2 motion; C or Pn: chaotic or high-order periodic motion. Black solid line indicates boundary curve for AD obtained by analytical criterion.