Fig. 6

Dynamics of the sinusoidally forced linear system (29)–(30) for $\alpha =1$, $ϵ=0.01$. a Projections of the phase-plane diagrams on the v–w plane for various representative values of the input frequency f. The fixed solid-red line and the solid-green line represent the v- and w-nullclines for the unforced system ($A_t=0$), respectively. The dashed-red lines represent the v-nullclines for $A_t=1$ (above-right) and $A_t=-1$ (below-left). The solid-blue lines represent the trajectories of the forced system for a single period ($T=1000$). b Impedance profile. c Phase profile. d Envelope-plane diagram. The solid-blue line represents the envelope curve: Each point on this curve is the maximum point on the limit cycle response to sinusoidal inputs parametrized by the input frequency f which increases from $f=0$ (blue-square at the intersection between upper dashed-red and green curves) to $f\to \mathrm{\infty }$ (blue dot at the origin). (The v-coordinates of the envelope curve are the impedance function Z, since $A_{\mathrm{in}}=1$.) Solid-red and -green lines: v- and w-nullclines for the unforced system ($A_t=0$). Dashed-red lines: v-nullclines for $A_t=1$ (above) and $A_t=-1$ (below)