Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies

We present a general method for analyzing mixed-mode oscillations (MMOs) in parametrically and externally excited systems with two low excitation frequencies (PEESTLEFs) for the case of arbitrary $m\text{:}n$ relation between the slow frequencies of excitations. The validity of the approach has been demonstrated using the equations of Duffing and van der Pol, separately. Our study shows that, by introducing a slow variable and finding the relation between the slow variable and the slow excitations, PEESTLEFs can be transformed into a fast-slow form with a single slow variable and therefore MMOs observed in PEESTLEFs can be understood by the classical machinery of fast subsystem analysis of the transformed fast-slow system.

Figure 1

Mixed-mode oscillations in the Duffing system (12) for (a) ${\omega }_1={\omega }_2=0.01$, (b) ${\omega }_1=0.01$ and ${\omega }_2=0.02$, (c) ${\omega }_1=0.01$ and ${\omega }_2=0.03$, (d) ${\omega }_1=0.01$ and ${\omega }_2=0.04$, (e) ${\omega }_1={\omega }_2=0.02$, and (f) ${\omega }_1=0.02$ and ${\omega }_2=0.04$. The insets in (a) and (b) are the enlargement of the rectangle areas and give a clear view of the oscillations. The circled region in (b) shows that the bursting in (b) exhibits different characteristics compared to the bursting in (a).

Figure 2

(a1) Bifurcation diagram of the fast subsystem (13), where $\gamma$ is used as the control parameter. Solid curves are stable equilibria, while the dashed curve indicates the unstable equilibrium. The FB denote fold bifurcations of the equilibria. (a2) Corresponding transformed phase diagram of MMOs, with the bifurcation diagram shown in (a1) superimposed, where the stable equilibria are highlighted with thick blue curves. (b1) and (b2) Same as in (a1) and (a2) for ${\omega }_1=0.01$ and ${\omega }_2=0.02$.

Figure 3

Same as in Fig. 2 for (a1) and (a2) ${\omega }_1=0.01$ and ${\omega }_2=0.03$ and (b1) and (b2) ${\omega }_1=0.01$ and ${\omega }_2=0.04$.

Figure 4

Bifurcation diagram of the fast subsystem for (a) ${\omega }_1=0.02$, (b) ${\omega }_1=0.03$, and (c) ${\omega }_1=0.04$. The frequency ${\omega }_2$ is fixed at ${\omega }_2=0.01$.

Figure 5

(a1) Time series of the solution for ${\omega }_1=0.02$ and ${\omega }_2=0.01$, (a2) transformed phase diagram of the solution and its overlay with the equilibrium curve in Fig. 4, (b1) and (b2) same as in (a1) and (a2) for ${\omega }_1=0.03$ and ${\omega }_2=0.01$, and (c1) and (c2) same as in (a1) and (a2) for ${\omega }_1=0.04$ and ${\omega }_2=0.01$. In (b2) and (c2) the FB indicate fold bifurcations of equilibria.

Figure 6

(a1) and (a2) Same as in Fig. 5 for ${\omega }_1=0.02$ and ${\omega }_2=0.03$, with the related fast subsystem shown in Table 2, row 4. (b1) and (b2) Same as in Fig. 5 for ${\omega }_1=0.03$ and ${\omega }_2=0.02$. The associated fast subsystem is shown in Table 2, row 5.

Figure 7

Mixed-mode oscillations in the van der Pol system (14) for (a) ${\omega }_1={\omega }_2=0.01$, (b) ${\omega }_1=0.01$ and ${\omega }_2=0.02$, (c) ${\omega }_1=0.02$ and ${\omega }_2=0.01$, and (d) ${\omega }_1=0.03$ and ${\omega }_2=0.01$.

Figure 8

(a1) Bifurcation diagram of the fast subsystem for ${\omega }_1={\omega }_2=0.01$, where $\gamma$ is used as the control parameter: solid (dashed) curve, stable (unstable) equilibrium; $H$, supercritical Hopf bifurcation; and $c_{\max }$ and $c_{\min }$, the maximum and minimum amplitudes of stable oscillations generated from $H$. (a2) Overlay of the bifurcation diagram with the transformed phase diagram of MMOs, where the stable equilibrium is highlighted with thick blue curves. (b1) and (b2) Same as in (a1) and (a2) for ${\omega }_1=0.01$ and ${\omega }_2=0.02$.

Figure 9

(a1) and (a2) Same as in Fig. 8 for ${\omega }_1=0.02$ and ${\omega }_2=0.01$ and (b1)–(b3) same as in Fig. 8 for ${\omega }_1=0.03$ and ${\omega }_2=0.01$, with (b3) the enlargement of (b2).

Figure 10

Complex periodic MMO pattern and its dynamical mechanism in the system (12) for ${\omega }_1=0.01$ and ${\omega }_2=0.1$. Since $\cos \left(0.1t\right)=f_{10}^{*}(\cos \left(0.01t\right))=[2{\cos }^2\left(0.01t\right)-1][256{\cos }^8\left(0.01t\right)-512{\cos }^6\left(0.01t\right)+304{\cos }^4\left(0.01t\right)-48{\cos }^2\left(0.01t\right)+1]$, we have $g\left(t\right)=\cos \left(0.01t\right), f_1\left(x\right)=x$, and $f_2\left(x\right)=(2x^2-1)(256x^8-512x^6+304x^4-48x^2+1)$, which leads to the fast subsystem $\ddot{x}+0.1\dot{x}-(0.5+\gamma )x+x^3=(2{\gamma }^2-1)(256{\gamma }^8-512{\gamma }^6+304{\gamma }^4-48{\gamma }^2+1)$, where $\gamma$ is the control parameter. (a) Bifurcation diagram of the fast subsystem, where $\gamma$ is the control parameter. The FB denote fold bifurcations of equilibria. (b) Overlay of the bifurcation diagram with the transformed phase diagram of the complex MMOs, where the stable branches of equilibrium are highlighted.

Figure 11

Transition from (a) and (b) complex periodic MMOs to (c) quasiperiodic oscillations for (a) ${\omega }_1=0.01$ and ${\omega }_2=0.1$ [time series of the MMOs in Fig. 10], (b) ${\omega }_1=0.01$ and ${\omega }_2=0.2$, and (c) ${\omega }_1=0.01$ and ${\omega }_2=1$.