Forced Duffing Oscillator

exb3
Variable and parameter:

u: state variable of an oscillator

$\omega$: frequency of the driving force (bifurcation parameter)

differential equation:

\begin{displaymath}\ddot u + {\scriptstyle{1 \over 25}} \dot u - {\scriptstyle{1... ...e{8 \over 15}} u^3 = {\scriptstyle{2 \over 5}} \cos \omega t. \end{displaymath}

We basically look for harmonic solutions, which are periodic with period $T=2\pi/\omega$.

Comments

Figure 1
Bifurcation diagram u(0) versus $\omega$. Solid curve: branch of symmetric orbits.

Figure 2
Bifurcation diagram $\dot u (0)$ versus $\omega$.

Figure 3
Detail of bifurcation diagram, u(0) versus $\omega$.

Figure 4
$(u,\dot u)$ phase-plane for $\omega=0.053$, symmetric oscillation.

Figure 5
u(t) for $\omega=0.053$, t is a normalized time, t=1 corresponds to $T=2\pi /\omega=118.55$.

Figure 6
$(u,\dot u)$ plane for $\omega=0.12861$, asymmetric oscillation.

Figure 7
Bifurcation diagram of a closed branch of asymmetric solutions, u(0) versus $\omega$.

Figure 8
Same as in Figure 7, but $\dot u (0)$ versus $\omega$.

Figure 9
Duffing equation, $(u, \dot u)$ plane, Poincaré set of the stroboscopic map, $\omega=0.2$, fractal dimension D=2.381.

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