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**Forced Duffing Oscillator**

exb3

Variable and parameter:
*u*: state variable of an oscillator
:
frequency of the driving force (bifurcation parameter)
differential equation:

We basically look for *harmonic * solutions, which are periodic with period
.

Figure 1

Bifurcation diagram *u*(0) versus .
Solid curve: branch of symmetric orbits.

Figure 2

Bifurcation diagram
versus .

Figure 3

Detail of bifurcation diagram, *u*(0) versus .

Figure 4

phase-plane for
,
symmetric oscillation.

Figure 5

*u*(*t*) for
,
*t* is a normalized time, *t*=1 corresponds to
.

Figure 6

plane for
,
asymmetric oscillation.

Figure 7

Bifurcation diagram of a closed branch of asymmetric solutions, *u*(0) versus .

Figure 8

Same as in Figure 7, but
versus .

Figure 9

Duffing equation,
plane, Poincaré set of the stroboscopic map,
,
fractal dimension *D*=2.381.

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Figure 9