Driven One Well Duffing Oscillator

Driving the Duffing Oscillator with either $\beta = 1$ or $\beta = -1$ produces incredibly complex behavior, leading to chaos for some range of the parameters.

At low driving strength the one well Duffing Oscillator exhibits attractive Limit Cycles. This is a trajectory in phase space toward which trajectories converge following a transient period. The region over which initial conditions lead to trajectories converging to the Limit Cycle is the Basin of Attraction. The following is a sketch of the attractive Limit Cycle for $A = 2.0$.

Exercise #10:

Draw the phase space portrait of this solution. Use the HO setup code to feed initial conditions in the general modular RK4 ODE solver program to solve the Duffing system. Use $\alpha = 1.0$, $\beta = 1$, $\mu = 0.1$, $A = 2.0$ and $\omega = 1$ in the code.

Note that the different trajectories seem to converge toward a common trajectory, a Limit Cycle

Exercise #11:

Refine the graph of the Limit Cycle. To do so, integrate the ODEs starting from an arbitrary initial conditions, i.e., $(x_0,v_0) = ( 1.0,0.0),$ for a large $t_{max}$ via the option -t 800, i.e., 800 periods of the external field.

Restart the integration from suitable initial conditions for another long integration time interval to refine the Limit Cycle calculation.

Do it again, i.e., run the ODE integrator again!

Plot $x(t)$ and $v(t)$ as functions of $t$. Note that the period of the Limit Cycle is that of the external forcing term.

Larger Driving Strengths

Larger driving strengths can produce very different solutions in the driven one well Duffing Oscillator. For instance the following sketch illustrates the two Limit Cycles at a driving strength $A=4.0$.

These two Limit Cycles are mirror images of each other. If a solution of a lesser symmetry than that of the potential exists, a solution of opposite symmetry should also exist.

Exercise #12:

Draw the phase space portrait of this solution. Use the HO setup code to feed initial conditions in the general modular RK4 ODE solver program to solve the Duffing system. Use $\alpha = 1.0$, $\beta = 1$, $\mu = 0.1$, $A=4.0$ and $\omega = 1$ in the code.

Note that the different trajectories seem to converge toward two possible trajectories, two Limit Cycles

Exercise #13:

Define better these Limit Cycles. To do so, integrate the ODEs starting from an arbitrary initial conditions, i.e., $(x_0,v_0) = ( 1.0,0.0),$ for a large $t_{max}$ via the option -t 800.

Restart the integration from suitable initial conditions for another long integration time interval to refine the two Limit Cycles calculations.

Exercise #14:

Repeat the calculations above for other initial conditions in the range $(x_0,v_0) = ( 0.9,0.0)$ down to $(0.3,0.0)$.

It is clear that each Limit Cycle has its own Basin of Attraction.

The following sketch illustrates the Limit Cycle generated by the Duffing Oscillator driven with a strength $A=6.0$.

Exercise #15:

Draw the phase space portrait of this solution. Use $\alpha = 1.0$, $\beta = 1$, $\mu = 0.1$, $A=6.0$ and $\omega = 1$ as parameters.