# Trying to find a differential equation that has multi time scale oscillatory behaviour

I am trying to find two differential equations of dynamical systems (like some non-linear oscillator, perhaps). My requirement is that they both should have the same initial condition (easy to impose) and same or similar early time dynamics and the same final state. They need to have different trajectories between early time and end state.

I have been playing around with the Van der Pol and Duffing oscillators but changing the parameters doesn't change the "middle" dynamics mainly because of their nature as initial value problems.

Van der Pol: $$k^2 x(t)+x''(t)+\delta \left(x(t)^2-1\right) x'(t)=\text{Am} \sin \left(\frac{2 \pi t}{\text{T1}}\right)$$

with $\delta =10,k=N[2 \pi ],\text{T1}=0.1,\text{Am}=0.$

Duffing: $$y''(t)+\delta y'(t)+\epsilon y(t)^3+y(t)=\gamma \sin \left(\frac{2 \pi t}{\text{T1}}\right) \sin \left(\frac{2 \pi t}{\text{T2}}\right)$$

with $\epsilon=5 ,\delta =1,\gamma =1,\text{T1}=1,\text{T2}=10$

The initial condition for either oscillator, as an example, could be $x(0)=1,x'(0)=0$ or $y(0)=1,y'(0)=0$

Mathematica code if solving:

Van der Pol:

Module[{tmax = 10., \[Mu] = 10, k = N[2 \[Pi]], T1 = 10, Am = 30.},
vdp = NDSolveValue[{x''[t] + \[Mu] (x[t]^2 - 1) x'[t] + k^2 x[t] ==
Am Sin[2 \[Pi] t/T1], x == 3, x' == 1}, x, {t, 0, tmax},
Method -> "LSODA"];
Plot[vdp[t], {t, 0., tmax},
PlotLabel ->
"Fig2. Stiff Van der Pol oscillator: appropriate solver",
PlotRange -> {{0, tmax}, {-5, 5}}]
]


Duffing:

Module[{tmax = 60, \[Epsilon], \[Delta] = 1, \[Gamma] = 1, T1 = 1,
T2 = 10},
\[Epsilon] = 5;
duffing5 =
NDSolveValue[{
y''[t] + \[Delta] y'[t] +
y[t] + \[Epsilon] (y[t])^3 == \[Gamma] Sin[2 \[Pi] t/T1] Sin[
2 \[Pi] t/T2], y == 1, y' == 0}, y, {t, 0, tmax}];
Plot[{duffing5[t]}, {t, 0., tmax},
PlotLabel ->
"Fig1. duffing oscillator with \[Epsilon]=5",
PlotRange -> {{0, tmax}, {-1, 1}},
PlotLegends -> {"\[Epsilon]=5"}]
]

• You could probably artificially construct a single oscillator with the long time behavior that you want and then modify it so that it passes through a metastable configuration (perhaps oscillating while it is in that metastable configuration) and still behaves the same along the way. – Ian Jul 28 '16 at 15:15
• You mean by detecting and changing a parameter at a certain time step? I assume there is no oscillator that is an IVP that has the behaviour that I mention (I would think it doesn't exist?) – dearN Jul 28 '16 at 15:19
• I mean that the equation would be different. Whether this can be done depends a bit on what exactly you mean by "oscillator". – Ian Jul 28 '16 at 15:21