x'' + P(x') x' + u(x) = F(t), P(x')>0, F(t+T)=F(t), ,<F(t)> = 0

Dear All,
I am doing a bibliographical research on the asymptotic behavior of a
periodically forced nonlinear oscillator with nonlinear (positive)
"damping":

$$ x'' + P(x') x' + u(x) = F(t) (EQ1)$$

with generic IV:
$$ x(0) = x_o x'(0) = v_o$$

with "null mean" forcing with (minimal and positive) period $T$:

$$ F(t+T)=F(t) $$

$$ \frac{1}{T}\int_0^T{F(t)dt}=0 $$

and positive and nonlinear (and nonconstant) damping:

$$ P(x') >0 \forall x' \in \R $$

Quite surprisingly for me, I did not find many recent papers on this so
physically relevant subject. Mainly I did find some (interesting but)
numerical study.

I did find some interesting work on the case of constant (positive)
damping, e.g. the paper



I wonder if some of you might give me some also non recent
bibliographical reference to review papers or book devoted to equation
(EQ1)

Ciao e grazie!
g.

 

Hello g.,

People have been playing around with those toy-systems a lot in chaos
theory, for example the van der Pol oscillator. Furthermore lots of
research has been done on kicked oscillators and coupled non-linear
oscillators. I don't know if this involves phase-dependent damping though.

HTH,

Maarten