Periodically forced system

1
Periodically forced system
Energy change
Integrate over half-period
E
Energy change over forcing period 2p
2p t
period energy change quasiperiodic motion
2
Forced Duffing equation
no forcing periodic motion b0, ?4, ?0.2
weak forcing quasiperiodic motion b0, ?4,
?0.2
3
Poincare maps for forced Duffing equation
Quasiperiodic motion b1, ?4, ?0.2
Frequency locking b10, ?4, ?0.2
Chaos b17, ?4, ?0.2
4
Forcing near a homoclinic orbit
non-dissipative system ????????d0, b0.2
initial conditions close to the saddle point of
the unforced system chaos
initial conditions away from the saddle point
quasiperiodic motion
5
Forcing near a homoclinic dissipative system
f(y)nyy3, g(y)?1, d0.01, b1
Dependence on initial conditions Left
relaxation to a stationary state Right chaotic
motion
6
Fractals
Hausdorf dimension
Cantor set
Serpinski carpet