Periodically forced system Energy change: Integrate over half-period E 2 t2 t Energy change over forcing period 2  period energy change = = quasiperiodic motion

Forced Duffing equation no forcing: periodic motion  = 0,  = 4,  =0. 2 weak forcing: quasiperiodic motion  = 0,  = 4,  =0. 2

Poincare maps for forced Duffing equation Quasiperiodic motion  = 1,  = 4,  =0. 2 Frequency locking:  = 10,  = 4,  =0. 2 Chaos:  = 17,  = 4,  =0. 2

Forcing near a homoclinic orbit initial conditions close to the saddle point of the unforced system: chaos initial conditions away from the saddle point: quasiperiodic motion non-dissipative system  = 0,  = 0.2

Forcing near a homoclinic: dissipative system Dependence on initial conditions Left: relaxation to a stationary state Right: chaotic motion f(y)= + y–y 3, g(y)=  1,  = 0.01,  = 1

Fractals Hausdorf dimension Cantor set Serpinski carpet