Unformatted text preview: x,y = x . (b) Consider the energy of the unperturbed ( = 0) system, E = ( x 2 + y 2 ) / 2. Find the region in phase space where E increases in time and the region where E decreases in time. (c) Introduce action-angle coordinates I = ( x 2 + y 2 ) / 2 , = arctan( y/x ) of the unperturbed harmonic oscillator. (d) Obtain the equations for I (which you should already have from 2b) and . (e) Average the equation for I with respect to (i.e. integrate the right-hand side with respect to , from = 0 to = 2 and divide by 2 ). Find out for which E the resulting averaged expression vanishes. (f) Numerically simulate the system obtained in 2a, using small . Plot trajectories in the phase space, starting from various initial condi-tions. Do you nd any limit cycles in the phase space? Where? 1...
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- Fall '08
- Simple Harmonic Motion, Dynamical systems, phase space, small forcing x2