Weakly dissipative system – local analysis Linear stability analysis orHopf bifurcation: S + /U + : upper state exists and is stable/unstable S – /U – : lower state exists and is stable/unstable intermediate state (unstable) exists at saddle–node Hopf (–) Hopf (+)

Weakly dissipative system – global analysis Trajectories at  = 0, Conserved energy: energy change : At small  compute the rate of energy change by averaging over the period T : integration limits are values of y at turning points where p vanishes

Weakly dissipative system – global analysis Condition for a stationary orbit: Value of  required for a stationarity: Bifurcation diagram (–) 2. O 3. O (+) 4. (+) 5. (+) o – 6. (–)(+) 7. (–) o + 8. (–) sniper