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Dependence of amplitude X of the cycles on driving frequency ; , . Solid (red and blue) lines correspond to the stable cycles and , respectively, and dashed black line refers to saddle cycle S. Thin dashed vertical lines denote the values and . Insets: the cycles in the state space for (upper left) and (lower right).
Dependence of energy J on duration for different phases : (a)for control force , (b) for control force . Markers + (red, upper set of markers) and × (blue, lower set of markers) correspond to transitions and , respectively.
(a) Trajectories during Pontryagin control and (b) respective control functions . (a) Points shown correspond to coordinates in the stroboscopic section taken after each period of the harmonic driving force. Markers + (red) and × (blue) reflect the transitions and , respectively. (b) Lower red line represents the truncated control function for transition and upper blue line refers to . The upper function is shifted by for an illustrative purpose.
Barrier as a function of driving frequency . Solid (red) and dashed (blue) lines correspond to the stable cycles and , respectively. Thin dashed vertical lines denote values and . The inset shows a magnified part of the figure.
Optimal values of the amplitude , duration and phase corresponding to the minimal energy and minimal power for different control functions. The first row specifies the type of control function and the initial state ( refers to arbitrary form as in Eq. (1)). define the ratio of to the minimal energy of Pontrygian control , calculated for each initial state separately. Values of correspond to the minimal power and to Pontryagin control. Ratio was calculated separately for each initial oscillatory state.
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