Controlling Hamiltonian chaos by medium perturbation in periodically driven systems

Local phase space control and interplay of classical and quantum effects in dissociation of a driven Morse oscillator

“…In other words the control field is of the form λ 2 cos(2ω F t) and, since λ 2 > 0, comes with a phase difference of π relative to the driving field. Interestingly, Wu et al in an earlier work [27] have suggested precisely the same control field characteristics for suppressing chaos in the driven Morse system. However, they were not clear about the mechanism for the suppression and this work yields the necessary insight in terms of the creation of local cantori barriers.…”

“…Second, at present the focus of researchers is increasingly shifting from gaining mechanistic insights to controlling [8,9,10,11] the various processes and in this regard a firm understanding of the underlying mechanisms is essential. Therefore, it is not entirely surprising that the driven Morse system has been studied in great detail from the quantum, classical, and semiclassical [12] perspectives and with an equally diverse choice for the field -monochromatic [13,14,15,16,17,18,19,20,21,22], bichromatic (with relative phase) [23,24,25,26,27,28,29,30], chirped [31,32,33,34], shaped pulses [21,26,35], and stochastic noise [36]. More recently, the dynamics of a Morse oscillator under the influence of external fields has become relevant in the context of models for quantum computing based on molecular vibrations [37].…”

Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos

“…Control of chaos in nonlinear dynamical systems has been achieved by applying small perturbations that effectively change the dynamics of the system around the region -typically a periodic orbit -that one wishes to stabilize [1]. This method has been successful in dissipative systems, but its extensions to control and targeting in Hamiltonian systems [2,3,4,5,6,7,8,9,10] have met various difficulties not present in the dissipative case: the absence of attracting sets, for instance, makes it hard to stabilize anything. In addition, these methods require that one know in advance what one wants to do; in particular, the orbit to be stabilized may have to be known in advance to a fair accuracy.…”

Topology of classical molecular optimal control landscapes in phase space

“…In this work, we consider the use of open loop control to find such an optimal trajectory, with its stability assessed post-facto. Other works [33][34][35][36] have considered using active feedback control to stabilize chaotic systems about an optimal trajectory. While there is no theoretical barrier to finding an optimally controlled trajectory in the chaotic regime, it can be numerically difficult due to chaotic systems' high sensitivity to initial conditions.…”