Resonant control of the Rössler system

Control and Identification of Chaotic Systems by Altering the Oscillation Energy

“…The phase of perturbation is used both to fine-tune the control or and to switch the drive direction on the opposite one, i.e. to switch between the directions of energy increase and decrease (Tereshko & Shchekinova, 1998). An important issue is the application of control in practice.…”

“…However, the independence of the perturbation from a system state leads to some limitations of above approach: the control by periodic perturbations relying only on their period and amplitude is not, in general, a goal-oriented technique (Shinbrot et al, 1993). On the other hand, the importance of a phase (Cao, 2005;Chacón, 1996;Chizhevsky & Corbalán, 1996;Chizhevsky et al, 1997;Dangoisse et al, 1997;Fronzoni et al, 1991;Meucci et al, 1994;Qu et al, 1995;Tereshko & Shchekinova, 1998) and even a shape (Azevedo & Rezende, 1991;Chacón, 1996;Chacón & Díaz Bejarano, 1993;Rödelsperger et al, 1995) of perturbation became evident. The utilization of extra parameters allows tuning the perturbation to a desired target shape more selectively.…”

“…Secondly, keeping the perturbation precisely in phase with the controlled signal ensures smallest control amplitudes, whereas dephasing can destroy the control. By changing only the perturbation phase, one can switch the trajectory from one controlled state to another (Tereshko & Shchekinova, 1998). In real-life systems, the existing uncontrolled drifts can spoil resonant conditions.…”

“…Typically, this method can require as many control forces, as there are dimensions of the system. In contrast, there are many examples of converting chaos to a periodic motion by exposing a system to only one, weak periodic force or weak parameter modulation (Alexeev & Loskutov, 1987;Braiman & Goldhirsch, 1991;Cao, 2005;Chacón, 1996;Chacón & Díaz Bejarano, 1993;Chizhevsky & Corbalán, 1996;Chizhevsky et al, 1997;Dangoisse et al, 1997;Fronzoni et al, 1991;Kivshar et al, 1994;Lima & Pettini, 1990;Liu & Leite, 1994;Meucci et al, 1994;Qu et al, 1995;Ramesh & Narayanan, 1999;Rödelsperger et al, 1995;Tereshko & Shchekinova, 1998). Typically, this approach utilizes only a period and an amplitude of perturbation (Alexeev & Loskutov, 1987;Braiman & Goldhirsch, 1991;Kivshar et al, 1994;Lima & Pettini, 1990;Liu & Leite, 1994;Ramesh & Narayanan, 1999).…”

Chaos Inducement and Enhancement in Two Particular Nonlinear Maps Using Weak Periodic/Quasiperiodic Perturbations

“…The basic idea is that, by way of weak perturbation, the modulated periodic orbit will either go through new bifurcations (such as period doubling cascade to chaos) or be destablized to give way to chaotic regime, therefore chaos is induced naturally. The perturbation we use here is either periodic or quasiperiodic, which has been employed otherwise to suppress or eliminate chaos [Braiman & Goldhirsch, 1991;Meucci et al, 1994;González et al, 1998;Tereshko & Shchekinova, 1998;Chacón, 1999] in dynamical systems.…”