Learning to control a complex multistable system

Space-dependent intermittent feedback can control birhythmicity

Birhythmicity is evident in many nonlinear systems, which include physical and biological systems. In some living systems, birhythmicity is necessary for response to the varying environment while unnecessary in some physical systems as it limits their efficiency. Therefore, its control is an important area of research. This paper proposes a space-dependent intermittent control scheme capable of controlling birhythmicity in various dynamical systems. We apply the proposed control scheme in five nonlinear systems from diverse branches of natural science and demonstrate that the scheme is efficient enough to control the birhythmic oscillations in all the systems. We derive the analytical condition for controlling birhythmicity by applying harmonic decomposition and energy balance methods in a birhythmic van der Pol oscillator. Further, the efficacy of the control scheme is investigated through numerical and bifurcation analyses in a wide parameter space. Since the proposed control scheme is general and efficient, it may be employed to control birhythmicity in several dynamical systems.

Feedforward attractor targeting for non-linear oscillators using a dual-frequency driving technique

A feedforward control technique is presented to steer a harmonically driven, non-linear system between attractors in the frequency-amplitude parameter plane of the excitation. The basis of the technique is the temporary addition of a second harmonic component to the driving. To illustrate this approach, it is applied to the Keller-Miksis equation describing the radial dynamics of a single spherical gas bubble placed in an infinite domain of liquid. This model is a second-order, non-linear ordinary differential equation, a non-linear oscillator. With a proper selection of the frequency ratio of the temporary dual-frequency driving and with the appropriate tuning of the excitation amplitudes, the trajectory of the system can be smoothly transformed between specific attractors; for instance, between period-3 and period-5 orbits. The transformation possibilities are discussed and summarized for attractors originating from the subharmonic resonances and the equilibrium state (absence of external driving) of the system.

Intermittent feedback induces attractor selection

We present a method for attractor selection in multistable dynamical systems. It involves a feedback term that is active only when the dynamics of the system is in a particular fraction of state space of the attractor. We implement this method first on a simplest symmetric chaotic flow and then on a bistable neuronal system. We find that adding this space-dependent feedback term to the dynamical equations of these systems will drive the dynamics to the desired attractor by annihilating the other. We further demonstrate that the attractor selection due to this feedback term can be used in construction of logic gates, which is one of the practical applications of the proposed method.

Reinforcement learning in complementarity game and population dynamics

We systematically test and compare different reinforcement learning schemes in a complementarity game [J. Jost and W. Li, Physica A 345, 245 (2005)] played between members of two populations. More precisely, we study the Roth-Erev, Bush-Mosteller, and SoftMax reinforcement learning schemes. A modified version of Roth-Erev with a power exponent of 1.5, as opposed to 1 in the standard version, performs best. We also compare these reinforcement learning strategies with evolutionary schemes. This gives insight into aspects like the issue of quick adaptation as opposed to systematic exploration or the role of learning rates.

Basin size evolution between dissipative and conservative limits

Recent methods for stabilizing systems like, e.g., loss-modulated CO2 lasers, involve inducing controlled monostability via slow parameter modulations. However, such stabilization methods presuppose detailed knowledge of the structure and size of basins of attraction. In this Brief Report, we numerically investigate basin size evolution when parameters are varied between dissipative and conservative limits. Basin volumes shrink fast as the conservative limit is approached, being well approximated by Gaussian profiles, independently of the period. Basin shrinkage and vanishing is due to the absence of bounded motions in the Hamiltonian limit. In addition, we find basin volume to remain essentially constant along a peculiar parameter path along which it is possible to recover the dissipation rate solely from metric properties of self-similar structures in phase-space.

Controlling chaos in unidimensional maps using macroevolutionary algorithms

We introduce a simple search algorithm that explores the parameter of periodically perturbed discrete maps in order to find desired orbits through chaos control. The method has been applied to one-dimensional maps but is easily extendable to higher-dimensional systems. Here, we consider two types of chaos control involving proportional pulses in the system variables [Phys. Rev. Lett. 72, 1455 (1994)] and constant feedback [Phys. Rev. E 51, 6239 (1995)], the first case being presented in detail. It is shown that our method allows a rapid exploration of parameter space and the finding of high-fitness (i.e., controlled) solutions close to the target orbits, even when high periodicities are required.