Stabilizing and tracking unknown steady States of dynamical systems

An adaptive dynamic state feedback controller for stabilizing and tracking unknown steady states of dynamical systems is proposed. We prove that the steady state can never be stabilized if the system and controller in sum have an odd number of real positive eigenvalues. For two-dimensional systems, this topological limitation states that only an unstable focus or node can be stabilized with a stable controller, and stabilization of a saddle requires the presence of an unstable degree of freedom in a feedback loop. The use of the controller to stabilize and track saddle points (as well as unstable foci) is demonstrated both numerically and experimentally with an electrochemical Ni dissolution system.     

Controlling a time-delay system using multiple delay feedback control

In this paper multiple delay feedback control (MDFC) with different and independent delay times is shown to be an efficient method for stabilizing fixed points in finite-dimensional dynamical systems. Whether MDFC can be applied to infinite-dimensional systems has been an open question. In this paper we find that for infinite-dimensional systems modelled by delay differential equations, MDFC works well for stabilizing (unstable) steady states in long-, moderate- and short-time delay regions, in particular for the hyperchaotic case.     

Oscillation death in coupled oscillators

We study dynamical behaviors in coupled nonlinear oscillators and find that under certain conditions, a whole coupled oscillator system can cease oscillation and transfer to a globally nonuniform stationary state [i.e., the so-called oscillation death (OD) state], and this phenomenon can be generally observed. This OD state depends on coupling strengths and is clearly different from previously studied amplitude death (AD) state, which refers to the phenomenon where the whole system is trapped into homogeneously steady state of a fixed point, which already exists but is unstable in the absence of coupling. For larger systems, very rich pattern structures of global death states are observed. These Turing-like patterns may share some essential features with the classical Turing pattern.      

Control of unstable steady states in neutral time-delayed systems

We present an analysis of time-delayed feedback control used to stabilize an unstable steady state of a neutral delay differential equation. Stability of the controlled system is addressed by studying the eigenvalue spectrum of a corresponding characteristic equation with two time delays. An analytic expression for the stabilizing control strength is derived in terms of original system parameters and the time delay of the control. Theoretical and numerical results show that the interplay between thecontrol strength and two time delays provides a number of regions in the parameter space where the time-delayed feedback control can successfully stabilize an otherwise unstable steady state.     

Controlled Hopf bifurcation of a storage-ring free-electron laser

Local bifurcation control is a topic of fundamental importance in the field of nonlinear dynamical systems. We discuss an original example within the context of storage-ring free-electron laser physics by presenting a new model that enables analytical insight into the system dynamics. The transition between the stable and the unstable regimes, depending on the temporal overlapping between the light stored in the optical cavity and the electrons circulating into the ring, is found to be a Hopf bifurcation. A feedback procedure is implemented and shown to provide an effective stabilization of the unstable steady state.     

Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators

We present some novel thermodynamic ideas based on the Maupertuis principle. By considering Hamiltonians written in terms of appropriate action-angle variables we show that thermal states can be characterized by the action variables and by their evolution in time when the system is nonintegrable. We propose dynamical definitions for the equilibrium temperature and entropy as well as an expression for the nonequilibrium entropy valid for isolated systems with many degrees of freedom. This entropy is shown to increase in the relaxation to equilibrium of macroscopic systems with short-range interactions, which constitutes a dynamical justification of the Second Law of Thermodynamics. Several examples are worked out to show that this formalism yields the right microcanonical (equilibrium) quantities. The relevance of this approach to nonequilibrium situations is illustrated with an application to a network of coupled oscillators (Kuramoto model). We provide an expression for the entropy production in this system finding that its positive value is directly related to dissipation at the steady state in attaining order through synchronization.     

Dynamical Creation of Entanglement and Steady Entanglement Between Two Spatially Separated Λ-Type Atoms

We report possibility of generating entanglement and steady entanglement between two identical atoms in free space with a very natural way when their spatial separation is on the order of wavelength or less. We show a dynamical creation of entanglement and steady entanglement due to the radiative coupling with different separable initial atomic states and study the entanglement properties about this atomic subsystem. Not only the creation of steady state entanglement is decided by the initial atomic states, but also the magnitude of the entanglement and the steady state entanglement are found to be strongly dependent on the initial states. We derive a master equation for the atomic subspace and solve it analytically to show how the spontaneous emission from the two atoms system induces entanglement and steady entanglement, the crossing coupling terms in master equation can enhance the entanglement value.     

Complex dynamics in delay-differential equations with large delay

We investigate the dynamical properties of delay differential equations with large delay. Starting from a mathematical discussion of the singular limit τ → ∞, we present a novel theoretical approach to the stability properties of stationary solutions in such systems. We introduce the notion of strong and weak instabilities and describe a method that allows us to calculate asymptotic approximations of the corresponding parts of the spectrum. The theoretical results are illustrated by several examples, including the control of unstable steady states of focus type by time delayed feedback control and the stability of external cavity modes in the Lang-Kobayashi system for semiconductor lasers with optical feedback.     

Entropy balance,time reversibility,and mass transport in dynamical systems

We review recent results concerning entropy balance in low-dimensional dynamical systems modeling mass (or charge) transport. The key ingredient for understanding entropy balance is the coarse graining of the local phase-space density. It mimics the fact that ever refining phase-space structures caused by chaotic dynamics can only be detected up to a finite resolution. In addition, we derive a new relation for the rate of irreversible entropy production in steady states of dynamical systems: It is proportional to the average growth rate of the local phase-space density. Previous results for the entropy production in steady states of thermostated systems without density gradients and of Hamiltonian systems with density gradients are recovered. As an extension we derive the entropy balance of dissipative systems with density gradients valid at any instant of time, not only in stationary states. We also find a condition for consistency with thermodynamics. A generalized multi-Baker map is used as an illustrative example. (c) 1998 American Institute of Physics.     

Nanoscale Spontaneous Patterning

The SMB equation describing nanoscale spontaneous patterning is studied both analytically and numerically. In contradiction to the claim in the original SMB paper [D. Srolovitz, A. Mazor, B. Bukiet, J. Vac. Sci. Technol. A6(4) (1988), 2371--2380.] that some steady states are stable, we found that all the steady states are unstable. A dynamical system reason for this is given. We also found that typical small initial data solutions undergo an exponential growth followed by an almost linear growth. Such a feature is consistent with the experimental data in the paper [J. Erlebacher et al., J. Vac. Sci. Technol., A18(1) (2000), 115--120, Figure 3]. On the other hand, we never observed the decay portion of the numerical solution reported in this paper. We invent an elegant energy principle which supports our findings.     

Chaos control using notch filter feedback

A method for stabilizing periodic orbits and steady states of chaotic systems is presented using specifically filtered feedback signals. The efficiency of this control technique is illustrated with simulations (R?ssler system, laser model) and a successful experimental application for stabilizing intensity fluctuations of an intracavity frequency-doubled Nd:YAG laser.     

Memory and Markov Blankets

In theoretical biology, we are often interested in random dynamical systems—like the brain—that appear to model their environments. This can be formalized by appealing to the existence of a (possibly non-equilibrium) steady state, whose density preserves a conditional independence between a biological entity and its surroundings. From this perspective, the conditioning set, or Markov blanket, induces a form of vicarious synchrony between creature and world—as if one were modelling the other. However, this results in an apparent paradox. If all conditional dependencies between a system and its surroundings depend upon the blanket, how do we account for the mnemonic capacity of living systems? It might appear that any shared dependence upon past blanket states violates the independence condition, as the variables on either side of the blanket now share information not available from the current blanket state. This paper aims to resolve this paradox, and to demonstrate that conditional independence does not preclude memory. Our argument rests upon drawing a distinction between the dependencies implied by a steady state density, and the density dynamics of the system conditioned upon its configuration at a previous time. The interesting question then becomes: What determines the length of time required for a stochastic system to ‘forget’ its initial conditions? We explore this question for an example system, whose steady state density possesses a Markov blanket, through simple numerical analyses. We conclude with a discussion of the relevance for memory in cognitive systems like us.     

Controlling chaos to unstable periodic orbits and equilibrium state solutions for the coupled dynamos system

In the case where the knowledge of goal states is not known, the controllers are constructed to stabilize unstable steady states for a coupled dynamos system. A delayed feedback control technique is used to suppress chaos to unstable focuses and unstable periodic orbits. To overcome the topological limitation that the saddle-type steady state cannot be stabilized, an adaptive control based on LaSalle's invariance principle is used to control chaos to unstable equilibrium (i.e. saddle point, focus, node, etc.). The control technique does not require any computer analysis of the system dynamics, and it operates without needing to know any explicit knowledge of the desired steady-state position.     

Nonlinear time-independent and time-dependent propagation of light in direct-gap semiconductors with paired excitons bound into biexcitons

We study a new class of nonlinear cooperative phenomena that occur when light propagates in direct-gap semiconductors. The nonlinearity here is due to a process, first discussed by A. L. Ivanov, L. V. Keldysh, and V. V. Panashchenko, in which two excitons are bound into a biexciton by virtue of their Coulomb interaction. For the geometry of a ring cavity, we derive a system of nonlinear differential equations describing the dynamical evolution of coherent excitons, photons, and biexcitons. For the time-independent case we arrive at the equation of state of optical bistability theory, and this equation is found to differ considerably from the equations of state in the two-level atom model and in the exciton region of the spectrum. We examine the stability of the steady states and determine the switchover times between the optical bistability branches. We also show that in the unstable sections of the equation of state, nonlinear periodic and chaotic self-pulsations may arise, with limit cycles and strange attractors being created in the phase space of the system. The scenario for the transition to the dynamical chaos mode is found. A computer experiment is used to study the dynamic optical bistability. Finally, we discuss the possibility of detecting these phenomena in experiments. Zh. éksp. Teor. Fiz. 112, 1778–1790 (November 1997)     

Tracking controlled chaos: Theoretical foundations and applications

Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. (c) 1997 American Institute of Physics.     

A separation principle for dynamical delayed output feedback control of chaos

In this Letter, a dynamical delayed output-feedback (DDOF) control strategy is proposed for stabilizing unstable periodic orbits (UPOs) of chaotic systems. Using the Floquet theory, a separation principle is established which gives a necessary and sufficient stability condition for DDOF UPO stabilizing control systems. The new principle shows that the so-called “odd number limitation” for delayed state-feedback control systems also applies to DDOF control.     

Patterned chaos in a system of three coupled identical unidirectional ring lasers

Three coupled identical unidirectional B-type ring lasers are considered and the dynamics for varying coupling parameter investigated. For given system parameters like pumping, cavity damping etc. the system is characterized by three distinct types of steady states. The linear stability of these steady states is investigated in detail which are unstable beyond a critical value of the coupling parameter. For a certain range of the coupling parameter chaotic behaviour of the system is demonstrated. The dynamical variables are mapped on to a complex variable reflecting the symmetry of the system. The plot of the real vs. the imaginary part of this complex variable exhibits patterns with threefold symmetry. The largest Lyapunov exponent as well as the power spectra corresponding to the chaotic attractors are also calculated.     

Controlling dynamical behavior of drive-response system through linear augmentation

Unidirectionally coupled chaotic systems give rise to driver induced bistability in response system under certain parameters setting. Such a system is studied here with augmented dynamics. A linear augmentation provides a controlled dynamical behavior of response system in two different ways: augmented drive system brings the stabilization of the steady state where as augmented response system is able to control the bistability. We present a detailed analysis of Lorenz–Rössler system with linear augmentation for controlled dynamical behavior.     

Suppression of dynamics and frequency synchronization in coupled slow and fast dynamical systems

We present our study on the emergent states of two interacting nonlinear systems withdiffering dynamical time scales. We find that the inability of the interacting systems tofall in step leads to difference in phase as well as change in amplitude. If the mismatchis small, the systems settle to a frequency synchronized state with constant phasedifference. But as mismatch in time scale increases, the systems have to compromise to astate of no oscillations. We illustrate this for standard nonlinear systems and identifythe regions of quenched dynamics in the parameter plane. The transition curves to thisstate are studied analytically and confirmed by direct numerical simulations. As animportant special case, we revisit the well-known model of coupled ocean-atmosphere systemused in climate studies for the interactive dynamics of a fast oscillating atmosphere andslowly changing ocean. Our study in this context indicates occurrence of multi stableperiodic states and steady states of convection coexisting in the system, with a complexbasin structure.