New Approximations and Tests of Linear Fluctuation-Response for Chaotic Nonlinear Forced-Dissipative Dynamical Systems

We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation–dissipation theorem. Unlike the earlier work in developing fluctuation–dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation–dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation–dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.     

Mathematical modeling, nonlinear equations of evolution, and the dynamics of nuclear reactors

The article presents some results on solvability and qualitative properties of solutions of the abstract Cauchy problem for a new class of nonlinear equations of evolution in Banach spaces. These results provide a unified framework for the analysis of a wide range of applied problems. Some applications to nonlinear mathematical problems of nuclear reactor dynamics are suggested. Supported by a grant from the Ministry of Science, Higher Education, and Technical Policy of the Russian Federation as part of the program, “Nonlinear Dynamic Systems: Qualitative Analysis and Control.” Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov, No. 2, pp. 76–90, 1994.     

Nonlinear generalized synchronization of chaotic systems by pure error dynamics and elaborate nondiagonal Lyapunov function

By applying pure error dynamics and elaborate nondiagonal Lyapunov function, the nonlinear generalized synchronization is studied in this paper. Instead of current mixed error dynamics in which master state variables and slave state variables are presented, the nonlinear generalized synchronization can be obtained by pure error dynamics without auxiliary numerical simulation. The elaborate nondiagonal Lyapunov function is applied rather than current monotonous square sum Lyapunov function deeply weakening the powerfulness of Lyapunov direct method. Both autonomous and nonautonomous double Mathieu systems are used as examples with numerical simulations.     

On a class of nonconvex and nonlinear optimal control problems

We propose two relaxation approaches for the existence of solutions of a nonconvex optimal control problem with a nonlinear dynamics and two fixed endpoints. The first approach adds to the functional a term depending on the final state; the second one introduces a new scalar control into both the functional and the dynamics. Our results generalize those obtained in the linear case. We assume that the integrand be concave w.r.t. the state variable, and provide an example showing that a strict concavity condition, in the nonlinear case, is essential. Finally, a result without relaxation is presented.     

Data-based prediction and causality inference of nonlinear dynamics

Natural systems are typically nonlinear and complex, and it is of great interest to be able to reconstruct a system in order to understand its mechanism, which cannot only recover nonlinear behaviors but also predict future dynamics. Due to the advances of modern technology, big data becomes increasingly accessible and consequently the problem of reconstructing systems from measured data or time series plays a central role in many scientic disciplines. In recent decades, nonlinear methods rooted in state space reconstruction have been developed, and they do not assume any model equations but can recover the dynamics purely from the measured time series data. In this review, the development of state space reconstruction techniques will be introduced and the recent advances in systems prediction and causality inference using state space reconstruction will be presented. Particularly, the cutting-edge method to deal with short-term time series data will be focused on. Finally, the advantages as well as the remaining problems in this field are discussed.     

A nonlinear inverse-dynamics flight control system

The theme of this paper is the use of differential-geometricalcontrol and its application, through the nonlinear inverse-dynamics(NID) methodology, to a nonlinear flight control system. Wepresent a way for generating a generic control model for a promisingcontrol device—the spoiler—from experimental data,and for combining the nonlinear spoiler model with nonlinearaircraft dynamics. A new design procedure concerning the useof the NID control techniques is developed and utilized forthe design of a nonlinear inverse-dynamics flight control systemwith various functional control modes. Simulation results foraircraft manoeuvres are presented, demonstrating the successof the design procedure and of the control effectiveness ofspoilers for the enhancement of aircraft manoeuvring and, inparticular, for alleviation of the effects of microbursts.     

Application of a hybrid method to the nonlinear dynamic analysis of a flexible rotor supported by a spherical gas-lubricated bearing system

This paper employs a hybrid numerical method combining the differential transformation method and the finite difference method to study the nonlinear dynamic behavior of a flexible rotor supported by a spherical gas-lubricated bearing system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, and quasi-periodic responses of the rotor center and the journal center. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass and bearing number are increased. The current analytical results are found to be in good agreement with those from other numerical methods. Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of spherical gas film rotor–bearing systems.     

Stabilization of Passive Nonlinear Stochastic Differential Systems by Bounded Feedback

Abstract

The purpose of this paper is to give a systematic method for global asymptotic stabilization in probability of nonlinear control stochastic differential systems the unforced dynamics of which are Lyapunov stable in probability. The approach developed in this paper is based on the concept of passivity for nonaffine stochastic differential systems together with the theory of Lyapunov stability in probability for stochastic differential equations. In particular, we prove that, as in the case of affine in the control stochastic differential systems, a nonlinear stochastic differential system is asymptotically stabilizable in probability provided its unforced dynamics are Lyapunov stable in probability and some rank conditions involving the affine part of the system coefficients are satisfied. Furthermore, for such systems, we show how a stabilizing smooth state feedback law can be designed explicitly. As an application of our analysis, we construct a dynamic state feedback compensator for a class of nonaffine stochastic differential systems.     

Algorithm for constructing an optimal control strategy in a nonlinear differential game with Lipschitz compactly supported payoff

For a zero-sum differential game, we consider an algorithm for constructing optimal control strategies with the use of backward minimax constructions. The dynamics of the game is not necessarily linear, the players’ controls satisfy geometric constraints, and the terminal payoff function satisfies the Lipschitz condition and is compactly supported. The game value function is computed by multilinear interpolation of grid functions. We show that the algorithm error can be arbitrarily small if the discretization step in time is sufficiently small and the discretization step in the state space has a higher smallness order than the time discretization step. We show that the algorithm can be used for differential games with a terminal set. We present the results of computations for a problem of conflict control of a nonlinear pendulum.     

On the dynamical equations of a system of linearly coupled nonlinear oscillators

We consider a system of differential equations that describes the dynamics of an infinite chain of linearly coupled nonlinear oscillators. Some results concerning the existence and uniqueness of global solutions of the Cauchy problem are obtained. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 723–729, June, 2006.     

Optimal control of constrained time lag systems: Necessary conditions and numerical treatment

We consider retarded optimal control problems with constant delays in state and control variables under mixed controlstate inequality constraints. First order necessary optimality conditions in the form of Pontryagin's minimum principle are presented and discussed as well as numerical methods based upon discretization techniques and nonlinear programming. The minimum principle for the considered problem class leads to a boundary value problem which is retarded in the state dynamics and advanced in the costate dynamics. It can be shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Robust controlling hyperchaos of the Rössler system subject to input nonlinearities by using sliding mode control

A sliding mode control is designed to stabilize the well-known hyperchaos of Rössler system to equilibrium points subject to sector nonlinear input. The proposed control law is robust against both the input nonlinearity and external disturbance. The error bound can be arbitrarily set by assigning the corresponding dynamics to the sliding surfaces when the desired state is not an equilibrium point. Simulation results show that the system state can be regulated to an equilibrium point in the state space. It is also seen that the system still possesses advantage of fast response and good transient performance even though the control input is nonlinear.     

Dynamic Mean Flow and Small-Scale Interaction through Topographic Stress

Summary. The equations describing the mean flow and small-scale interaction of a barotropic flow via topographic stress with layered topography are studied here through the interplay of theory and numerical experiments. Both a viewpoint toward atmosphere—ocean science and one toward chaotic nonlinear dynamics are emphasized. As regards atmosphere—ocean science, we produce prototype topographic blocking patterns without damping or driving, with topographic stress as the only transfer mechanism; these patterns and their chaos bear some qualitative resemblance to those observed in recent laboratory experiments on topographic blocking. As regards nonlinear dynamics, it is established that the equations for mean flow and small-scale interaction with layered anisotropic topography form a novel Hamiltonian system with rich regimes of intrinsic conservative chaos, which include both global and weak homoclinic stochasticity, as well as other regimes with complete integrability involving complex heteroclinic structure. Received August 7, 1997; second revision received December 18, 1997     

The Hamiltonian dynamics of self-gravitating liquid and gas ellipsoids

The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).      

The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

We consider nonlinear aspects of the flow of an inviscid two-dimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a long-wave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a well-posed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finite-time rupture singularity. This has been established numerically for rather general initial conditions.     

Well-Posedness and Regularity for a Parabolic-Hyperbolic Penrose-Fife Phase Field System

This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The system couples a nonlinear parabolic equation for the absolute temperature with a nonlinear hyperbolic equation for the phase variable χ, which is characterized by the presence of an inertial term multiplied by a small positive coefficient μ. This feature is the main consequence of supposing that the response of χ to the generalized force (which is the functional derivative of a free energy potential and arises as a consequence of the tendency of the free energy to decay towards a minimum) is subject to delay. We first obtain well-posedness for the resulting initial-boundary value problem in which the heat flux law contains a special function of the absolute temperature ϑ, i.e. α(ϑ) ∼ ϑ − 1/ϑ. Then we prove convergence of any family of weak solutions of the parabolic-hyperbolic model to a weak solution of the standard Penrose-Fife model as μ ↘ 0. However, the main novelty of this paper consists in proving some regularity results on solutions of the parabolic-hyperbolic system (including also estimates of Moser type) that could be useful for the study of the longterm dynamics.     

Functional classical mechanics and rational numbers

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility problem. A “functional” formulation of classical mechanics is suggested. The physical meaning is attached in this formulation not to an individual trajectory but only to a “beam” of trajectories, or the distribution function on phase space. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values and there are corrections to the Newton trajectories. We give a construction of probability density function starting from the directly observable quantities, i.e., the results of measurements, which are rational numbers.     

An investigation of open-loop and inverse simulation as nonlinear model validation tools for helicopter flight mechanics

The nonlinear model validation techniques of open-loop and inverse simulation are introduced. The methodologies are explained and examples are given. The paper presents the results of an investigation into the use of open-loop and inverse simulation to help in the development of a nonlinear real-time helicopter model. The individual rigid body state equations in the model are simulated with the aim of producing insight into the cause of inaccuracies in the model. A suspected source of inaccuracy is verified using partial open-loop simulation. Unmodelled dynamics are represented by using the relevant flight data as an open-loop input to the simulation thus revealing the effect of incorporating those dynamics. After localising the cause of inaccuracies in the simulation model, modifications and improvements are verified using closed-loop simulation. The improvements are then evaluated by comparing results in normal and inverse simulation modes.     

External Estimates for Reachable Sets of a Control System with Uncertainty and Combined Nonlinearity

The problem of estimating the trajectory tubes of a nonlinear control dynamic system with uncertainty in the initial data is studied. It is assumed that the dynamic system has a special structure in which the nonlinear terms are defined by quadratic forms in the state coordinates and the values of uncertain initial states and admissible controls are subject to ellipsoidal constraints. The matrix of the linear terms in the velocities of the system is not known exactly; it belongs to a given compact set in the corresponding space. Thus, the dynamics of the system is complicated by the presence of bilinear components in the righthand sides of the differential equations of the system. We consider a complicated case and generalize the author’s earlier results. More exactly, we assume the simultaneous presence in the dynamics of the system of bilinear functions and quadratic forms (without the assumption of their positive definiteness) and we also take into account the uncertainty in the initial data and the impact of the control actions, which may also be treated here as undefined additive disturbances. The presence of all these factors greatly complicates the study of the problem and requires an adequate analysis, which constitutes the main purpose of this study. The paper presents algorithms for estimating the reachable sets of a nonlinear control system of this type. The results are illustrated by examples.