Valentin N. Tkhai

Model with coupled subsystems

Consideration was given to the model with coupled subsystems. In the absence of relations between the subsystems, the MIS falls down into independent systems of autonomous ordinary differential equations. In the structure of the entire system, the subsystems make up hierarchical levels. Sun-planets-satellites, interacting moving objects, and so on exemplify the models with coupled subsystems. The problem of studying dynamics of such models was posed. The following natural approach to their analysis was proposed: classification of the subsystems by types (dynamic properties), specification of various bundles of subsystems, and subsequent analysis of these bundles. Realization of the approach to oscillations, stability, stabilization, bifurcation, and resonance was given. These problems were solved for the model with coupled subsystems having two second-kind subsystems in the basic combination of the oscillation modes in the subsystem.

Oscillations in the autonomous model containing coupled subsystems

Consideration was given to the model containing coupled subsystems (MCCS). In the absence of coupling, MCCS falls down into independent subsystems represented by autonomous ordinary differential equations (ODE). In the structure of the entire system its subsystems make up hierarchical levels. The autonomous MCCS was studied. The “cycle or family of periodic motions” alternative was shown to be always realized by an individual system in a nondegenerate situation. For the main mode of oscillations, a scenario was given for bifurcation of the family of all families of periodic solutions arising with generation of the MCCS cycles. Consideration was given to stability of cycles, and the problem of their stabilization was solved.

Basic oscillation mode in the coupled-subsystems model

Consideration was given to the model obeying a system of ordinary differential equations where the subsystems are systems of autonomous ordinary differential equations. If the coupling parameter ɛ = 0, then the model falls apart into decoupled subsystems. For a model consisting of coupled subsystems, considered was the main mode for which the problems of oscillations, bifurcation, and stability were solved, and the results obtained before for the case of two second-order subsystems were generalized.

Oscillations and stability in quasiautonomous system. II. Critical point of the one-parameter family of periodic motions

Consideration was given to the single-frequency oscillations of a periodic system allied to the nonlinear autonomous system. The publications of the present author demonstrated that the period on the family of oscillations of the autonomous system usually depends only on a single parameter. At that, the points of the family are divided into the ordinary (the derivative with respect to the period in parameter is other than zero) and critical (this derivative vanishes) points. Origination of oscillations at the critical point was studied. It was established that at least two resonance oscillations are generated. The first part of the paper considered the ordinary point.

Stabilizing the oscillations of an autonomous system

For a dynamical system that admits a family of oscillations, we propose a small smooth autonomous control that corrects the model itself and at the same time stabilizes the oscillations of the controlled system. We consider a separate system, a set of dynamical systems, and a dynamical model containing weakly coupled subsystem (MCCS). For the MCCS, we give a solution of the stabilization problem for oscillations of the system itself. We use an idea that goes back to Pontryagin’s work on the limit cycle for a system close to Hamiltonian.

Oscillation family in weakly coupled identical systems

Consideration was given to an autonomous model with weakly coupled identical subsystems. Existence of a family of periodic solutions which is similar to the family in a subsystem was established. A scenario of bifurcations of the characteristic exponents was given, and the stabilization problem was solved. An example was given.

Oscillations, stability and stabilization in the model containing coupled subsystems with cycles

Consideration was given to the model containing coupled subsystems (MCCS). With no connection between the subsystems, the MCCS falls down into independent subsystems, that is, systems of autonomous ordinary differential equations. MCCS with cycles in the subsystems were examined. For the autonomous MCCS, conditions were established for cycles and their stability, and a cycle stabilization by the control explicitly independent of time was proposed. For the periodic MCCS, generation of isolated periodic solutions was proved, and the problem of their stability and stabilization was solved.

On models structurally stable in periodic motion

The results obtained for a class of models that are structurally stable in periodic motion were presented, and, in particular, the notion of structural stability was introduced. Feasibility of such structurally stable model was proved. For a model featuring space-time symmetry, the sufficient structural stability conditions were established, and examples were presented.

Stabilization of oscillations from a monoparametric family of the autonomous system

The problem of oscillation stabilization is solved for the autonomous system in the case when the oscillations form a monoparametric family. For each possible point type of the family the explicit formulae that solve the problem are obtained. It follows from those formulae that the dissipation is a necessary condition of the stabilization. The obtained sufficient conditions of the stabilizations coinside with the appropriate necessary conditions to within the sign of unstrict inequality.

On the Model Containing Coupled Subsystems

The model containing coupled subsystems (MCCS), which decouples into indepen- dent subsystems as the coupling vanishes, is considered. The subsystems of the MCCS are described by autonomous systems of ordinary differential equations. Each subsystem is sup- posed to admit a single parameter family of periodic motions. The general characteristics of the MCCS are presented. Some problems concerning the MCCS are stated, i.e. the problems of oscillations existence, bifurcations, stability, stabilization, and resonance. The classification of MCCS is proposed, the currently investigated classes of MCCS are announced. One class of MCCS is studied in detail, the corresponding results are presented. In this paper we study the model containing coupled subsystems. This model is described by a system of ordinary differential equations (ODE), where subsystems are autonomous ODEs. The coupling is characterized by a numeric parameter e, which can be either scalar or vector. The subsystems become independent as e = 0. If e is vector then the MCCS can be of hierarchical structure, where subsystems are coupled on various levels. In the general case subsystems are either linear or nonlinear, of arbitrary order, and of various nature. The N-planet problem serves as an example of the MCCS with single-leveled subsystems. The following are some more examples of the MCCS: multi-link pendulum, chain of spring oscillators, Sun-planets-satellites system (a system of two levels), translationally and rota- tionally moving space vehicles, in particular, artificial Earth satellites, robot systems with cross-coupling, model of DNA oscillations, coupled neurons, wind turbine (electromechan- ical system), mechatronic systems, etc.