Suppressing chaos via Lyapunov–Krasovskii’s method

An algorithm for suppressing the chaotic oscillations in non-linear dynamical systems with singular Jacobian matrices is developed using a linear feedback control law based upon the Lyapunov–Krasovskii (LK) method. It appears that the LK method can serve effectively as a generalised method for the suppression of chaotic oscillations for a wide range of systems. Based on this method, the resulting conditions for undisturbed motions to be locally or globally stable are sufficient and conservative. The generalized Lorenz system and disturbed gyrostat equations are exemplified for the validation of the proposed feedback control rule.     

A qualitative analysis of the subharmonic oscillations of a parametrically excited flexible rod

A non-autonomous non-linear dynamical system with a small parameter that describes the parametric oscillations of a flexible rod with three static equilibrium positions is obtained. The generating equation of this model is a dynamical system in a plane with a separatrix loop. The qualitative analysis presented includes an investigation of the stability and bifurcation of subharmonic motions at resonance energy levels.     

Mode interactions and resonances of an elastic fluid-conveying tube

We consider the non-linear two-dimensional oscillations of a fluid conveying tube using dynamical bifurcation theory. The tube is clamped at the upper end, and at its free lower end a point mass is fixed. The tube is assumed to be slender and flexurally elastic, and its transversal motion is constrained by two symmetrically arranged springs. The flow rate of the incompressible fluid is used as a distinguished parameter in the problem. By determining the stability regions in parameter space, it is investigated whether Hopf and/or steady-state bifurcations may occur, as it was found for similar cases in previous works [1,3]. The non-linear behaviour close to the bifurcation points is analyzed. Of specific interest are low-order resonances. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Near-resonace transverse oscillations in an elastic layer. Steady-state solutions

The solutions of the equations of the non-linear evolution of transverse oscillations in a layer of an incompressible elastic medium under conditions close to resonance conditions are investigated qualitatively and using analytical methods. The oscillations are created by a small periodic motion of one of the boundaries in its plane, with a period that is close to the period of the natural oscillations of the layer. It is assumed that the medium can possess slight anisotropy and that the amplitude of the oscillations which arise is small. Previously obtained differential equations are used, which describe the slow evolution of the wave pattern of non-linear transverse waves. Two possible formulations of problems for these equations are considered. In the first formulation, it is determined what the external action must be in order that the non-linear evolution of oscillations or periodic oscillations occurs according to a (previously specified) desired law. In the second formulation it is assumed that the periodic motion of one of the boundaries is given. It is shown that a steady-state solution, that does not vary from period to period, can be represented by a continuous solution and also by a solution which contains discontinuities in the strain and velocity components. The mechanism of the overturn of a non-linear wave during its evolution and the formation of a discontinuity are qualitatively described.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

Exact solutions of the problem of the vibro-impact oscillations of a discrete system with two degrees of freedom

Non-smooth time transformations are used to investigate strongly non-linear periodic free oscillations of a vibro-impact system with two degrees of freedom. Allowance for the boundary conditions at collision times enables the singularities induced by these transformations to be eliminated. The smoothed equations of motion turn out to be linear. Investigation of the periodic solutions reveals vibro-impact states with one- an two-sided collisions, including localized states (only one of the masses experiences collisions with stopping devices), and their bifurcation structure.     

On vibrations in non-linear,forced, friction-excited systems

To accelerate the development of non-linear, friction-excited systems, i.e. automotive friction brakes, novel test procedures are being developed. These include the study of the system's forced vibrations. However, there is a lack of knowledge concerning feasible test procedures, the assessment of the gathered data, and the possible vibration phenomena. In this context, the contribution at hand discusses vibrations in non-linear, forced, friction-excited systems. In the pre-flutter regime proposed criteria for identification of parameter regions in which limit cycle oscillations can occur are challenged. In the post-flutter regime frequency spectra for different excitation conditions are studied. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The momentum of waves and their effect on the motion of lumped objects along one-dimensional elastic systems

Taking the example of the small longitudinal oscillations of a rod, it is shown that, in order to answer the question concerning wave momentum and its action on an obstacle, the problem of the wave motion in the medium has to be solved in a non-linear formulation. The variational formulation of problems in the dynamics of one-dimensional elastic systems with moving clampings and loads is improved taking account of non-linear factors. The equations of motion and the natural boundary conditions are obtained. The small longitudinal-transverse oscillations of a string and the motion of a bead sliding along it are considered.     

Variational setting of the problem on the oscillations of a fluid in a vessel with an elastic inclusion

The problem on the oscillations of an ideal incompressible fluid in a moving rectangular vessel is studied. One wall of the vessel contains an elastic inclusion. The problem involves two free boundaries—the free surface of the fluid and the surface of the elastic inclusion. It is suggested to solve this problem by using a functional whose variation leads to differential equations with nonlinear kinematic and dynamical conditions on the free boundaries. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

Arithmetic coding as a non-linear dynamical system

In order to perform source coding (data compression), we treat messages emitted by independent and identically distributed sources as imprecise measurements (symbolic sequence) of a chaotic, ergodic, Lebesgue measure preserving, non-linear dynamical system known as Generalized Luröth Series (GLS). GLS achieves Shannon’s entropy bound and turns out to be a generalization of arithmetic coding, a popular source coding algorithm, used in international compression standards such as JPEG2000 and H.264. We further generalize GLS to piecewise non-linear maps (Skewed-nGLS). We motivate the use of Skewed-nGLS as a framework for joint source coding and encryption.     

Semistability of switched dynamical systems,Part II: Non-linear system theory

In a companion paper [Q. Hui, W.M. Haddad, Semistability of switched dynamical systems. Part I: Linear system theory, Non-linear Anal. Hybrid Syst. 3 (3) (2009) 343–353] semistability and uniform semistability results for switched linear systems were developed. In this paper we develop semistability analysis results for non-linear switched systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. The main results of the paper involve sufficient conditions for semistability using multiple Lyapunov functions and integral-type inequalities.     

Bifurcations of traveling wave solutions in a coupled non-linear wave equation

By using the theory of planar dynamical systems to a coupled non-linear wave equation, the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Modeling chemical reactions by forced limit-cycle oscillator: synchronization phenomena and transition to chaos

The lattice limit-cycle (LLC) model is introduced as a minimal mean-field scheme which can model reactive dynamics on lattices (low dimensional supports) producing non-linear limit cycle oscillations. Under the influence of an external periodic force the dynamics of the LLC may be drastically modified. Synchronization phenomena, bifurcations and transitions to chaos are observed as a function of the strength of the force. Taking advantage of the drastic change on the dynamics due to the periodic forcing, it is possible to modify the output/product or the production rate of a chemical reaction at will, simply by applying a periodic force to it, without the need to change the support properties or the experimental conditions.     

Change-of-bases abstractions for non-linear hybrid systems

We present abstraction techniques that transform a given non-linear dynamical system into a linear system, or more generally, an algebraic system described by polynomials of bounded degree, so that invariant properties of the resulting abstraction can be used to infer invariants for the original system. The abstraction techniques rely on a change-of-bases transformation that associates each state variable of the abstract system with a function involving the state variables of the original system. We present conditions under which a given change-of-bases transformation for a non-linear system can define an abstraction. Furthermore, the techniques developed here apply to continuous systems defined by Ordinary Differential Equations (ODEs), discrete systems defined by transition systems and hybrid systems that combine continuous as well as discrete subsystems.The techniques presented here allow us to discover, given a non-linear system, if a change-of-bases transformation involving degree-bounded polynomials yielding an algebraic abstraction exists. If so, our technique yields the resulting abstract system, as well. Our techniques enable the use of analysis techniques for linear systems to infer invariants for non-linear systems. We present preliminary evidence of the practical feasibility of our ideas using a prototype implementation.     

Relaxation oscillations in spruce–budworm interactions

We review and complement the study of the mathematical properties of the predator–prey dynamical system for spruce–budworm interactions studied in Ludwig et al. [D. Ludwig, D. Jones, C.S. Holling, Qualitative analysis of insect outbreak systems: the spruce budworm and forest, Journal of Animal Ecology 47 (1978), 315–332]. We use the singular perturbation method to identify parameter regimes which permit relaxation oscillations. The leading order contribution to the period of these oscillations is computed by means of this method and compared with the outcome of direct numerical simulations.     

Inverse problem of the calculus of variation,its application to integration of an ordinary non-linear second order differential equation

Necessary and sufficient conditions for the existence of a functional for a strongly non-linear differential equation of the second order in ordinary derivatives are established. The Cauchy problem is solved. The integration method is illustrated by solving problems of theoretical mechanics, relativity theory, quasilinear oscillations and strength of materials.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 3, pp. 413–418, March, 1990.     

Decomposition and suboptimal control in dynamical systems

A non-linear controllable dynamical system described by Lagrange equations is considered. The problem of constructing bounded controlling forces which steer the system to a given state in a finite time is investigated. Sufficient conditions are indicated for the problem to be solvable. Under these conditions, the initial system splits into subsystems, each with the degree of freedom. On the basis of this decomposition, using a game-theoretic approach, a feedback control law is proposed which solves the problem posed above and is nearly time-optimal. It is shown that the control must be constructed with proper allowance for the maximum values of the non-linear terms and perturbations in the equations of motion. The perturbations may be ignored only if the ratio of the maximum level of the perturbation to that of the control does not exceed the “golden section”.     

Vibration suppression of non-linear system via non-linear absorber

In this paper, the coupled non-linear differential equations of the non-linear dynamical two-degree-of-freedom vibrating system including quadratic and cubic non-linearities are studied. The system consists of the main system and the absorber. The absorber is used to control the main system vibrations when subjected to multi-external excitation forces at simultaneous primary and internal resonance. This system represents many applications in machine tools, ultrasonic cutting process, etc. The method of multiple scales perturbation technique (MSPT) is applied throughout to determine the solution up to third order approximations. The different resonance cases are reported and studied numerically. Stability is studied applying frequency response functions. The effects of different parameters of the system are studied numerically. Optimum working conditions for the absorber where obtained at internal resonance ratio 1:3. This means smaller mass for the absorber which solves the problem of space limitation. A comparison is made with the available published work.