The theory of parametric oscillations

The correctness of the existing definitions of the parametric oscillations of linear and non-linear systems is discussed. The possibility of an erroneous choice of the mathematical parametric model instead of the autooscillatory model, connected with the existence in such systems of the same periodic solutions, is pointed out. Some non-local properties of parametric oscillations in Hamiltonian systems are established. It is shown, in particular, that stability regions are convex with respect to the frequency of the parametric excitation (i.e., all the points between the boundaries of neighbouring instability regions correspond to stable solutions). At the critical frequencies of parametric resonance the well-known Rayleigh and Zhuravlev theorems on the behaviour of the frequencies of natural oscillations when the stiffness and inertia changes are generalized. Some additional assertions on the limits of the first instability region for the Hill vector equations are established.     

Self-excited oscillations of a third-order system

A third-order self-excited oscillatory system in the neighbourhood of a stable local integral manifold is investigated. A periodic manifold and a corresponding system in amplitude-phase variables are constructed with approximately the required accuracy. Using the procedure of separation of variables (averaging) the conditions for the existence, uniqueness and stability of self-excited oscillation modes are established, and critical cases of the degeneracy of these conditions are considered. A thermomechanical model of the self-excitation of oscillations inherent in gas-dynamic systems with a heat source is taken as an example. The bifurcation pattern of self-excited oscillations in the space of the governing parameters of the system is investigated in the second approximation.     

On the boundaries of the parametric resonance domain

The problem of stability for a system of linear differential equations with coefficients which are periodic in time and depend on the parameters is considered. The singularities of the general position arising at the boundaries of the stability and instability (parametric resonance) domains in the case of two and three parameters are listed. A constructive approach is proposed which enables one, in the first approximation, to determine the stability domain in the neighbourhood of a point of the boundary (regular or singular) from the information at this point. This approach enables one to eliminate a tedious numerical analysis of the stability region in the neighbourhood of the boundary point and can be employed to construct the boundaries of parametric resonance domains. As an example, the problem of the stability of the oscillations of an articulated pipe conveying fluid with a pulsating velocity is considered. In the space of three parameters (the average fluid velocity and the amplitude and frequency of pulsations) a singularity of the boundary of the stability domain of the “dihedral angle” type is obtained and the tangential cone to the stability domain is calculated.     

Mathematical modeling of time fractional reaction–diffusion systems

We study a fractional reaction–diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction–diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

Dynamical stability of a viscoelastic bar

In the paper the dynamic stability of a simply supported viscoelastic bar of the Zener material model is investigated. The bar is subjected to a parametric excitation of a periodic nature. The physical model of the system is of the continuous type. However, the proposed approach yields a transformation of the mathematical model from the partial differential equation to an ordinary one. Such a transformation is made possible by the use of an approximation function for the bending line of the bar. This way, the system is governed by a homogeneous, ordinary differential equation of the third order with periodic coefficients. For stability studies the Floquet theory is applied. A sensitivity analysis of a parametric periodic system is discussed, i.e. the influence of stiffness and damping coefficient of the Zener model on stability of the differential equation that describes the vibration of the viscoelastic bar. Furthermore the stabilization process of an unstable parametric system was realized. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Periodic motions of a reversible second-order mechanical system: Application to the Sitnikov problem

A theory of the symmetric periodic motions (SPMs) of a reversible second-order system is presented which covers both oscillations and rotations. The structural stability property of the generating autonomous reversible system, which lies in the fact that the presence or absence of SPMs in a perturbed system is independent of the actual form of the “reversible” perturbations, is established. Both the case of the generation of SPMs from the family of SPMs of the generating system and birth cycle from the equilibrium state are investigated. Criteria of Lyapunov stability in a non-degenerate situation are obtained for the SPMs which are generated (in case of small values of the parameter). A method is proposed for constructing and investigating the Lyapunov stability of all the SPMs. The conditions for the existence of a cycle (symmetric and asymmetric) in the neighbourhood of a support “almost” resonance SPM are established for all cases of resonances. The theoretical results are applied to a study of the motion of a particle along a straight line which passes through the centre of mass of the system perpendicular to the plane of the identical attracting and simultaneously radiating main bodies (an extension of the Sitnikov problem) in the photogravitational version of the three-body problem. The circular problem is analysed and two different series of families of SPMs are found in the weakly elliptic problem. The instability of the equilibrium state is proved in the case of parametric resonance and the stability (and instability) domains are distinguished for arbitrary values of the eccentricity. All the SPMs with a period of 2π are constructed and the property of Lyapunov stability is investigated for these motions.     

Stability,bifurcation and global existence of a Hopf-bifurcating periodic solution for a class of three-neuron delayed network models

In this paper a system of three delay differential equations representing a Hopfield type general model for three neurons with two-way (bidirectional) time delayed connections between the neurons and time delayed self-connection from each neuron to itself is studied. Delay independent and delay dependent sufficient conditions for linear stability, instability and the occurrence of a Hopf bifurcation about the trivial equilibrium are addressed. The partition of the resulting parametric space into regions of stability, instability, and Hopf bifurcation in the absence of self-connection is realized. To extend the local Hopf branches for large delay values a particular bidirectional delayed tri-neuron model without self-connection is investigated. Sufficient conditions for global existence of multiple non-constant periodic solutions are obtained for such a model using the global Hopf-bifurcation theorem for functional differential equations due to J. Wu and the Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney, and following the approach developed by Wei and Li.     

Coherent Swing Instability of Power Grids

We interpret and explain a phenomenon in short-term swing dynamics of multi-machine power grids that we term the Coherent Swing Instability (CSI). This is an undesirable and emergent phenomenon of synchronous machines in a power grid, in which most of the machines in a sub-grid coherently lose synchronism with the rest of the grid after being subjected to a finite disturbance. We develop a minimal mathematical model of CSI for synchronous machines that are strongly coupled in a loop transmission network and weakly connected to the infinite bus. This model provides a dynamical origin of CSI: it is related to the escape from a potential well, or, more precisely, to exit across a separatrix in the dynamical system for the amplitude of the weak nonlinear mode that governs the collective motion of the machines. The linear oscillations between strongly coupled machines then act as perturbations on the nonlinear mode. Thus we reveal how the three different mode oscillations??local plant, inter-machine, and inter-area modes??interact to destabilize a power grid. Furthermore, we present a phenomenon of short-term swing dynamics in the New England (NE) 39-bus test system, which is a well-known benchmark model for power grid stability studies. Using a partial linearization of the nonlinear swing equations and the proper orthonormal decomposition, we show that CSI occurs in the NE test system, because it is a dynamical system with a nonlinear mode that is weak relative to the linear oscillatory modes.     

On the instability of a periodic system in the case of internal resonance

The stability of periodic motion is studied in the critical case of n pairs of purely imaginary characteristic indices. It is shown that in the case of resonance, when the ratio of the modulus of one of the characteristic indices to the frequency of the unperturbed motion is an integer, instability usually occurs. The results obtained are used to study the free oscillations of an autonomous quasilinear system when the Andronov-Witt criterion /1/ cannot be used. The instability of free oscillations of the Froude pendulum at the bifurcation point is proved.     

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.     

Linearized stability and instability of nonconstant periodic solutions of Lagrangian equations

This paper is motivated by the stability problem of nonconstant periodic solutions of time‐periodic Lagrangian equations, like the swing and the elliptic Sitnikov problem. As a beginning step, we will study the linearized stability and instability of nonconstant periodic solutions that are bifurcated from those of autonomous Lagrangian equations. Applying the theory for Hill equations, we will establish a criterion for linearized stability. The criterion shows that the linearized stability depends on the temporal frequencies of the perturbed systems in a delicate way.     

Iterative algorithm of determining the stability of an equation of oscillations with damping

A problem of studying parametric oscillations with damping is reduced to a spectral problem for a linear bundle of operators in a Hilbert space. This spectral problem is solved by an efficient algorithm. The boundaries of the first stability domain for different values of a damping factor and a special form of a periodic function in the equation are calculated.     

Effect of time delay on a harvested predator-prey model

The predator-prey systems with harvesting have received a great deal of attentions for last few decades. Incorporating discrete time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of harvesting and discrete time delay on the dynamics of a predator-prey model. A comparative analysis is provided for stability behaviour in absence as well as in presence of time delay. The length of discrete time delay to preserve stability of the model system is obtained. Existence of Hopf-bifurcating small amplitude periodic solutions is derived by taking discrete time delay as a bifurcation parameter.     

Bifurcation analysis of a delayed mathematical model for tumor growth

In this study, we present a modified mathematical model of tumor growth by introducing discrete time delay in interaction terms. The model describes the interaction between tumor cells, healthy tissue cells (host cells) and immune effector cells. The goal of this study is to obtain a better compatibility with reality for which we introduced the discrete time delay in the interaction between tumor cells and host cells. We investigate the local stability of the non-negative equilibria and the existence of Hopf-bifurcation by considering the discrete time delay as a bifurcation parameter. We estimate the length of delay to preserve the stability of bifurcating periodic solutions, which gives an idea about the mode of action for controlling oscillations in the tumor growth. Numerical simulations of the model confirm the analytical findings.     

Stability regions of linear canonical systems with periodic coefficients

A new definition of the index of the stability region of a canonical system of linear differential equations with periodic coefficients is proposed. A simple proof of the Gel'fand-Lidskii theorem [1] on the structure of stability regions is given and a theorem on the directional convexity of such regions is proved. It follows from this theorem, in particular, that stability regions of parametric oscillations in a system with a sign-definite Hamiltonian are convex with respect to the frequency of parametric perturbation.     

Forced oscillations of a non-linear system with a repulsive positional force

Non-linear systems with one degree of freedom, in which the positional force is directed away from the equilibrium position of the system, are considered. The existence of forced periodic oscillations, their Lyapunov stability, and the behaviour of amplitude-frequency characteristics are investigated. It is shown that stable periodic oscillations are possible in the case when the positional force has non-monotonic properties. Forced oscillations of a pendulum with respect to the upper equilibrium position are considered as an example.     

The stability of the parametric oscillations of second-order non-linear systems

The parametric oscillations of strongly non-linear systems with one degree of freedom are considered using a more general definition of these oscillations than the generally accepted definition. Stability criteria, that are verifiable using the signs of the derivatives of the amplitude-frequency characteristics, are found for the two families of periodic solutions corresponding to the fundamental parametric resonance. A condition is indicated under which the latter are monotonic and, as a result, one of the families is stable and the other is unstable. It is shown that, in a system with a concave non-monotonic elastic characteristic, the stable family loses stability for fairly large amplitudes and this effect is not revealed by the well-known analytical methods of non-linear mechanics.     

On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.     

Spatiotemporal Dynamic Analysis in a Time-space Discrete Brusselator Mode

In this paper, we study the spatiotemporal patterns of a Brusselator model with discrete time-space by using the coupled mapping lattice (CML) model. The existence and stability conditions of the equilibrium point are obtained by using linear stability analysis. Then, applying the center manifold reduction theorem and the bifurcation theory, the parametric conditions of the flip and the Neimark-Sacker bifurcation are described respectively. Under space diffusion, the model admits the Turing instability at stable homogeneous solutions under some certain conditions. Two nonlinear mechanisms, including flip-Turing instability and Neimark-Sacker-Turing instability, are presented. Through numerical simulation, periodic windows, invariant circles, chaotic phenomenon and some interesting spatial patterns are found.