Stability Regions of Nonlinear Dynamical Systems

A topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented. The stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points (and/or closed orbits) on the stability boundary. Several necessary and sufficient conditions are derived to determine whether a given equilibrium point (or closed orbit) is on the stability boundary. A method for finding the stability region on the basis of these results is proposed. The method, when feasible, will find the exact stability region, rather than a subset of it as in the Lyapunov theory approach. Several examples are given to illustrate the theoretical prediction. >

Stability regions of nonlinear dynamical systems may suffer drastic changes as a consequence of parameter variation. These changes are triggered by local or global bifurcations of the vector field. In this chapter, these changes are studied for two types of local bifurcations on the stability boundary: saddle-node bifurcations and Hopf bifurcations on the stability boundary. Local and global characterizations of the stability boundary (the topological boundary of stability region) will be developed at the bifurcation points and the behavior (changes) of stability boundaries and stability regions at these bifurcations will be studied.

In recent developments of direct methods for power system transient stability analysis, the task of finding the controlling unstable equilibrium point (u.e.p.) of the original system is performed on an artificial, dimension-reduction system for easily computing the controlling u.e.p. It has been shown that if the stable and unstable manifolds of a family of parametrized systems, which relates the original power system and an artificial, dimension-reduction system, satisfy the transversality condition as the parameter varies, then the artificial, dimension-reduction system contains the same set of u.e.p.'s on the stability boundary. In this paper, instead of direct verification of the transversality conditions, we show that under small mechanical power injections, the original power system model and the artificial, dimension-reduction system contain the same unstable equilibrium points on the stability boundary for one machine and two machine power system models.

A complete characterization of the boundary of the stability region of nonlinear autonomous dynamical systems is developed admitting the existence of a particular type of non-hyperbolic equilibrium point on the stability boundary, the supercritical Hopf equilibrium points. Under condition of transversality, it is shown that the stability boundary is comprised of all stable manifolds of the hyperbolic equilibrium points lying on the stability boundary union with the center-stable and\or center manifolds of the type-k, k ≥ 1, supercritical Hopf equilibrium points on the stability boundary.

The stability of nonlinear dynamical system of relative rotation is studied. Firstly, the dynamics equation of relative rotation autonomous nonlinear dynamical system with commonly damped force and forced excitation is deduced. Secondly, the stability of relative rotation nonlinear dynamical system is studied. For the nonlinear dynamical system, it is proved that the closed orbit bifurcation can occur under some conditions. Finally, The approximate solution of the equation under forced excitation is obtained by the method of multiple scales.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

On observer-based fault detection for nonlinear systems

In this paper finite-time stabilization of nonlinear affine systems, is considered. First we provide a formula for a finite-time stabilizing state feedback control law, for the last class of system. Then we extend the result to the case of a norm-bounded uncertain system and a polytopic uncertain system stabilization are given. Some numerical examples are given to illustrate the main result.

This paper deals with some improvements of the Waveform Relaxation method for large power systems transient stability analysis. In this context, the classical Waveform Relaxation iterative method is usually not efficient due to its slow convergence. The convergence speed depends on a lot of parameters such as the model, the disturbance nature, the number of subdomains created by the decomposition, the initialization, the time domain length, etc. In order to make the method competitive with usual and mastered sequential methods, some important points are here investigated such as an initialization of nonlinear integrations with linear solutions and a preconditioning technique to deal with a large number of subsystems. These innovative methods can bring interesting speedups for large power systems as illustrated in the presented numerical experiments on a large and realistic European power system.

This paper proposes a decoupling based direct method to analyze the post-contingency transient stability for a general multi-machine power system. A linear decoupling transformation is used to construct the same number of independent single machine infinite bus (SMIB) power systems as oscillation modes. Each SMIB system carries stability information regarding one oscillation mode of the original system at the equilibrium. Then, the transient energy function method is applied on all decoupled SMIB systems to calculate stability margins, the smallest of which indicates the stability margin of the original system. Case studies on an IEEE 9-bus power system and a WECC 179-bus power system demonstrate the validity of the proposed method.

In this paper, we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space. Through introducing two new concepts-the Lip constant of a nonlinear operator and strongly equivalent metrics of a norm-we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics, or if and only if the Lip constant of the nonlinear operator is less than one. In the latter case, we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the Lip constant. Based on the obtained results, we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the Lip constant and strongly equivalent metrics. The obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new, useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems.

The paper considers the issues of control and stabilization of nonlinear dynamic systems described by a set of differential and difference equations, the latter of which contain a control vector. The states of these systems have both continuous and discrete components, so such systems are called continuous-discrete or hybrid. Necessary and sufficient features of controllability of nonlinear hybrid systems with a constant discretization step are established, which imply a transition from these systems to equivalent, in the natural sense, nonlinear discrete dynamic systems. A transformation is presented that allows reducing a linear discrete system to the canonical Brunovsky form and constructing a stabilizing control on its basis for the corresponding continuous-discrete system with scalar control. An algorithm for reducing a first approximation system of a nonlinear discrete system with scalar control to the canonical Brunovsky form and an algorithm for constructing a stabilizing control for nonlinear hybrid systems with scalar control are developed and illustrated with examples. Sufficient signs of stabilization of nonlinear hybrid systems are presented both without and with the feedback controller.

In this chapter we investigate the dynamics of classical nonlinear Hamiltonian systems, which—a priori—are examples of continuous dynamical systems. As in the discrete case (see examples in Chap. 2), we are interested in the classification of their dynamics. After a short review of the basic concepts of Hamiltonian mechanics, we define integrability (and therewith regular motion) in Sect. 3.4. The non-integrability property is then discussed in Sect. 3.5. The addition of small non-integrable parts to the Hamiltonian function (Sects. 3.6.1 and 3.7) leads us to the formal theory of canonical perturbations, which turns out to be a highly valuable technique for the treatment of systems with one degree of freedom and shows profound difficulties when applied to realistic systems with more degrees of freedom. We will interpret these problems as the seeds of chaotic motion in general. A key result for the understanding of the transition from regular to chaotic motion is the KAM theorem (Sect. 3.7.4), which assures the stability in nonlinear systems that are not integrable but behave approximately like them. Within the framework of the surface of section technique, chaotic motion is discussed from a phenomenological point of view in Sect. 3.8. More quantitative measures of local and global chaos are finally presented in Sect. 3.9.

Modeling of Microblogging Social Networks: Dynamical System vs. Random Dynamical System

The 2011 issue of Nonlinear Dynamics presents articles on stability of non-linear dynamic structures and systems. Alijani and co-researchers investigate geometrically nonlinear vibrations of rectangular plates with Functionally Graded Materials (FGM) in thermal environments via a multi-modal energy approach. Demeio and co-researchers investigate the primary non-linear resonance response of a one-dimensional continuous system, which can be regarded as a model for semiinfinite cables resting on an elastic substrate reacting in compression only, and subjected, to a constant distributed load and to a small harmonic displacement applied to the finite boundary. Stoykov and Ribeiro investigate the geometrically non-linear periodic vibrations of beams with rectangular cross section under harmonic forces using a p-version finite element method. Krylov and co-researchers investigate the feasibility of two-directional switching of an initially curved or prebuckled electrostatically actuated micro-beams using a single electrode fabricated from the same structural layer.