Generalized quadratic embeddings for nonlinear dynamics using deep learning

Quadratic Dynamical Systems and Algebras

In this paper, we present a theoretical analysis of non linear quadratic systems using higher order cumulants (HOC). In the one hand, we develop the equations linking the second and third order cumulants with the impulse response of quadratic non linear systems. In the other hand, these relationships are used to develop an extension of linear algorithm based on third order cumulants to non linear algorithm for identification of quadratic systems. The proposed algorithm is tested using different quadratic models for various values of signal to noise ratio (SNR).

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.

In this paper, an effective method is proposed called the “Dynamic Routh’s stability Criterion (DRSC)”, which is developed using the Dynamic Pole Motion (DPM) approach (DPMA). This innovative technique extends the classical Routh’s stability criterion—traditionally limited to linear time-invariant (LTI) systems—to encompass a broader range of systems, including linear time-varying and nonlinear dynamic systems. By incorporating the behavior of pole trajectories over time, the proposed method offers a more comprehensive framework for analyzing system stability in more complex and realistic scenarios. DPM approach describes the notion of dynamic poles (DP) in a three dimensional ‘g-plane’, which is an extension to the two dimensional ‘s-plane’. The s-domain approach is based upon Laplace transform and has been used for linear time-invariant (LTI) system only. This novel g-plane framework is versatile enough to be applied to both linear and nonlinear dynamic systems. For the stability of dynamic systems, the DP must present in the left-hand side (LHS) of the g-plane. For nonlinear systems, the locations of DP are the function of system states; and these systems states are the function of initial conditions with amplitude and frequency (AnF) of input signals. Therefore, in nonlinear systems, stability is influenced not only by the initial conditions but also by the AnF of the input signals. For example, for given amplitude of the input signal, system may be unstable at low frequency; however, it may become stable at high frequency or vice-versa. These stability conditions are illustrated by several examples.

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.

Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach

New results about the existence of chaotic dynamics in the quadratic 3D systems are derived. These results are based on the method allowing studying dynamics of 3D system of autonomous quadratic differential equations with the help of reduction of this system to the special 2D quadratic system of differential equations.

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.

The stability of stochastic dynamic systems have widespread application prospects and greater theoretical significance. Its theory are mainly applied to engineering control, communication equipment, military technology, biology, finance, etc. In this article, we consider the stability of a Class of nonlinear impulsive neutral stochastic differential dynamic systems. A new set of conditions proving the mean square stability of this nonlinear impulsive neutral stochastic dynamic systems are derived by means of the Banach fixed point theorem. Some well-known results are improved and generalized.

SUMMARYWe consider the pole assignment problems for time‐invariant linear and quadratic control systems, with time‐delay in the control. Closed‐loop eigenvectors in X = [x1, x2, ⋯ ] are chosen from their corresponding invariant subspaces, possibly optimizing some robustness measure, and explicit expressions for the feedback matrices are given in terms of X. Condition of the problems is also investigated. Our approach extends the well‐known Kautsky, Nichols, and Van Dooren algorithm. Consequently, the results are similar to those for systems without time‐delay, except for the presence of the ‘secondary’ eigenvalues and the condition of the problems. Simple illustrative numerical examples are given. Copyright © 2011 John Wiley & Sons, Ltd.

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.

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

The authors establish a generalized frequency-domain criterion for checking families of polynomials for root confinement in open subsets of the complex plane. The authors show how this criterion reduces to checking certain curves in the complex plane for zero confinement. Moreover, in some special cases, it further reduces to some complex functions with pointwise phase differences that are always less than pi in magnitude. Most of the currently available results on the robust stability of linear systems with parametric uncertainties can be viewed within the unifying frequency-domain framework presented. The framework encapsulates not just finite-dimensional systems, but any linear-time-invariant (LTI) system that can be characterized by transfer functions of a single variable. It also covers robust stability of LTI systems under passive feedback. >

In this note, we deal with the stability and stabilization problems for nonlinear quadratic discrete-time periodic systems. By using the quadratic Lyapunov function, and a so called periodic invariant set, delay-independent sufficient conditions for local stability and local stabilization for nonlinear quadratic discrete-time periodic systems are derived in terms of linear matrix inequalities (LMIs). Based on these sufficient conditions, iterative linear matrix inequality algorithms are proposed for maximizing the stability regions of the systems. Finally, two examples are given to illustrate the effectiveness of the methods presented in this paper.

Stability of dynamical systems is a central topic with applications in widespread areas such as economy, biology, physics and mechanical engineering. The dynamics of nonlinear systems may completely change due to perturbations forcing the solution to jump from a safe state into another, possibly dangerous, attractor. Such phenomena cannot be traced by the widespread local stability and resilience measures, based on linearizations, accounting only for arbitrary small perturbations. Using numerical estimates of the size and shape of the basin of attraction, as well as the systems returntime to the attractor after given a perturbation, we construct simple nonlocal stability and resilience measures that record a systems ability to tackle both large and small perturbations. We demonstrate our approach on the Solow–Swan model of economic growth, an electro-mechanical system, a stage-structured population model as well as on a high-dimensional system, and conclude that the suggested measures detect dynamic behavior, crucial for a systems stability and resilience, which can be completely missed by local measures. The presented measures are also easy to implement on a standard laptop computer. We believe that our approach will constitute an important step toward filling a current gap in the literature by putting forward and explaining simple ideas and methods, and by delivering explicit constructions of several promising nonlocal stability and resilience measures.