An approach to solving nonlinear algebraic systems. 2

New methods of solving nonlinear algebraic systems in two variables are suggested, which make it possible to find all zero-dimensional roots without knowing initial approximations. The first method reduces the solution of nonlinear algebraic systems to eigenvalue problems for a polynomial matrix pencil. The second method is based on the rank factorization of a two-parameter polynomial matrix, allowing, us to compute the GCD of a set of polynomials and all zero-dimensional roots of the GCD. Bibliography: 10 titles. Translated by V. N. Kublanovskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 202, 1992, pp. 71–96     

Designing torus-doubling solutions to discrete time systems by hybrid projective synchronization

Doubling of torus occurs in high dimensional nonlinear systems, which is related to a certain kind of typical second bifurcations. It is a nontrivial task to create a torus-doubling solution with desired dynamical properties based on the classical bifurcation theories. In this paper, dead-beat hybrid projective synchronization is employed to build a novel method for designing stable torus-doubling solutions into discrete time systems with proper properties to achieve the purpose of utilizing bifurcation solutions as well as avoiding the possible conflict of physical meaning of the created solution. Although anti-controls of bifurcation and chaos synchronizations are two different topics in nonlinear dynamics and control, the results imply that it is possible to develop some new interdisciplinary methods between chaos synchronization and anti-controls of bifurcations.     

Controlling based method for modelling chaotic dynamical systems from time series

A simple method is introduced for modelling chaotic dynamical systems from the time series, based on the concept of controlling of chaos by constant bias. In this method, a modified system is constructed by including some constants (controlling constants) into the given (original) system. The system parameters and the controlling constants are determined by solving a set of implicit nonlinear simultaneous algebraic equations which is obtained from the relation connecting original and modified systems. The method is also extended to find the form of the evolution equation of the system itself. The important advantage of the method is that it needs only a minimal number of time series data and is applicable to dynamical systems of any dimension. It also works extremely well even in the presence of noise in the time series. The method is illustrated in some specific systems of both discrete and continuous cases.     

New Approximations and Tests of Linear Fluctuation-Response for Chaotic Nonlinear Forced-Dissipative Dynamical Systems

We develop and test two novel computational approaches for predicting the mean linear response of a chaotic dynamical system to small change in external forcing via the fluctuation–dissipation theorem. Unlike the earlier work in developing fluctuation–dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, the new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. The new methods take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian formula of the fluctuation–dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. A simple new low-dimensional chaotic nonlinear forced-dissipative model is used to study the response of both linear and nonlinear functions to small external forcing in a range of dynamical regimes with an adjustable degree of chaos. We demonstrate that the two new methods are remarkably superior to the classical fluctuation–dissipation formula with quasi-Gaussian approximation in weakly and moderately chaotic dynamical regimes, for both linear and nonlinear response functions. One straightforward algorithm gives excellent results for short-time response while the other algorithm, based on systematic rational approximation, improves the intermediate and long time response predictions.     

Polynomial Chaos for Analysing Periodic Processes of Differential Algebraic Equations with Random Parameters

Mathematical models of dynamical systems typically include technical parameters. Assuming an uncertainty, some parameters are replaced by random variables and the solution of the time–dependent system becomes a stochastic process. We consider forced oscillators, which are modelled by systems of differential algebraic equations. Consequently, periodic boundary conditions are imposed on the system. We apply the technique of the generalised polynomial chaos to resolve the stochastic model. Numerical simulations based on the electric circuit of a transistor amplifier are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An algebraic multifrontal preconditioner that exploits the low‐rank property

We present an algebraic structured preconditioner for the iterative solution of large sparse linear systems. The preconditioner is based on a multifrontal variant of sparse LU factorization used with nested dissection ordering. Multifrontal factorization amounts to a partial factorization of a sequence of logically dense frontal matrices, and the preconditioner is obtained if structured factorization is used instead. This latter exploits the presence of low numerical rank in some off‐diagonal blocks of the frontal matrices. An algebraic procedure is presented that allows to identify the hierarchy of the off‐diagonal blocks with low numerical rank based on the sparsity of the system matrix. This procedure is motivated by a model problem analysis, yet numerical experiments show that it is successful beyond the model problem scope. Further aspects relevant for the algebraic structured preconditioner are discussed and illustrated with numerical experiments. The preconditioner is also compared with other solvers, including the corresponding direct solver. Copyright © 2015 John Wiley & Sons, Ltd.     

Recurrence and symmetry of time series: Application to transition detection

The study of transitions in low dimensional, nonlinear dynamical systems is a complex problem for which there is not yet a simple, global numerical method able to detect chaos–chaos, chaos–periodic bifurcations and symmetry-breaking, symmetry-increasing bifurcations. We present here for the first time a general framework focusing on the symmetry concept of time series that at the same time reveals new kinds of recurrence. We propose several numerical tools based on the symmetry concept allowing both the qualification and quantification of different kinds of possible symmetry. By using several examples based on periodic symmetrical time series and on logistic and cubic maps, we show that it is possible with simple numerical tools to detect a large number of bifurcations of chaos–chaos, chaos–periodic, broken symmetry and increased symmetry types.     

Controlling chaotic behaviour for spin generator and Rossler dynamical systems with feedback control

Different methods are proposed to control chaotic behaviour of the Nuclear Spin Generator (NSG) and Rossler continuous dynamical systems. Linear and nonlinear feedback control techniques are used to suppress chaos. The stabilization of unstable fixed point or unstable periodic solution of chaotic behaviour is achieved. The controlled system is stable under some conditions on the parameters of the system. Stability of the controlled system is determined by the Routh–Hurwitz criterion and Lyapunov direct method. Numerical simulation results are included to show the control process.     

The rank of a sequence as an indicator of chaos in discrete nonlinear dynamical systems

An alternative technique for clocking the convergence of iterative chaotic maps is proposed in this paper. It is based on the concept of the Hankel rank of a solution of the discrete nonlinear dynamical system. Computation and visualization of pseudoranks in the space of system’s parameters and initial conditions provides the insight into the fractal nature of the dynamical attractor and reveals the stable, the unstable manifold and the convergence properties of the system. All these manifolds are produced by a simple and a straightforward computational rule and are intertwined in one figure. On the other hand, the computation of ranks of subsequences of solutions helps to identify and assess the sensitivity of the system to initial conditions and can be used as a simple and effective numerical tool for qualitative investigation of discrete iterative maps.     

Stability Analysis for Discrete Biological Models Using Algebraic Methods

This paper is concerned with stability analysis of biological networks modeled as discrete and finite dynamical systems. We show how to use algebraic methods based on quantifier elimination, real solution classification and discriminant varieties to detect steady states and to analyze their stability and bifurcations for discrete dynamical systems. For finite dynamical systems, methods based on Gr?bner bases and triangular sets are applied to detect steady states. The feasibility of our approach is demonstrated by the analysis of stability and bifurcations of several discrete biological models using implementations of algebraic methods.     

An Approach to Solving Inverse Eigenvalue Problems for Matrices

The paper considers different formulations of inverse eigenvalue problems for matrices whose entries either polynomially or rationally depend on unknown parameters. An approach to solving inverse problems together with numerical algorithms is suggested. The solution of inverse problems is reduced to the problem of finding the so-called discrete solutions of nonlinear algebraic systems. The corresponding systems are constructed using the method of traces, and their discrete roots are found by applying the algorithms for solving nonlinear algebraic systems in several variables previously suggested by the author. Bibliography: 30 titles.     

A theory for n-dimensional nonlinear dynamics on continuous vector fields

This paper systematically presents a theory for n-dimensional nonlinear dynamics on continuous vector fields. In this paper, a different view to look into the fundamental theory in dynamics is presented. The ideas presented herein are less formal and rigorous in an informal and lively manner. The ideas may give some inspirations in the field of nonlinear dynamics. The concepts of local and global flows are introduced to interpret the complexity of flows in nonlinear dynamic systems. Further, the global tangency and transversality of flows to the separatrix surface in nonlinear dynamical systems are discussed, and the corresponding necessary and sufficient conditions for such global tangency and transversality are presented. The ε-domains of flows in phase space are introduced from the first integral manifold surface. The domain of chaos in nonlinear dynamic systems is also defined, and such a domain is called a chaotic layer or band. The first integral quantity increment is introduced as an important quantity. Based on different reference surfaces, all possible expressions for the first integral quantity increment are given. The stability of equilibriums and periodic flows in nonlinear dynamical systems are discussed through the first integral quantity increment. Compared to the Lyapunov stability conditions, the weak stability conditions for equilibriums and periodic flows are developed. The criteria for resonances in the stochastic and resonant, chaotic layers are developed via the first integral quantity increment. To discuss the complexity of flows in nonlinear dynamical systems, the first integral manifold surface is used as a reference surface to develop the mapping structures of periodic and chaotic flows. The invariant set fragmentation caused by the grazing bifurcation is discussed. The global grazing bifurcation is a key to determine the global transversality to the separatrix. The local grazing bifurcation on the first integral manifold surface in a single domain without separatrix is a mechanism for the transition from one resonant periodic flow to another one. Such a transition may occur through chaos. The global grazing bifurcation on the separatrix surface may imply global chaos. The complexity of the global chaos is measured by invariant sets on the separatrix surface. The invariant set fragmentation of strange attractors on the separatrix surface is central to investigate the complexity of the global chaotic flows in nonlinear dynamical systems. Finally, the theory developed herein is applied to perturbed nonlinear Hamiltonian systems as an example. The global tangency and tranversality of the perturbed Hamiltonian are presented. The first integral quantity increment (or energy increment) for 2n-dimensional perturbed nonlinear Hamiltonian systems is developed. Such an energy increment is used to develop the iterative mapping relation for chaos and periodic motions in nonlinear Hamiltonian systems. Especially, the first integral quantity increment (or energy increment) for two-dimensional perturbed nonlinear Hamiltonian systems is derived, and from the energy increment, the Melnikov function is obtained under a certain perturbation approximation. Because of applying the perturbation approximation, the Melnikov function only can be used for a rough estimate of the energy increment. Such a function cannot be used to determine the global tangency and transversality to the separatrix surface. The global tangency and transversality to the separatrix surface only can be determined by the corresponding necessary and sufficient conditions rather than the first integral quantity increment. Using the first integral quantity increment, limit cycles in two-dimensional nonlinear systems is discussed briefly. The first integral quantity of any n-dimensional nonlinear dynamical system is very crucial to investigate the corresponding nonlinear dynamics. The theory presented in this paper needs to be further developed and to be treated more rigorously in mathematics.     

Resonance,stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback

This paper examines dynamical behavior of a nonlinear oscillator with a symmetric potential that models a quarter-car forced by the road profile. The primary, superharmonic and subharmonic resonances of a harmonically excited nonlinear quarter-car model with linear time delayed active control are investigated. The method of multiple scales is utilized to obtain first order approximation of response. We focus on the influence of delay in the system. This naturally gives rise to a delay deferential equation (DDE) model of the system. The effect of time delay and feedback gains of the steady state responses of primary, superharmonic and subharmonic resonances are investigated. By means of Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. We describe a method to identify the critical forcing function and time delay above which the system becomes unstable. It is found that proper selection of time-delay shows optimum dynamical behavior. The accuracy of the method is obtained from the fractal basin boundaries.     

基于交互耦合网络的项目组合决策模型研究

随着项目活动进入“大尺度”时代,复杂性成为现代化项目组合管理中的突出问题。在项目组合决策系统复杂性分析基础上,提出了交互耦合网络视角下的项目组合决策系统表征方法;借鉴非线性动力学建模方法构建项目组合决策系统复杂动力网络模型,结合模型的稳定解和稳定条件将项目组合决策系统划分为竞争型、共生型、强依存型和弱依存型,并通过数值仿真方法对系统的稳定域、分岔和混沌进行分析。研究表明,项目组合决策系统的复杂性和稳定性依赖于系统内交互关系作用,改善协作关系,避免过分竞争,以系统整体为先优化配置有利于项目组合目标实现。     

Stochastic optimal control and algorithm of the trajectory of horizontal wells

This paper presents a nonlinear, multi-phase and stochastic dynamical system according to engineering background. We show that the stochastic dynamical system exists a unique solution for every initial state. A stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. This model can be converted into a parametric nonlinear stochastic programming by integrating the state equation. It is discussed here that the local optimal solution depends in a continuous way on the parameters. A revised Hooke–Jeeves algorithm based on this property has been developed. Computer simulation is used for this paper, and the numerical results illustrate the validity and efficiency of the algorithm.     

A tau method for nonlinear dynamical systems

In this work we present a new Tau method for the solution of nonlinear systems of differential equations which are linear in the derivative of highest order and polynomial in the remaining. We avoid the linearization of the problem by associating to it a nonlinear algebraic system and combine a forward substitution with the Tau method. We develop an adaptive step by step version of this alternative nonlinear tau method and we apply it to several nonlinear dynamical systems.     

An adaptive Newton-method based on a dynamical systems approach

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.     

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced,time-delayed,hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.     

求全局最优解的新算法

利用局部极大值点与动力系统的稳定奇点的对应性，计算代数方程的根、无约束极大值点、有约束极大值点、非线性规划解、及最小二乘解．我们采用了常微分方程数值解的Euler算法及网格初始点的循序迭代算法，并以具体的例子和程序说明创立的方法具有通用性，同时考虑了一些存在的问题以便在理论和算法上作进一步的改进。     

Dynamical analysis of turbulence in fusion plasmas and nonlinear waves

Turbulence is one of the key problems of classical physics, and it has been the object of intense research in the last decades in a large spectrum of problems involving fluids, plasmas, and waves. In order to review some advances in theoretical and experimental investigations on turbulence a mini-symposium on this subject was organized in the Dynamics Days South America 2010 Conference. The main goal of this mini-symposium was to present recent developments in both fundamental aspects and dynamical analysis of turbulence in nonlinear waves and fusion plasmas. In this paper we present a summary of the works presented at this mini-symposium. Among the questions to be addressed were the onset and control of turbulence and spatio-temporal chaos.