Modified slow-fast analysis method for slow-fast dynamical systems with two scales in frequency domain

A modified slow-fast analysis method is presented for the periodically excited non-autonomous dynamical system with an order gap between the exciting frequency and the natural frequency. By regarding the exciting term as a slow-varying parameter, a generalized autonomous fast subsystem can be defined, the equilibrium branches as well as the bifurcations of which can be employed to account for the mechanism of the bursting oscillations by combining the transformed phase portrait introduced. As an example, a typical periodically excited Hartley model is used to demonstrate the validness of the method, in which the exciting frequency is far less than the natural frequency. The equilibrium branches and their bifurcations of the fast subsystem with the variation of the slow-varying parameter are presented. Bursting oscillations for two typical cases are considered, which reveals that, fold bifurcation may cause the the trajectory to jump between different equilibrium branches, while Hopf bifurcation may cause the trajectory to oscillate around the stable limit cycle.     

Error estimates for averaging method in multifrequency systems with constant delay

We prove new error estimates for the averaging method for multifrequency systems of higher approximations with delay in slow and fast variables. The main assumption is imposed on the vector of frequencies and its derivatives along the solution of the averaged system of zero approximation. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 233–243, April–June, 2006.     

On higher-order averaging of time-periodic systems: reconciliation of two averaging techniques

Nonlinear Dynamics - In this paper we show how higher-order averaging can be used to remedy serious technical issues with the direct application of the averaging theorem. While doing so, we...     

Generalized canonical transformation for second-order Birkhoffian systems on time scales

The theory of time scales, which unifies continuous and discrete analysis, provides a powerful mathematical tool for the study of complex dynamic systems. It enables us to understand more clearly the essential problems of continuous systems and discrete systems as well as other complex systems. In this paper, the theory of generalized canonical transformation for second-order Birkhoffian systems on time scales is proposed and studied, which extends the canonical transformation theory of Hamilton canonical equations. First, the condition of generalized canonical transformation for the second-order Birkhoffian system on time scales is established.Second, based on this condition, six basic forms of generalized canonical transformation for the second-order Birkhoffian system on time scales are given. Also, the relationships between new variables and old variables for each of these cases are derived. In the end, an example is given to show the application of the results.     

Stability analysis of uncertain systems on time scales

A method for the (in)stability analysis of the zero solution of an uncertain system on a time scale is proposed. The method is based on the matrix-valued Lyapunov function and its Δ-derivative on a time scale     

Solutions for a class of Hamiltonian systems on time scales with non-local boundary conditions

In this work, the solvability of a class of second-order Hamiltonian systems on time scales is generalized to non-local boundary conditions. The measurements obtained by non-local conditions are more accurate than those given by local conditions in some problems. Compared with the known results, this work establishes the variational structure in an appropriate Sobolev’s space. Then, by applying the mountain pass theorem and symmetric mountain pass theorem, the existence and multiplicity of the s...     

A frequency-domain method for solving linear time delay systems with constant coefficients

In an active control system, time delay will occur due to processes such as signal acquisition and transmission, calculation, and actuation. Time delay systems are usually described by delay differential equations (DDEs). Since it is hard to obtain an analytical solution to a DDE, numerical solution is of necessity. This paper presents a frequency-domain method that uses a truncated transfer function to solve a class of DDEs. The theoretical transfer function is the sum of infinite items expressed in terms of poles and residues. The basic idea is to select the dominant poles and residues to truncate the transfer function, thus ensuring the validity of the solution while improving the efficiency of calculation. Meanwhile, the guideline of selecting these poles and residues is provided. Numerical simulations of both stable and unstable delayed systems are given to verify the proposed method, and the results are presented and analysed in detail.     

Cluster synchronization of coupled delayed competitive neural networks with two time scales

Nonlinear Dynamics - This paper investigates the cluster synchronization problem of coupled delayed competitive neural networks (CNNs) with two time scales. Each CNN contains short- and long-term...     

An enriched multiple scales method for harmonically forced nonlinear systems

This article explores enrichment to the method of Multiple Scales, in some cases extending its applicability to periodic solutions of harmonically forced, strongly nonlinear systems. The enrichment follows from an introduced homotopy parameter in the system governing equation, which transitions it from linear to nonlinear behavior as the value varies from zero to one. This same parameter serves as a perturbation quantity in both the asymptotic expansion and the multiple time scales assumed solution form. Two prototypical nonlinear systems are explored. The first considered is a classical forced Duffing oscillator for which periodic solutions near primary resonance are analyzed, and their stability is assessed, as the strengths of the cubic term, the forcing, and a system scaling factor are increased. The second is a classical forced van der Pol oscillator for which quasiperiodic and subharmonic solutions are analyzed. For both systems, comparisons are made between solutions generated using (a) the enriched Multiple Scales approach, (b) the conventional Multiple Scales approach, and (c) numerical simulations. For the Duffing system, important qualitative and quantitative differences are noted between solutions predicted by the enriched and conventional Multiple Scales. For the van der Pol system, increased solution flexibility is noted with the enriched Multiple Scales approach, including the ability to seek subharmonic (and superharmonic) solutions not necessarily close to the linear natural frequency. In both nonlinear systems, comparisons to numerical simulations show strong agreement with results from the enriched technique, and for the Duffing case in particular, even when the system is strongly nonlinear.     

Estimation of the relative error for the averaging method

A quasilinear ordinary differential system is considered. It and the corresponding averaged system have the same equilibrium point. With some assumptions of regularity and stability, it is shown that the relative error due to the averaging method, tends to zero when the independent variable tends to infinite. The capital point lies in the choice of the two solutions which are compared.     

An estimate for the small parameter in the asymptotic analysis of non-linear systems by the method of averaging

The Method of Averaging is an asymptotic method that can be used to obtain approximate solutions for many parameter dependent non-linear systems. The resulting approximate solutions are as accurate as desired provided the system parameter is sufficiently small. In this paper, a general technique is developed to obtain a relationship between the magnitude of the small parameter and the permissible error in the approximate solution. The technique is then demonstrated by its application to two examples.     

A generalization of wieghardt soil for two - dimensional foundation structures

Summary On the basis of a mechanical interpretation (see Reference[1]) the problem of generalization of Wieghardt soils for two-dimensional foundation structures is tackled. The results of experiments agree well with the schematization arrived at. The problem of the rigid circular plate on the thus schematized elastic soil is solved.

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Sommario Premesse alcune considerazioni di carattere generale sulla schematizzazione proposta dal Wieghardt per la legge carichi-spostamenti di un letto elastico di fondazione di un solido monodimensionale si affronta, al lume di un'interpretazione meccanica proposta dall'A. in un articolo precedente[1], il problema della generalizzazione di tale legge per i problemi di fondazione bidimensionali. Si perviene così in forma semplice alla formulazione della legge carichi-spostamenti per un generico piano di fondazione e si dimostra l'ottima rispondenza di tale schematizzazione con i risultati dedotti da prove sperimentali reali. Viene risolto in via completa il problema della piastra circolare rigida sul suolo elastico così schematizzato.

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First published in Italian in La ricerca scientifica, 35 (II-A), 1290–1304, Roma, 1965.     

On the theorems for local volume averaging of multiphase systems

Proofs of the theorems of local volume averaging which relate averages of derivatives to derivatives of averages are presented. A distribution function whose derivatives are proportional to the Dirac delta function is used in the development and the proofs demonstrated are intended to be simpler than those found in the literature.     

时滞线性系统振动主动控制的滑移模态方法

直接从时滞微分方程求解控制律，对时滞线性系统振动主动控制的滑移模态方法进行了研究，并给出了控制具体实现过程。推导出的切换面和控制律表达式中，除了包含有当前的状态反馈外，还包含有前若干步控制的线性组合。算例结果显示，所提控制方法能够较好地控制系统的峰值响应。且能够保证控制系统的稳定性；若按无时滞控制设计对地滞系统进行振动控制，系统响应有可能出现发散。     

Solving systems of nonlinear difference equations by the multiple scales perturbation method

In this paper, we apply an improved version of the multiple scales perturbation method to a system of weakly nonlinear, regularly perturbed ordinary difference equations. Such systems arise as a result of the discretization of a system of nonlinear differential equations, or as a result in the stability analysis of nonlinear oscillations. In our procedure, asymptotic approximations of the solutions of the difference equations will be constructed which are valid on long iteration scales.     

A single adaptive controller with one variable for synchronizing two identical time delay hyperchaotic Lorenz systems with mismatched parameters

Time delays are ubiquitous in real world and are often sources of complex behaviors of dynamical systems. This paper addresses the problem of parameter identification and synchronization of uncertain hyperchaotic time-delayed systems. Based on the Lyapunov stability theory and the adaptive control theory, a single adaptive controller with one variable for synchronizing two identical time-delay hyperchaotic Lorenz systems with mismatch parameters is proposed. The parameter update laws and sufficient conditions of the scheme are obtained for both linear feedback and adaptive control approaches. Numerical simulations are also given to show the effectiveness of the proposed method.