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JournalISSN: 0924-090X

Nonlinear Dynamics 

About: Nonlinear Dynamics is an academic journal. The journal publishes majorly in the area(s): Nonlinear system & Control theory. It has an ISSN identifier of 0924-090X. Over the lifetime, 9237 publication(s) have been published receiving 203311 citation(s). more

Topics: Nonlinear system, Control theory, Chaotic more
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Journal ArticleDOI
Igor Mezic1Institutions (1)
01 Aug 2005-Nonlinear Dynamics
Abstract: In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system with the properties and possibilities for model reduction. We review some elements of the spectral theory of dynamical systems. We apply this theory to obtain a decomposition of the process that utilizes spectral properties of the linear Koopman operator associated with the asymptotic dynamics on the attractor. This allows us to extract the almost periodic part of the evolving process. The remainder of the process has continuous spectrum. The second topic we discuss is that of model validation, where the original, possibly high-dimensional dynamics and the dynamics of the reduced model – that can be deterministic or stochastic – are compared in some norm. Using the “statistical Takens theorem” proven in (Mezic, I. and Banaszuk, A. Physica D, 2004) we argue that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection. more

854 citations

Journal ArticleDOI
01 Aug 2005-Nonlinear Dynamics
Abstract: Modal analysis is used extensively for understanding the dynamic behavior of structures. However, a major concern for structural dynamicists is that its validity is limited to linear structures. New developments have been proposed in order to examine nonlinear systems, among which the theory based on nonlinear normal modes is indubitably the most appealing. In this paper, a different approach is adopted, and proper orthogonal decomposition is considered. The modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data. The utility of the method for dynamic characterization and order reduction of linear and nonlinear mechanical systems is demonstrated in this study. more

724 citations

Journal ArticleDOI
Weihua Deng1, Weihua Deng2, Changpin Li2, Jinhu Lu3Institutions (3)
09 Mar 2007-Nonlinear Dynamics
Abstract: In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R+)n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined. more

621 citations

Journal ArticleDOI
Igor Podlubny1, Ivo Petras1, Blas M. Vinagre, Paul O'Leary2  +1 moreInstitutions (2)
01 Jul 2002-Nonlinear Dynamics
Abstract: An approach to the design of analogue circuits, implementingfractional-order controllers, is presented. The suggestedapproach is based on the use of continued fraction expansions;in the case of negative coefficients in a continued fractionexpansion, the use of negative impedance converters is proposed.Several possible methods for obtaining suitable rational appromixationsand continued fraction expansions are discussed. An exampleof realization of a fractional-order Iλ controlleris presented and illustrated by obtained measurements.The suggested approach can be used for the control of veryfast processes, where the use of digital controllers isdifficult or impossible. more

602 citations

Journal ArticleDOI
Carl F. Lorenzo1, Tom T. Hartley2Institutions (2)
01 Feb 2002-Nonlinear Dynamics
Abstract: Many physical processes appear to exhibit fractional order behavior that may vary with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. This paper develops the concept of variable and distributed order fractional operators. Definitions based on the Riemann-Liouville definitions are introduced and behavior of the operators is studied. Several time domain definitions that assign different arguments to the order q in the Riemann-Liouville definition are introduced. For each of these definitions various characteristics are determined. These include: time invariance of the operator, operator initialization, physical realization, linearity, operational transforms. and memory characteristics of the defining kernels. A measure (m2) for memory retentiveness of the order history is introduced. A generalized linear argument for the order q allows the concept of "tailored" variable order fractional operators whose a, memory may be chosen for a particular application. Memory retentiveness (m2) and order dynamic behavior are investigated and applications are shown. The concept of distributed order operators where the order of the time based operator depends on an additional independent (spatial) variable is also forwarded. Several definitions and their Laplace transforms are developed, analysis methods with these operators are demonstrated, and examples shown. Finally operators of multivariable and distributed order are defined in their various applications are outlined. more

575 citations

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Journal's top 5 most impactful authors

Ali H. Nayfeh

59 papers, 4.1K citations

Ahmed A. Shabana

33 papers, 1.2K citations

Pengjian Shang

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Ju H. Park

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J. A. Tenreiro Machado

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