Applications of Centre Manifold Theory

Book•

Applications of Centre Manifold Theory

28 Dec 2011-

TL;DR: In this paper, the authors present an approach for solving the panel flutter problem using a Second Order Equation (SOPE) and a Semigroup Theory. But their approach is limited to the case when the case is 1 < 0 and the case where 0 < 0.

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Abstract: 0.- 4.5. The Case ?1 < 0.- 4.6. More Scaling.- 4.7. Completion of the Phase Portraits.- 4.8. Remarks and Exercises.- 4.9. Quadratic Nonlinearities.- 5. Application to a Panel Flutter Problem.- 5.1. Introduction.- 5.2. Reduction to a Second Order Equation.- 5.3. Calculation of Linear Terms.- 5.4. Calculation of the Nonlinear Terms.- 6. Infinite Dimensional Problems.- 6.1. Introduction.- 6.2. Semigroup Theory.- 6.3. Centre Manifolds.- 6.4. Examples.- References.

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Journal Article•DOI•

Functional and effective connectivity: a review.

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Karl J. Friston¹•Institutions (1)

Wellcome Trust Centre for Neuroimaging¹

01 Jun 2011-Brain connectivity

TL;DR: The inception of this journal has been foreshadowed by an ever-increasing number of publications on functional connectivity, causal modeling, connectomics, and multivariate analyses of distributed patterns of brain responses.

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Abstract: Over the past 20 years, neuroimaging has become a predominant technique in systems neuroscience. One might envisage that over the next 20 years the neuroimaging of distributed processing and connectivity will play a major role in disclosing the brain's functional architecture and operational principles. The inception of this journal has been foreshadowed by an ever-increasing number of publications on functional connectivity, causal modeling, connectomics, and multivariate analyses of distributed patterns of brain responses. I accepted the invitation to write this review with great pleasure and hope to celebrate and critique the achievements to date, while addressing the challenges ahead.

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2,822 citations

Cites background from "Applications of Centre Manifold The..."

...In practice, there are various theorems such as the center manifold theorem* and slaving principle, which means one can reduce the effective number of hidden states substantially but still retain the underlying dynamical structure of the system (Ginzburg and Landau, 1950; Carr, 1981; Haken, 1983; Kopell and Ermentrout, 1986)....
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...…theorems such as the center manifold theorem* and slaving principle, which means one can reduce the effective number of hidden states substantially but still retain the underlying dynamical structure of the system (Ginzburg and Landau, 1950; Carr, 1981; Haken, 1983; Kopell and Ermentrout, 1986)....
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Book•

Asymptotic Behavior of Dissipative Systems

[...]

Jack K. Hale

31 Dec 1988

TL;DR: In this article, the authors consider a continuous dynamical system with a global attractor and describe the properties of the flow on the attractor asymptotically smooth and Morse-Smale maps.

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Abstract: Discrete dynamical systems: Limit sets Stability of invariant sets and asymptotically smooth maps Examples of asymptotically smooth maps Dissipativeness and global attractors Dependence on parameters Fixed point theorems Stability relative to the global attractor and Morse-Smale maps Dimension of the global attractor Dissipativeness in two spaces Continuous dynamical systems: Limit sets Asymptotically smooth and $\alpha$-contracting semigroups Stability of invariant sets Dissipativeness and global attractors Dependence on parameters Periodic processes Skew product flows Gradient flows Dissipativeness in two spaces Properties of the flow on the attractor Applications: Retarded functional differential equations Sectorial evolutionary equations A scalar parabolic equation The Navier-Stokes equation Neutral functional differential equations Some abstract evolutionary equations A one-dimensional damped wave equation A three-dimensional damped wave equation Remarks on other applications Dependence on parameters and approximation of the attractor.

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2,639 citations

Cites background or methods from "Applications of Centre Manifold The..."

...However, with considerable effort, one can show that they are C* - 1 , 1 ; that is, they are represented by a function which is C~ with the k — 1 first derivatives being Lipschitz (see, for example, Carr [1981], Sijbrand [1985])....
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...However, with considerable effort, one can show that they are C* - 1 , 1 ; that is, they are represented by a function which is C~ with the k — 1 first derivatives being Lipschitz (see, for example, Carr [1981], Sijbrand [1985]). One must then use some other method to show that the manifolds actually are C. A convenient way is to use the following lemma of Henry [1983] based on a remark in Hirsch, Pugh, and Shub [1977, p....
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Journal Article•DOI•

Control of chained systems application to path following and time-varying point-stabilization of mobile robots

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Claude Samson

01 Jan 1995-IEEE Transactions on Automatic Control

TL;DR: Application to the control of nonholonomic wheeled mobile robots is described by considering the case of a car pulling trailers, and globally stabilizing time-varying feedbacks are derived.

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Abstract: Chain form systems have recently been introduced to model the kinematics of a class of nonholonomic mechanical systems. The first part of the study is centered on control design and analysis for nonlinear systems which can be converted to the chain form. Solutions to various control problems (open-loop steering, partial or complete state feedback stabilization) are either recalled, generalized, or developed. In particular, globally stabilizing time-varying feedbacks are derived, and a discussion of their convergence properties is provided. Application to the control of nonholonomic wheeled mobile robots is described in the second part of the study by considering the case of a car pulling trailers. >

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1,094 citations

Cites methods from "Applications of Centre Manifold The..."

...For example: f (t) = (1 + t)- 3 , in the case of the control considered in 1281, as it may be rigorously established either by applying Center Manifold techniques [ 7 ] or by invoking two-time scale techniques, as done in [14]....
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Book•

Nonlinear Vibrations and Stability of Shells and Plates

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Marco Amabili

01 Aug 2014

TL;DR: In this article, a comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells is presented. But the authors do not consider the effect of boundary conditions on the large-amplitude vibrations of circular cylinders.

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Abstract: Introduction. 1. Nonlinear theories of elasticity of plates and shells 2. Nonlinear theories of doubly curved shells for conventional and advanced materials 3. Introduction to nonlinear dynamics 4. Vibrations of rectangular plates 5. Vibrations of empty and fluid-filled circular cylindrical 6. Reduced order models: proper orthogonal decomposition and nonlinear normal modes 7. Comparison of different shell theories for nonlinear vibrations and stability of circular cylindrical shells 8. Effect of boundary conditions on a large-amplitude vibrations of circular cylindrical shells 9. Vibrations of circular cylindrical panels with different boundary conditions 10. Nonlinear vibrations and stability of doubly-curved shallow-shells: isotropic and laminated materials 11. Meshless discretization of plates and shells of complex shapes by using the R-functions 12. Vibrations of circular plates and rotating disks 13. Nonlinear stability of circular cylindrical shells under static and dynamic axial loads 14. Nonlinear stability and vibrations of circular shells conveying flow 15. Nonlinear supersonic flutter of circular cylindrical shells with imperfections.

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862 citations

Journal Article•DOI•

Dynamical behavior of epidemiological models with nonlinear incidence rates

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Wei-min Liu¹, Herbert W. Hethcote², Simon A. Levin³•Institutions (3)

Cornell University¹, University of Iowa², Ithaca College³

01 Jan 1987-Journal of Mathematical Biology

TL;DR: Epidemiological models with nonlinear incidence rates λIpSq show a much wider range of dynamical behaviors than do those with bilinear incidence ratesλIS, and these behaviors are determined mainly by p and λ, and secondarily by q.

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Abstract: Epidemiological models with nonlinear incidence rates λIpSqshow a much wider range of dynamical behaviors than do those with bilinear incidence rates λIS. These behaviors are determined mainly by p and λ, and secondarily by q. For such models, there may exist multiple attractive basins in phase space; thus whether or not the disease will eventually die out may depend not only upon the parameters, but also upon the initial conditions. In some cases, periodic solutions may appear by Hopf bifurcation at critical parameter values.

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747 citations

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Applications of Centre Manifold Theory

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