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In the present chapter we use a method for the qualitative asymptotic analysis of singularly perturbed differential equations by reducing the order of the differential system under consideration. The method relies on the theory of... more
In the present chapter we use a method for the qualitative asymptotic analysis of singularly perturbed differential equations by reducing the order of the differential system under consideration. The method relies on the theory of integral manifolds. It essentially replaces the original system by another system on an integral manifold with a lower dimension that is equal to that of the slow subsystem. The emphasis in this chapter is on the study of autonomous systems.
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Page 1. ISSN 0005-1179, Automation and Remote Control, 2006, Vol. 67, No. 8, pp. 1185–1193. c © Pleiades Publishing, Inc., 2006. Original Russian Text c © NV Voropaeva, VA Sobolev, 2006, published in Avtomatika i Telemekhanika, 2006, No.... more
Page 1. ISSN 0005-1179, Automation and Remote Control, 2006, Vol. 67, No. 8, pp. 1185–1193. c © Pleiades Publishing, Inc., 2006. Original Russian Text c © NV Voropaeva, VA Sobolev, 2006, published in Avtomatika i Telemekhanika, 2006, No. 8, pp. 3–11. ...
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Methods of the geometric theory of singular perturbations are used to reduce the dimensions of problems in chemical kinetics. The methods are based on using slow invariant manifolds. As a result, the original system is replaced by one on... more
Methods of the geometric theory of singular perturbations are used to reduce the dimensions of problems in chemical kinetics. The methods are based on using slow invariant manifolds. As a result, the original system is replaced by one on an invariant manifold, whose dimension coincides with that of the slow subsystem. Explicit and implicit representations of slow invariant manifolds are applied. The mathematical apparatus described is used to develop N.N. Semenov’s fundamental ideas related to the method of quasi-stationary concentrations and is used to study particular problems in chemical kinetics.
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In this chapter the first number in the title of a section denotes the dimension of the slow variable, the second one denotes the dimension of the fast variable. A series of examples, of increasing complexity, are given to illustrate the... more
In this chapter the first number in the title of a section denotes the dimension of the slow variable, the second one denotes the dimension of the fast variable. A series of examples, of increasing complexity, are given to illustrate the theoretical concepts. The main examples come from applications in enzyme kinetics. These examples illustrate the effectiveness of the order reduction method.
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The PD-regulators design problem for the singularly perturbed control system is considered in the paper. It is shown that this problem can be reduced to the P-regulators design problems for two subsystems of the lower dimension.
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We propose sufficient conditions for existence of topologically stable periodic canard solutions in non-smooth slow-fast systems.
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The theory and applications of singularly perturbed systems of differential equations, traditionally connected with the problems of fluid dynamics and nonlinear mechanics, has been developed intensively and the methods are applied... more
The theory and applications of singularly perturbed systems of differential equations, traditionally connected with the problems of fluid dynamics and nonlinear mechanics, has been developed intensively and the methods are applied actively to the solution of a wide range of problems from other areas of natural science. This can be explained by the fact that such singularly perturbed systems appear naturally in the process of modelling various processes, that are characterized by slow and fast motions simultaneously. In many cases it is necessary to consider the behaviour of the system as a whole and not of separate trajectories, and to investigate the system by means of a qualitative analysis. This is what is done in this talk. In the present talk asymptotic and geometrical techniques of analysis are combined for the investigation of singularly perturbed systems. The essence of this approach consists in separating out the slow motions of the system under investigation. Then the order of the differential system decreases, but the reduced system, of lesser order, inherits the essential elements of the qualitative behaviour of the original system in the corresponding domain when the slow integral manifold is attracting. The construction of simplified models is achieved and these simpler models reflect the behaviour of the original models to a high order of accuracy. A mathematical justification of this method can be given by means of the theory of integral manifolds for singularly perturbed systems Note that the pioneering papers were published during the period 1957–1970 by K. Zadiraka [4], V. Fodchuk and Y. Baris, who followed on the work of N. Bogolyubov and Y. Mitropolsky. The existence of slow integral manifolds, stable, unstable and conditionally stable, occur in these papers.
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In constructing the asymptotic expansions of slow integral manifolds it is assumed that the degenerate equation (\(\varepsilon = 0\)) allows one to find the slow surface explicitly. In many problems this is not possible due to the fact... more
In constructing the asymptotic expansions of slow integral manifolds it is assumed that the degenerate equation (\(\varepsilon = 0\)) allows one to find the slow surface explicitly. In many problems this is not possible due to the fact that the degenerate equation is either a high degree polynomial or transcendental. In this situation many authors suggest the use of numerical methods. However, in many problems the slow surface can be described in parametric form, and then the slow integral manifold can be found in parametric form as asymptotic expansions. If this is not possible, it is necessary to use an implicit slow surface and obtain asymptotic representations for the slow integral manifold in an implicit form. Model examples, as well as examples borrowed from combustion theory, are treated.
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An International Workshop on Hysteresis & Multi-scale Asymptotics was held at University College Cork, Ireland on March 1721, 2004. It brought together about 40 active scientists in the areas of dynamical systems with hysteresis and... more
An International Workshop on Hysteresis & Multi-scale Asymptotics was held at University College Cork, Ireland on March 1721, 2004. It brought together about 40 active scientists in the areas of dynamical systems with hysteresis and singular perturbations to analyse ...
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We are interested in singular perturbation problems and hysteresis as common strongly nonlinear phenomena that occur in many industrial, physical and economic systems. The wording `strongly nonlinear' means that linearization will... more
We are interested in singular perturbation problems and hysteresis as common strongly nonlinear phenomena that occur in many industrial, physical and economic systems. The wording `strongly nonlinear' means that linearization will not encapsulate the observed phenomena. Often these two types of phenomena are manifested for different stages of the same or similar processes. A number of fundamental hysteresis models can be considered as limit cases of time relaxation processes, or admit an approximation by a differential equation which is singular with respect to a particular parameter. However, the amount of interaction between practitioners of theories of systems with time relaxation and systems with hysteresis (and between the `relaxation' and `hysteresis' research communities) is still low, and cross-fertilization is small. In recent years Ireland has become a home for a series of prestigious International Workshops in Singular Perturbations and Hysteresis: International Workshop on Multi-rate Processes and Hysteresis (University College Cork, Ireland, 3–8 April 2006). Proceedings are published in Journal of Physics: Conference Series, volume 55. See further information at http://euclid.ucc.ie/murphys2008.htmInternational Workshop on Hysteresis and Multi-scale Asymptotics (University College Cork, Ireland, 17–21 March 2004). Proceedings are published in Journal of Physics: Conference Series, volume 22. See further information at http://euclid.ucc.ie/murphys2006.htmInternational Workshop on Relaxation Oscillations and Hysteresis (University College Cork, Ireland, 1–6 April 2002). The related collection of invited lectures, was published as a volume Singular Perturbations and Hysteresis, SIAM, Philadelphia, 2005. See further information at http://euclid.ucc.ie/hamsa2004.htmInternational Workshop on Geometrical Methods of Nonlinear Analysis and Semiconductor Laser Dynamics (University College Cork, Ireland, 5–5 April 2001). A collection of invited papers has been published as a special issue of Proceedings of the Russian Academy of Natural Sciences: Nonlinear dynamics of laser and reacting systems, and is available online at http://www.ins.ucc.ie/roh2002.htm. See further information at http://www.ins.ucc.ie/roh2002.htm Among the aims of these workshops were to bring together leading experts in singular perturbations and hysteresis phenomena in applied problems; to discuss important problems in areas such as reacting systems, semiconductor lasers, shock phenomena in economic modelling, fluid mechanics, etc with an emphasis on hysteresis and singular perturbations; to learn and to share modern techniques in areas of common interest. The `International Workshop on Multi-Rate Processes and Hysteresis' (University College Cork, Ireland, April 3–8, 2006) brought together more than 70 scientists (including more than 10 students), actively researching in the areas of dynamical systems with hysteresis and singular perturbations, to analyze those phenomena that occur in many industrial, physical and economic systems. The countries represented at the Workshop included Czech Republic, England, France, Germany, Hungary, Ireland, Israel, Italy, Poland, Romania, Russia, Scotland, South Africa, Switzerland and USA. All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing. The Workshop has been sponsored by Science Foundation Ireland (SFI), KE Consulting group, Drexel University, Philadelphia, USA, University College Cork (UCC), Boole Centre for Research in Informatics, UCC, Cork, School of Mathematical Sciences, UCC, Cork, Irish Mathematical Society, Tyndall National Institute, Cork, University of Limerick, Cork Institute of Technology, and Heineken. The supportive affiliation of the European Geophysics Society, International Association of Hydrological Sciences, and Laboratoire Poncelet is gratefully acknowledged. The Editors and the Organizers of the Workshop wish to place on record their sincere gratitude to Mr Andrew Zhezherun and Mr Alexander Pimenov of University College Cork for both the assistance which he provided to all the presenters at the Workshop, and for the careful formatting of all the manuscripts prior to their being forwarded to the Publisher. More information about the Workshop can be found at http://euclid.ucc.ie/murphys2006.htmMichael P Mortell, Robert E O'Malley Jr, Alexei Pokrovskii, Dmitrii Rachinskii and Vladimir Sobolev Editors
Research Interests: Engineering, Geography, Physics, Fluid Mechanics, Nonlinear dynamics, and 15 moreStatistical Physics, Natural Science, Peer Review, Czech Republic, Hysteresis, South Africa, Nonlinear Analysis, Economic System, Physical sciences, Oscillations, Linearization, Nonlinear system, Economic Modelling, Differential equation, and Dynamic System
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In this paper we present sufficient conditions for stability of quasi-static paths of finite dimensional systems that have a smooth behavior. The concept of stability of quasi-static paths used here is essentially a continuity property... more
In this paper we present sufficient conditions for stability of quasi-static paths of finite dimensional systems that have a smooth behavior. The concept of stability of quasi-static paths used here is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces (which plays here the role of the small parameter in singular perturbation problems). These conditions are applied to a mechanical system with a convex potential energy.
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ABSTRACT A concept of stability of quasi-static paths is discussed in this paper that takes into account the existence of fast (dynamic) and slow (quasi-static) time scales in the evolution of many mechanical systems. The proposed concept... more
ABSTRACT A concept of stability of quasi-static paths is discussed in this paper that takes into account the existence of fast (dynamic) and slow (quasi-static) time scales in the evolution of many mechanical systems. The proposed concept is essentially a continuity property with respect to the smallness of the initial perturbations (as in Lyapunov stability) and the smallness of the quasi-static loading rate (that plays the role of the small parameter in singular perturbation problems). A related concept of attractiveness is also proposed. Several examples illustrate the relevance of the definitions. Sufficient conditions for attractiveness or for instability of quasi-static paths of smooth systems are proved. Copyright © 2005 John Wiley & Sons, Ltd.
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Introduction.- Slow Integral Manifolds.- The Book of Numbers.- Representations of Slow Integral Manifolds.- Singular Singularly Perturbed Systems.- Reduction Methods for Chemical Systems.- Specific Cases.- Canards and Black Swans.-... more
Introduction.- Slow Integral Manifolds.- The Book of Numbers.- Representations of Slow Integral Manifolds.- Singular Singularly Perturbed Systems.- Reduction Methods for Chemical Systems.- Specific Cases.- Canards and Black Swans.- Appendix: Proofs.
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We consider a simple model of a passive dynamic biped robot with point feet and legs without knee. The model is a switched system, which includes an inverted double pendulum. We present an asymptotic solution of the model. The first... more
We consider a simple model of a passive dynamic biped robot with point feet and legs without knee. The model is a switched system, which includes an inverted double pendulum. We present an asymptotic solution of the model. The first correction to the zero order approximation is shown to agree with the numerical solution with high degree of accuracy for a limited parameter range.
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The paper is concerned on solitary waves for singularly perturbed generalized KdV equation with high order nonlinear terms. We firstly give the phase portraits of system related to the unperturbed equation under various cases by theory of... more
The paper is concerned on solitary waves for singularly perturbed generalized KdV equation with high order nonlinear terms. We firstly give the phase portraits of system related to the unperturbed equation under various cases by theory of planar dynamical system. Then by using geometric singular perturbation theory and Melnikov's method, the existence of solitary wave solutions of generalized KdV equations with high order nonlinear terms is established. It is proven that some solitary wave solutions with particular wave speeds will persist under small perturbations.
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We consider a simple model of a passive dynamic biped robot walker with point feet and legs without knee. The model is a switched system, which includes an inverted double pendulum. Robot's gait and its stability depend on parameters... more
We consider a simple model of a passive dynamic biped robot walker with point feet and legs without knee. The model is a switched system, which includes an inverted double pendulum. Robot's gait and its stability depend on parameters such as the slope of the ramp, the length of robot's legs, and the mass distribution along the legs. We present an asymptotic solution of the model. The first correction to the zero order approximation is shown to agree with the numerical solution for a limited parameter range.