Smooth And Nonsmooth High Dimensional Chaos And The Melnikov Type Methods

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Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods

Author : Jan Awrejcewicz
Publisher : World Scientific
ISBN 13 : 981270910X
Total Pages : 318 pages
Book Rating : 4.8/5 (127 download)

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Book Synopsis Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods by : Jan Awrejcewicz

Download or read book Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods written by Jan Awrejcewicz and published by World Scientific. This book was released on 2007 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the development of Melnikov-type methods applied to high dimensional dynamical systems governed by ordinary differential equations. Although the classical Melnikov's technique has found various applications in predicting homoclinic intersections, it is devoted only to the analysis of three-dimensional systems (in the case of mechanics, they represent one-degree-of-freedom nonautonomous systems). This book extends the classical Melnikov's approach to the study of high dimensional dynamical systems, and uses simple models of dry friction to analytically predict the occurrence of both stick-slip and slip-slip chaotic orbits, research which is very rarely reported in the existing literature even on one-degree-of-freedom nonautonomous dynamics. This pioneering attempt to predict the occurrence of deterministic chaos of nonlinear dynamical systems will attract many researchers including applied mathematicians, physicists, as well as practicing engineers. Analytical formulas are explicitly formulated step-by-step, even attracting potential readers without a rigorous mathematical background. Sample Chapter(s). Chapter 1: A Role of the Melnikov-Type Methods in Applied Sciences (137 KB). Contents: A Role of the Melnikov-Type Methods in Applied Sciences; Classical Melnikov Approach; Homoclinic Chaos Criterion in a Rotated Froude Pendulum with Dry Friction; Smooth and Nonsmooth Dynamics of a Quasi-Autonomous Oscillator with Coulomb and Viscous Frictions; Application of the MelnikovOCoGruendler Method to Mechanical Systems; A Self-Excited Spherical Pendulum; A Double Self-excited Duffing-type Oscillator; A Triple Self-Excited Duffing-type Oscillator. Readership: Graduate students and researchers in dynamical systems.

Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems

Author : Michal Fečkan
Publisher : Academic Press
ISBN 13 : 0128043644
Total Pages : 260 pages
Book Rating : 4.1/5 (28 download)

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Book Synopsis Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems by : Michal Fečkan

Download or read book Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems written by Michal Fečkan and published by Academic Press. This book was released on 2016-06-07 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: Poincaré-Andronov-Melnikov Analysis for Non-Smooth Systems is devoted to the study of bifurcations of periodic solutions for general n-dimensional discontinuous systems. The authors study these systems under assumptions of transversal intersections with discontinuity-switching boundaries. Furthermore, bifurcations of periodic sliding solutions are studied from sliding periodic solutions of unperturbed discontinuous equations, and bifurcations of forced periodic solutions are also investigated for impact systems from single periodic solutions of unperturbed impact equations. In addition, the book presents studies for weakly coupled discontinuous systems, and also the local asymptotic properties of derived perturbed periodic solutions. The relationship between non-smooth systems and their continuous approximations is investigated as well. Examples of 2-, 3- and 4-dimensional discontinuous ordinary differential equations and impact systems are given to illustrate the theoretical results. The authors use so-called discontinuous Poincaré mapping which maps a point to its position after one period of the periodic solution. This approach is rather technical, but it does produce results for general dimensions of spatial variables and parameters as well as the asymptotical results such as stability, instability, and hyperbolicity. Extends Melnikov analysis of the classic Poincaré and Andronov staples, pointing to a general theory for freedom in dimensions of spatial variables and parameters as well as asymptotical results such as stability, instability, and hyperbolicity Presents a toolbox of critical theoretical techniques for many practical examples and models, including non-smooth dynamical systems Provides realistic models based on unsolved discontinuous problems from the literature and describes how Poincaré-Andronov-Melnikov analysis can be used to solve them Investigates the relationship between non-smooth systems and their continuous approximations

Bifurcation and Chaos in Discontinuous and Continuous Systems

Author : Michal Fečkan
Publisher : Springer Science & Business Media
ISBN 13 : 3642182690
Total Pages : 378 pages
Book Rating : 4.6/5 (421 download)

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Book Synopsis Bifurcation and Chaos in Discontinuous and Continuous Systems by : Michal Fečkan

Download or read book Bifurcation and Chaos in Discontinuous and Continuous Systems written by Michal Fečkan and published by Springer Science & Business Media. This book was released on 2011-05-30 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Bifurcation and Chaos in Discontinuous and Continuous Systems" provides rigorous mathematical functional-analytical tools for handling chaotic bifurcations along with precise and complete proofs together with concrete applications presented by many stimulating and illustrating examples. A broad variety of nonlinear problems are studied involving difference equations, ordinary and partial differential equations, differential equations with impulses, piecewise smooth differential equations, differential and difference inclusions, and differential equations on infinite lattices as well. This book is intended for mathematicians, physicists, theoretically inclined engineers and postgraduate students either studying oscillations of nonlinear mechanical systems or investigating vibrations of strings and beams, and electrical circuits by applying the modern theory of bifurcation methods in dynamical systems. Dr. Michal Fečkan is a Professor at the Department of Mathematical Analysis and Numerical Mathematics on the Faculty of Mathematics, Physics and Informatics at the Comenius University in Bratislava, Slovakia. He is working on nonlinear functional analysis, bifurcation theory and dynamical systems with applications to mechanics and vibrations.

Deterministic Chaos in One-Dimensional Continuous Systems

Author : Jan Awrejcewicz
Publisher : World Scientific
ISBN 13 : 9814719714
Total Pages : 576 pages
Book Rating : 4.8/5 (147 download)

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Book Synopsis Deterministic Chaos in One-Dimensional Continuous Systems by : Jan Awrejcewicz

Download or read book Deterministic Chaos in One-Dimensional Continuous Systems written by Jan Awrejcewicz and published by World Scientific. This book was released on 2016-03-14 with total page 576 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the computational analysis of nonlinear vibrations of structural members (beams, plates, panels, shells), where the studied dynamical problems can be reduced to the consideration of one spatial variable and time. The reduction is carried out based on a formal mathematical approach aimed at reducing the problems with infinite dimension to finite ones. The process also includes a transition from governing nonlinear partial differential equations to a set of finite number of ordinary differential equations. Beginning with an overview of the recent results devoted to the analysis and control of nonlinear dynamics of structural members, placing emphasis on stability, buckling, bifurcation and deterministic chaos, simple chaotic systems are briefly discussed. Next, bifurcation and chaotic dynamics of the Euler–Bernoulli and Timoshenko beams including the geometric and physical nonlinearity as well as the elastic–plastic deformations are illustrated. Despite the employed classical numerical analysis of nonlinear phenomena, the various wavelet transforms and the four Lyapunov exponents are used to detect, monitor and possibly control chaos, hyper-chaos, hyper-hyper-chaos and deep chaos exhibited by rectangular plate-strips and cylindrical panels. The book is intended for post-graduate and doctoral students, applied mathematicians, physicists, teachers and lecturers of universities and companies dealing with a nonlinear dynamical system, as well as theoretically inclined engineers of mechanical and civil engineering. Contents:Bifurcational and Chaotic Dynamics of Simple Structural Members:BeamsPlatesPanelsShellsIntroduction to Fractal Dynamics:Cantor Set and Cantor DustKoch Snowflake1D MapsSharkovsky's TheoremJulia SetMandelbrot's SetIntroduction to Chaos and Wavelets:Routes to ChaosQuantifying Chaotic DynamicsSimple Chaotic Models:IntroductionAutonomous SystemsNon-Autonomous SystemsDiscrete and Continuous Dissipative Systems:IntroductionLinear FrictionNonlinear FrictionHysteretic FrictionImpact DampingDamping in Continuous 1D SystemsEuler-Bernoulli Beams:IntroductionPlanar BeamsLinear Planar Beams and Stationary Temperature FieldsCurvilinear Planar Beams and Stationary Temperature and Electrical FieldsBeams with Elasto-Plastic DeformationsMulti-Layer BeamsTimoshenko and Sheremetev-Pelekh Beams:The Timoshenko BeamsThe Sheremetev-Pelekh BeamsConcluding RemarksPanels:Infinite Length PanelsCylindrical Panels of Infinite LengthPlates and Shells:Plates with Initial ImperfectionsFlexible Axially-Symmetric Shells Readership: Post-graduate and doctoral students, applied mathematicians, physicists, mechanical and civil engineers. Key Features:Includes fascinating and rich dynamics exhibited by simple structural members and by the solution properties of the governing 1D non-linear PDEs, suitable for applied mathematicians and physicistsCovers a wide variety of the studied PDEs, their validated reduction to ODEs, classical and non-classical methods of analysis, influence of various boundary conditions and control parameters, as well as the illustrative presentation of the obtained results in the form of colour 2D and 3D figures and vibration type charts and scalesContains originally discovered, illustrated and discussed novel and/or modified classical scenarios of transition from regular to chaotic dynamics exhibited by 1D structural members, showing a way to control chaotic and bifurcational dynamics, with directions to study other dynamical systems modeled by chains of nonlinear oscillators

Nonsmooth Dynamics of Contacting Thermoelastic Bodies

Author : Jan Awrejcewicz
Publisher : Springer Science & Business Media
ISBN 13 : 0387096531
Total Pages : 256 pages
Book Rating : 4.3/5 (87 download)

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Book Synopsis Nonsmooth Dynamics of Contacting Thermoelastic Bodies by : Jan Awrejcewicz

Download or read book Nonsmooth Dynamics of Contacting Thermoelastic Bodies written by Jan Awrejcewicz and published by Springer Science & Business Media. This book was released on 2008-12-10 with total page 256 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work is devoted to an intensive study in contact mechanics, treating the nonsmooth dynamics of contacting bodies. Mathematical modeling is illustrated and discussed in numerous examples of engineering objects working in different kinematic and dynamic environments. Topics covered in five self-contained chapters examine non-steady dynamic phenomena which are determined by key factors: i.e., heat conduction, thermal stresses, and the amount of wearing. New to this monograph is the importance of the inertia factor, which is considered on par with thermal stresses. Nonsmooth Dynamics of Contacting Thermoelastic Bodies is an engaging accessible practical reference for engineers (civil, mechanical, industrial) and researchers in theoretical and applied mechanics, applied mathematics, physicists, and graduate students.

New Advances in Mechanisms, Mechanical Transmissions and Robotics

Author : Burkhard Corves
Publisher : Springer
ISBN 13 : 3319454501
Total Pages : 497 pages
Book Rating : 4.3/5 (194 download)

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Book Synopsis New Advances in Mechanisms, Mechanical Transmissions and Robotics by : Burkhard Corves

Download or read book New Advances in Mechanisms, Mechanical Transmissions and Robotics written by Burkhard Corves and published by Springer. This book was released on 2016-09-30 with total page 497 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume presents the proceedings of the Joint International Conference of the XII International Conference on Mechanisms and Mechanical Transmissions (MTM) and the XXIII International Conference on Robotics (Robotics ’16), that was held in Aachen, Germany, October 26th-27th, 2016. It contains applications of mechanisms and transmissions in several modern technical fields such as mechatronics, biomechanics, machines, micromachines, robotics and apparatus. In connection with these fields, the work combines the theoretical results with experimental testing. The book presents reviewed papers developed by researchers specialized in mechanisms analysis and synthesis, dynamics of mechanisms and machines, mechanical transmissions, biomechanics, precision mechanics, mechatronics, micromechanisms and microactuators, computational and experimental methods, CAD in mechanism and machine design, mechanical design of robot architecture, parallel robots, mobile robots, micro and nano robots, sensors and actuators in robotics, intelligent control systems, biomedical engineering, teleoperation, haptics, and virtual reality.

Asymptotic Multiple Scale Method in Time Domain

Author : Jan Awrejcewicz
Publisher : CRC Press
ISBN 13 : 100058125X
Total Pages : 411 pages
Book Rating : 4.0/5 (5 download)

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Book Synopsis Asymptotic Multiple Scale Method in Time Domain by : Jan Awrejcewicz

Download or read book Asymptotic Multiple Scale Method in Time Domain written by Jan Awrejcewicz and published by CRC Press. This book was released on 2022-05-10 with total page 411 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book offers up novel research which uses analytical approaches to explore nonlinear features exhibited by various dynamic processes. Relevant to disciplines across engineering and physics, the asymptotic method combined with the multiple scale method is shown to be an efficient and intuitive way to approach mechanics. Beginning with new material on the development of cutting-edge asymptotic methods and multiple scale methods, the book introduces this method in time domain and provides examples of vibrations of systems. Clearly written throughout, it uses innovative graphics to exemplify complex concepts such as nonlinear stationary and nonstationary processes, various resonances and jump pull-in phenomena. It also demonstrates the simplification of problems through using mathematical modelling, by employing the use of limiting phase trajectories to quantify nonlinear phenomena. Particularly relevant to structural mechanics, in rods, cables, beams, plates and shells, as well as mechanical objects commonly found in everyday devices such as mobile phones and cameras, the book shows how each system is modelled, and how it behaves under various conditions. It will be of interest to engineers and professionals in mechanical engineering and structural engineering, alongside those interested in vibrations and dynamics. It will also be useful to those studying engineering maths and physics.

Replication of Chaos in Neural Networks, Economics and Physics

Author : Marat Akhmet
Publisher : Springer
ISBN 13 : 3662475006
Total Pages : 457 pages
Book Rating : 4.6/5 (624 download)

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Book Synopsis Replication of Chaos in Neural Networks, Economics and Physics by : Marat Akhmet

Download or read book Replication of Chaos in Neural Networks, Economics and Physics written by Marat Akhmet and published by Springer. This book was released on 2015-08-13 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents detailed descriptions of chaos for continuous-time systems. It is the first-ever book to consider chaos as an input for differential and hybrid equations. Chaotic sets and chaotic functions are used as inputs for systems with attractors: equilibrium points, cycles and tori. The findings strongly suggest that chaos theory can proceed from the theory of differential equations to a higher level than previously thought. The approach selected is conducive to the in-depth analysis of different types of chaos. The appearance of deterministic chaos in neural networks, economics and mechanical systems is discussed theoretically and supported by simulations. As such, the book offers a valuable resource for mathematicians, physicists, engineers and economists studying nonlinear chaotic dynamics.

Advanced Topics on Cellular Self-organizing Nets and Chaotic Nonlinear Dynamics to Model and Control Complex Systems

Author : Riccardo Caponetto
Publisher : World Scientific
ISBN 13 : 9812814051
Total Pages : 208 pages
Book Rating : 4.8/5 (128 download)

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Book Synopsis Advanced Topics on Cellular Self-organizing Nets and Chaotic Nonlinear Dynamics to Model and Control Complex Systems by : Riccardo Caponetto

Download or read book Advanced Topics on Cellular Self-organizing Nets and Chaotic Nonlinear Dynamics to Model and Control Complex Systems written by Riccardo Caponetto and published by World Scientific. This book was released on 2008 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the research topics investigated during the three-year research project funded by the Italian Ministero dell'Istruzione, dell'Universite e della Ricerca (MIUR: Ministry of Education, University and Research) under the FIRB project RBNE01CW3M. With the aim of introducing newer perspectives of the research on complexity, the final results of the project are presented after a general introduction to the subject. The book is intended to provide researchers, PhD students, and people involved in research projects in companies with the basic fundamentals of complex systems and the advanced project results recently obtained.

Ordinary Differential Equations and Mechanical Systems

Author : Jan Awrejcewicz
Publisher : Springer
ISBN 13 : 3319076590
Total Pages : 614 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Ordinary Differential Equations and Mechanical Systems by : Jan Awrejcewicz

Download or read book Ordinary Differential Equations and Mechanical Systems written by Jan Awrejcewicz and published by Springer. This book was released on 2014-09-17 with total page 614 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book applies a step-by-step treatment of the current state-of-the-art of ordinary differential equations used in modeling of engineering systems/processes and beyond. It covers systematically ordered problems, beginning with first and second order ODEs, linear and higher-order ODEs of polynomial form, theory and criteria of similarity, modeling approaches, phase plane and phase space concepts, stability optimization and ending on chaos and synchronization. Presenting both an overview of the theory of the introductory differential equations in the context of applicability and a systematic treatment of modeling of numerous engineering and physical problems through linear and non-linear ODEs, the volume is self-contained, yet serves both scientific and engineering interests. The presentation relies on a general treatment, analytical and numerical methods, concrete examples and engineering intuition. The scientific background used is well balanced between elementary and advanced level, making it as a unique self-contained source for both theoretically and application oriented graduate and doctoral students, university teachers, researchers and engineers of mechanical, civil and mechatronic engineering.

Bifurcations in Piecewise-smooth Continuous Systems

Author : David John Warwick Simpson
Publisher : World Scientific
ISBN 13 : 9814293857
Total Pages : 255 pages
Book Rating : 4.8/5 (142 download)

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Book Synopsis Bifurcations in Piecewise-smooth Continuous Systems by : David John Warwick Simpson

Download or read book Bifurcations in Piecewise-smooth Continuous Systems written by David John Warwick Simpson and published by World Scientific. This book was released on 2010 with total page 255 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1. Fundamentals of piecewise-smooth, continuous systems. 1.1. Applications. 1.2. A framework for local behavior. 1.3. Existence of equilibria and fixed points. 1.4. The observer canonical form. 1.5. Discontinuous bifurcations. 1.6. Border-collision bifurcations. 1.7. Poincaré maps and discontinuity maps. 1.8. Period adding. 1.9. Smooth approximations -- 2. Discontinuous bifurcations in planar systems. 2.1. Periodic orbits. 2.2. The focus-focus case in detail. 2.3. Summary and classification -- 3. Codimension-two, discontinuous bifurcations. 3.1. A nonsmooth, saddle-node bifurcation. 3.2. A nonsmooth, Hopf bifurcation. 3.3. A codimension-two, discontinuous Hopf bifurcation -- 4. The growth of Saccharomyces cerevisiae. 4.1. Mathematical model. 4.2. Basic mathematical observations. 4.3. Bifurcation structure. 4.4. Simple and complicated stable oscillations -- 5. Codimension-two, border-collision bifurcations. 5.1. A nonsmooth, saddle-node bifurcation. 5.2. A nonsmooth, period-doubling bifurcation -- 6. Periodic solutions and resonance tongues. 6.1. Symbolic dynamics. 6.2. Describing and locating periodic solutions. 6.3. Resonance tongue boundaries. 6.4. Rotational symbol sequences. 6.5. Cardinality of symbol sequences. 6.6. Shrinking points. 6.7. Unfolding shrinking points -- 7. Neimark-Sacker-like bifurcations. 7.1. A two-dimensional map. 7.2. Basic dynamics. 7.3. Limiting parameter values. 7.4. Resonance tongues. 7.5. Complex phenomena relating to resonance tongues. 7.6. More complex phenomena

2-D Quadratic Maps and 3-D ODE Systems

Author :
Publisher :
ISBN 13 : 9814464791
Total Pages : pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis 2-D Quadratic Maps and 3-D ODE Systems by :

Download or read book 2-D Quadratic Maps and 3-D ODE Systems written by and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

The Chaotic Pendulum

Author :
Publisher :
ISBN 13 : 9814464244
Total Pages : pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis The Chaotic Pendulum by :

Download or read book The Chaotic Pendulum written by and published by . This book was released on with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Control of Chaos in Nonlinear Circuits and Systems

Author : Wing-Kuen Ling
Publisher : World Scientific
ISBN 13 : 9812790578
Total Pages : 281 pages
Book Rating : 4.8/5 (127 download)

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Book Synopsis Control of Chaos in Nonlinear Circuits and Systems by : Wing-Kuen Ling

Download or read book Control of Chaos in Nonlinear Circuits and Systems written by Wing-Kuen Ling and published by World Scientific. This book was released on 2009 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, leading researchers present their current work in the challenging area of chaos control in nonlinear circuits and systems, with emphasis on practical methodologies, system design techniques and applications. A combination of overview, tutorial and technical articles, the book describes state-of-the-art research on significant problems in this area. The scope and aim of this book are to bridge the gap between chaos control methods and circuits and systems. It is an ideal starting point for anyone who needs a fundamental understanding of controlling chaos in nonlinear circuits and systems.

Classical Mechanics

Author : Jan Awrejcewicz
Publisher : Springer Science & Business Media
ISBN 13 : 1461437407
Total Pages : 475 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Classical Mechanics by : Jan Awrejcewicz

Download or read book Classical Mechanics written by Jan Awrejcewicz and published by Springer Science & Business Media. This book was released on 2012-07-12 with total page 475 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second volume of three books devoted to Mechanics. In this book, dynamical and advanced mechanics problems are stated, illustrated, and discussed, including a few novel concepts in comparison to standard text books and monographs. Apart from being addressed to a wide spectrum of graduate students, postgraduate students, researchers, and teachers from the fields of mechanical and civil engineering, this volume is also intended to be used as a self-contained material for applied mathematicians and physical scientists and researchers.

Numerical Simulation Of Waves And Fronts In Inhomogeneous Solids

Author : Gerard A Maugin
Publisher : World Scientific
ISBN 13 : 9814470481
Total Pages : 236 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Numerical Simulation Of Waves And Fronts In Inhomogeneous Solids by : Gerard A Maugin

Download or read book Numerical Simulation Of Waves And Fronts In Inhomogeneous Solids written by Gerard A Maugin and published by World Scientific. This book was released on 2008-06-13 with total page 236 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book shows the advanced methods of numerical simulation of waves and fronts propagation in inhomogeneous solids and introduces related important ideas associated with the application of numerical methods for these problems. Great care has been taken throughout the book to seek a balance between the thermomechanical analysis and numerical techniques. It is suitable for advanced undergraduate and graduate courses in continuum mechanics and engineering. Necessary prerequisites for this text are basic continuum mechanics and thermodynamics. Some elementary knowledge of numerical methods for partial differential equations is also preferable.

A Gallery of Chua Attractors

Author : Eleonora Bilotta
Publisher : World Scientific
ISBN 13 : 9812790624
Total Pages : 607 pages
Book Rating : 4.8/5 (127 download)

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Book Synopsis A Gallery of Chua Attractors by : Eleonora Bilotta

Download or read book A Gallery of Chua Attractors written by Eleonora Bilotta and published by World Scientific. This book was released on 2008 with total page 607 pages. Available in PDF, EPUB and Kindle. Book excerpt: Chaos is considered as one of the most important concepts in modern science. It originally appeared only in computer simulation (the famous Lorenz equation of 1963), but this changed with the introduction of Chua's oscillator (1986) — a simple electronic circuit with the ability to generate a vast range of chaotic behaviors. With Chua's circuit, chaos became a physical phenomenon, readily understood and represented in mathematical language. Yet, even so, it is still difficult for the non-specialist to appreciate the full variety of behaviors that the system can produce.This book aims to bridge the gap. A gallery of nearly 900 “chaotic attractors” — some generated by Chua's physical circuit, the majority through computer simulation of the circuit and its generalizations — are illustrated as 3D color images, time series and fast Fourier transform algorithms. For interested researchers, also presented is the information necessary to replicate the behaviors and images. Finally, how the fractal richness can be plied to artistic ends in generating music and interesting sounds is shown; some examples are included in the DVD-ROM which comes with the book.The contents have also appeared in the International Journal of Bifurcation and Chaos (2007).