Download or read book Differential Geometry Applied To Dynamical Systems With Cd rom written by Jean-marc Ginoux and published by World Scientific. This book was released on 2009-04-03 with total page 341 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book aims to present a new approach called Flow Curvature Method that applies Differential Geometry to Dynamical Systems. Hence, for a trajectory curve, an integral of any n-dimensional dynamical system as a curve in Euclidean n-space, the curvature of the trajectory — or the flow — may be analytically computed. Then, the location of the points where the curvature of the flow vanishes defines a manifold called flow curvature manifold. Such a manifold being defined from the time derivatives of the velocity vector field, contains information about the dynamics of the system, hence identifying the main features of the system such as fixed points and their stability, local bifurcations of codimension one, center manifold equation, normal forms, linear invariant manifolds (straight lines, planes, hyperplanes).In the case of singularly perturbed systems or slow-fast dynamical systems, the flow curvature manifold directly provides the slow invariant manifold analytical equation associated with such systems. Also, starting from the flow curvature manifold, it will be demonstrated how to find again the corresponding dynamical system, thus solving the inverse problem.
Download or read book Differential Geometry and Topology written by Keith Burns and published by CRC Press. This book was released on 2005-05-27 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow. Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models. The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow. The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field.
Download or read book Dynamical Systems and Differential Geometry via MAPLE written by Constantin Udriste and published by Cambridge Scholars Publishing. This book was released on 2021-10-01 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: The area of dynamical systems and differential geometry via MAPLE is a field which has become exceedingly technical in recent years. In the field, everything is structured for the benefit of optimizing evolutionary geometric aspects that describe significant physical or engineering phenomena. This book is structured in terms of the importance, accessibility and impact of theoretical notions capable of shaping a future mathematician-computer scientist possessing knowledge of evolutionary dynamical systems. It provides a self-contained and accessible introduction for graduate and advanced undergraduate students in mathematics, engineering, physics, and economic sciences. This book is suitable for both self-study for students and professors with a background in differential geometry and for teaching a semester-long introductory graduate course in dynamical systems and differential geometry via MAPLE.
Download or read book Dynamical systems and differential geometry written by and published by . This book was released on 2000 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Dynamical Systems and Differential Geometry Via MAPLE written by Constantin Udriste and published by . This book was released on 2021-10 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: The area of dynamical systems and differential geometry via MAPLE is a field which has become exceedingly technical in recent years. In the field, everything is structured for the benefit of optimizing evolutionary geometric aspects that describe significant physical or engineering phenomena. This book is structured in terms of the importance, accessibility and impact of theoretical notions capable of shaping a future mathematician-computer scientist possessing knowledge of evolutionary dynamical systems. It provides a self-contained and accessible introduction for graduate and advanced undergraduate students in mathematics, engineering, physics, and economic sciences. This book is suitable for both self-study for students and professors with a background in differential geometry and for teaching a semester-long introductory graduate course in dynamical systems and differential geometry via MAPLE.
Download or read book Advances in Discrete Differential Geometry written by Alexander I. Bobenko and published by Springer. This book was released on 2016-08-12 with total page 441 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture. This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing.
Download or read book Dynamical Systems and Differential Geometry written by and published by . This book was released on 2006 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Modeling by Nonlinear Differential Equations written by Paul E. Phillipson and published by World Scientific. This book was released on 2009 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This book aims to provide mathematical analyses of nonlinear differential equations, which have proved pivotal to understanding many phenomena in physics, chemistry and biology. Topics of focus are autocatalysis and dynamics of molecular evolution, relaxation oscillations, deterministic chaos, reaction diffusion driven chemical pattern formation, solitons and neuron dynamics. Included is a discussion of processes from the viewpoints of reversibility, reflected by conservative classical mechanics, and irreversibility introduced by the dissipative role of diffusion. Each chapter presents the subject matter from the point of one or a few key equations, whose properties and consequences are amplified by approximate analytic solutions that are developed to support graphical display of exact computer solutions."--back cover.
Download or read book Applied Differential Geometry written by William L. Burke and published by Cambridge University Press. This book was released on 1985-05-31 with total page 440 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.
Download or read book Fractional Order Systems Modeling And Control Applications written by Riccardo Caponetto and published by World Scientific. This book was released on 2010-02-10 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book aims to propose the implementation and application of Fractional Order Systems (FOS). It is well known that FOS can be utilized in control applications and systems modeling, and their effectiveness has been proven in many theoretical works and simulation routines. A further and mandatory step for FOS real world utilization is their hardware implementation and applications on real systems modeling. With this viewpoint, introductory chapters are included on the definition of stability region of Fractional Order PID Controller and Chaotic FOS, followed by the practical implementation based on Microcontroller, Field Programmable Gate Array, Field Programmable Analog Array and Switched Capacitor. Another section is dedicated to FO modeling of Ionic Polymeric Metal Composite (IPMC). This new material will have applications in robotics, aerospace and biomedicine.
Download or read book Discrete Systems with Memory written by Ramon Alonso-Sanz and published by World Scientific. This book was released on 2011 with total page 478 pages. Available in PDF, EPUB and Kindle. Book excerpt: Memory is a universal function of organized matter. What is the mathematics of memory? How does memory affect the space-time behaviour of spatially extended systems? Does memory increase complexity? This book provides answers to these questions. It focuses on the study of spatially extended systems, i.e., cellular automata and other related discrete complex systems. Thus, arrays of locally connected finite state machines, or cells, update their states simultaneously, in discrete time, by the same transition rule. The classical dynamics in these systems is Markovian: only the actual configuration is taken into account to generate the next one. Generalizing the conventional view on spatially extended discrete dynamical systems evolution by allowing cells (or nodes) to be featured by some trait state computed as a function of its own previous state-values, the transition maps of the classical systems are kept unaltered, so that the effect of memory can be easily traced. The book demonstrates that discrete dynamical systems with memory are not only priceless tools for modeling natural phenomena but unique mathematical and aesthetic objects.
Download or read book 2 D Quadratic Maps and 3 D ODE Systems written by Elhadj Zeraoulia and published by World Scientific. This book was released on 2010 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hnon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hnon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincar map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincar mapping in addition to other analytical methods.
Download or read book A Nonlinear Dynamics Perspective of Wolfram s New Kind of Science written by Leon O. Chua and published by World Scientific. This book was released on 2011-03-30 with total page 405 pages. Available in PDF, EPUB and Kindle. Book excerpt: Annotation This text introduces cellular automata from a rigorous nonlinear dynamics perspective. It supplies the missing link between nonlinear differential and difference equations to discrete symbolic analysis. It provides an analysis, and classification of the empirical results presented in Wolfram's 'New Kind of Science'.
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
Download or read book Mathematical Mechanics written by Ellis D. Cooper and published by World Scientific. This book was released on 2011 with total page 390 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unprecedented book offers all the details of the mathematical mechanics underlying state-of-the-art modeling of skeletal muscle contraction. The aim is to provide an integrated vision of mathematics, physics, chemistry and biology for this one understanding. The method is to take advantage of modern mathematical technology — Eilenberg-Mac Lane category theory, Robinson infinitesimal calculus and Kolmogorov probability theory — to examine a succession of distinguishable universes of particles, and continuous, thermodynamic, chemical, and molecular bodies, all with a focus on proofs by algebraic calculation without set theory. Also provided are metaphors and analogies, and careful distinction between representational pictures, mental model drawings, and mathematical diagrams.High school mathematics teachers, undergraduate and graduate college students, and researchers in mathematics, physics, chemistry, and biology may use this integrated publication to broaden their perspective on science, and to experience the precision that mathematical mechanics brings to understanding the muscular mechanism of nearly all animal behavior.
Download or read book Physarum Machines written by Andrew Adamatzky and published by World Scientific. This book was released on 2010 with total page 281 pages. Available in PDF, EPUB and Kindle. Book excerpt: A Physarum machine is a programmable amorphous biological computer experimentally implemented in the vegetative state of true slime mould Physarum polycephalum. It comprises an amorphous yellowish mass with networks of protoplasmic veins, programmed by spatial configurations of attracting and repelling gradients. This book demonstrates how to create experimental Physarum machines for computational geometry and optimization, distributed manipulation and transportation, and general-purpose computation. Being very cheap to make and easy to maintain, the machine also functions on a wide range of substrates and in a broad scope of environmental conditions. As such a Physarum machine is a 'green' and environmentally friendly unconventional computer. The book is readily accessible to a nonprofessional reader, and is a priceless source of experimental tips and inventive theoretical ideas for anyone who is inspired by novel and emerging non-silicon computers and robots. An account on Physarum Machines can be viewed at http: //www.youtube.com/user/PhysarumMachines.
Download or read book Nonlinear Dynamics Perspective Of Wolfram s New Kind Of Science A Volume Iii written by Leon O Chua and published by World Scientific. This book was released on 2009-08-11 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: Volume III continues the author's quest for developing a pedagogical, self-contained, yet rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. Using carefully conceived and illuminating color graphics, the global dynamical behaviors of the 50 (out of 256) local rules that have not yet been covered in Volumes I and II are exposed via their stunningly revealing basin tree diagrams. The Bernoulli στ-shift dynamics discovered in Volume II is generalized to hold for all 50 (or 18 globally equivalent) local rules via complex and hyper Bernoulli wave dynamics. Explicit global state transition formulas derived for rules 60, 90, 105, and 150 reveal a new scale-free phenomenon. The most surprising new result unveiled in this volume is the “Isle of Eden” found hidden in most (almost 90%) of the 256 local rules. Readers are challenged to hunt for long-period, isolated Isles of Eden. These are rare gems waiting to be discovered.