Stephen Lynch Dynamical Systems with Applications using MATLAB® Stephen Lynch Department of Computing and Mathematics, Manchester Metropolitan University School of Computing, Mathematics & Digital Technology, Manchester, UK ISBN 978-3-319-06819-0 e-ISBN 978-3-319-06820-6 DOI 10.1007/978-3-319-06820-6 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2014941703 Mathematics Subject Classification (2010): 37-01, 49K15, 78A60, 28A80, 80A30, 34H10, 34K18, 70K05, 34C07, 34D06, 92B20, 94C05 © Springer International Publishing Switzerland 2014 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) Preface Since the first printing of this book in 2004, MATLAB ® ;  has evolved from MATLAB version to MATLAB version R2014b, where presently, the package is updated twice a year Accordingly, the second edition has been thoroughly updated and new material has been added In this edition, there are many more applications, examples, and exercises, all with solutions, and new sections on series solutions of ordinary differential equations, perturbation methods, normal forms, Gröbner bases, and chaos synchronization have been added There are also new chapters on image processing and binary oscillator computing This book provides an introduction to the theory of dynamical systems with the aid of MATLAB, the Symbolic Math Toolbox TM , the Image Processing Toolbox TM , and Simulink TM It is written for both senior undergraduates and graduate students Chapter provides a tutorial introduction to MATLAB—new users should go through this chapter carefully whilst those moderately familiar and experienced users will find this chapter a useful source of reference Chapters – deals with discrete systems, Chaps.  – 17 are devoted to the study of continuous systems using ordinary differential equations and Chaps.  18 – 21 deal with both continuous and discrete systems Chapter 22 lists three MATLAB-based examinations to be sat in a computer laboratory with access to MATLAB and Chap.  23 lists answers to all of the exercises given in the book It should be pointed out that dynamical systems theory is not limited to these topics but also encompasses partial differential equations, integral and integro-differential equations, stochastic systems, and time-delay systems, for instance References [1–5] given at the end of the Preface provide more information for the interested reader The author has gone for breadth of coverage rather than fine detail and theorems with proofs are kept at a minimum The material is not clouded by functional analytic and group theoretical definitions and so is intelligible to readers with a general mathematical background Some of the topics covered are scarcely covered elsewhere Most of the material in Chaps. 7, 10, 11, 15–19, and 21 is at postgraduate level and has been influenced by the author’s own research interests There is more theory in these chapters than in the rest of the book since it is not easily accessed anywhere else It has been found that these chapters are especially useful as reference material for senior undergraduate project work The theory in other chapters of the book is dealt with more comprehensively in other texts, some of which may be found in the references section of the corresponding chapter The book has a very hands-on approach and takes the reader from the basic theory right through to recently published research material MATLAB is extremely popular with a wide range of researchers from all sorts of disciplines, it has a very user-friendly interface and has extensive visualization and numerical computation capabilities It is an ideal package to adopt for the study of nonlinear dynamical systems; the numerical algorithms work very quickly, and complex pictures can be plotted within seconds The Simulink accessory to MATLAB is used for simulating dynamical processes It is as close as one can get to building apparatus and investigating the output for a given input without the need for an actual physical model For this reason, Simulink is very popular in the field of engineering Note that the latest student version of MATLAB comes with a generous additional ten toolboxes including the Symbolic Math Toolbox, the Image Processing Toolbox, and Simulink The first chapter provides an efficient tutorial introduction to MATLAB New users will find the tutorials will enable them to become familiar with MATLAB within a few hours Both engineering and mathematics students appreciate this method of teaching and I have found that it generally works well with one staff member to about twenty students in a computer laboratory In most cases, I have chosen to list the MATLAB program files at the end of each chapter, this avoids unnecessary cluttering in the text The MATLAB programs have been kept as simple as possible and should run under later versions of the package The MATLAB program files and Simulink model files (including updates) can even be downloaded from the Web at http://​www.​mathworks.​com/​matlabcentral/​ fileexchange/​authors/​6536 Readers will find that they can reproduce the figures given in the text, and then it is not too difficult to change parameters or equations to investigate other systems Readers may be interested to hear that the MATLAB and Simulink model files have been downloaded over 30,000 times since they were uploaded in 2002 and that these files were selected as the MATLAB Central Pick of the Week in July 2013 Chapters – deal with discrete dynamical systems Chapter starts with a general introduction to iteration and linear recurrence (or difference) equations The bulk of the chapter is concerned with the Leslie model used to investigate the population of a single species split into different age classes Harvesting and culling policies are then investigated and optimal solutions are sought Nonlinear discrete dynamical systems are dealt with in Chap.  Bifurcation diagrams, chaos, intermittency, Lyapunov exponents, periodicity, quasiperiodicity, and universality are some of the topics introduced The theory is then applied to real-world problems from a broad range of disciplines including population dynamics, biology, economics, nonlinear optics, and neural networks Chapter is concerned with complex iterative maps; Julia sets and the now-famous Mandelbrot set are plotted Basins of attraction are investigated for these complex systems As a simple introduction to optics, electromagnetic waves and Maxwell’s equations are studied at the beginning of Chap.  Complex iterative equations are used to model the propagation of light waves through nonlinear optical fibers A brief history of nonlinear bistable optical resonators is discussed and the simple fiber ring resonator is analyzed in particular Chapter is devoted to the study of these optical resonators and phenomena such as bistability, chaotic attractors, feedback, hysteresis, instability, linear stability analysis, multistability, nonlinearity, and steady states are discussed The first and second iterative methods are defined in this chapter Some simple fractals may be constructed using pencil and paper in Chap.  , and the concept of fractal dimension is introduced Fractals may be thought of as identical motifs repeated on ever-reduced scales Unfortunately, most of the fractals appearing in nature are not homogeneous but are more heterogeneous, hence the need for the multifractal theory given later in the chapter It has been found that the distribution of stars and galaxies in our universe is multifractal, and there is even evidence of multifractals in rainfall, stock markets, and heartbeat rhythms Applications in materials science, geoscience, and image processing are briefly discussed Chapter provides a brief introduction to the Image Processing Toolbox TM which is being used more and more by a diverse range of scientific disciplines, especially medical imaging The fast Fourier transform is introduced and has a wide range of applications throughout the realms of science Chapters – 17 deal with continuous dynamical systems Chapters and cover some theory of ordinary differential equations and applications to models in the real world are given The theory of differential equations applied to chemical kinetics and electric circuits is introduced in some detail The memristor is introduced and one of the most remarkable stories in the history of mathematics is relayed Chapter ends with the existence and uniqueness theorem for the solutions of certain types of differential equations A variety of numerical procedures are available in MATLAB when solving stiff and nonstiff systems when an analytic solution does not exist or is extremely difficult to find The theory behind the construction of phase plane portraits for two-dimensional systems is dealt with in Chap.  Applications are taken from chemical kinetics, economics, electronics, epidemiology, mechanics, and population dynamics The modeling of the populations of interacting species is discussed in some detail in Chap.  10 and domains of stability are discussed for the first time Limit cycles, or isolated periodic solutions, are introduced in Chap.  11 Since we live in a periodic world, these are the most common type of solution found when modeling nonlinear dynamical systems They appear extensively when modeling both the technological and natural sciences Hamiltonian, or conservative, systems and stability are discussed in Chaps. 12, and 13 is concerned with how planar systems vary depending upon a parameter Bifurcation, bistability, multistability, and normal forms are discussed The reader is first introduced to the concept of chaos in continuous systems in Chaps.  14 and 15 , where three-dimensional systems and Poincaré maps are investigated These higher-dimensional systems can exhibit strange attractors and chaotic dynamics One can rotate the three-dimensional objects in MATLAB and plot time series plots to get a better understanding of the dynamics involved Once again, the theory can be applied to chemical kinetics (including stiff systems), electric circuits, and epidemiology; a simplified model for the weather is also briefly discussed Chapter 15 deals with Poincaré first return maps that can be used to untangle complicated interlacing trajectories in higher-dimensional spaces A periodically driven nonlinear pendulum is also investigated by means of a nonautonomous differential equation Both local and global bifurcations are investigated in Chap.  16 The main results and statement of the famous second part of David Hilbert’s sixteenth problem are listed in Chap.  17 In order to understand these results, Poincaré compactification is introduced The study of continuous systems ends with one of the author’s specialities—limit cycles of Liénard systems There is some detail on Liénard systems, in particular, in this part of the book, but they have a ubiquity for systems in the plane A brief introduction to the enticing field of neural networks is presented in Chap.  18 Imagine trying to make a computer mimic the human brain One could ask the question: In the future will it be possible for computers to think and even be conscious? The human brain will always be more powerful than traditional, sequential, logic-based digital computers and scientists are trying to incorporate some features of the brain into modern computing Neural networks perform through learning and no underlying equations are required Mathematicians and computer scientists are attempting to mimic the way neurons work together via synapses; indeed, a neural network can be thought of as a crude multidimensional model of the human brain The expectations are high for future applications in a broad range of disciplines Neural networks are already being used in machine learning and pattern recognition (computer vision, credit card fraud, prediction and forecasting, disease recognition, facial and speech recognition), the consumer home entertainment market, psychological profiling, predicting wave over-topping events, and control problems, for example They also provide a parallel architecture allowing for very fast computational and response times In recent years, the disciplines of neural networks and nonlinear dynamics have increasingly coalesced and a new branch of science called neurodynamics is emerging Lyapunov functions can be used to determine the stability of certain types of neural network There is also evidence of chaos, feedback, nonlinearity, periodicity, and chaos synchronization in the brain Chapter 19 is devoted to the new and exciting theory behind chaos control and synchronization For most systems, the maxim used by engineers in the past has been “stability good, chaos bad,” but more and more nowadays this is being replaced with “stability good, chaos better.” There are exciting and novel applications in cardiology, communications, engineering, laser technology, and space research, for example Chapter 20 focuses on binary oscillator computing, the subject of UK, International and Taiwanese patents The author and his coinventor, Jon Borresen, came up with the idea when modeling connected biological neurons Binary oscillator technology can be applied to the design of arithmetic logic units (ALUs), memory, and other basic computing components It has the potential to provide revolutionary computational speedup, energy saving, and novel applications and may be applicable to a variety of technological paradigms including biological neurons, complementary metal–oxide–semiconductor (CMOS), memristors, optical oscillators, and superconducting materials The research has the potential for MMU and industrial partners to develop super fast, low-power computers and may provide an assay for neuronal degradation for brain malfunctions such as Alzheimer’s, epilepsy, and Parkinson’s disease! Examples of Simulink models, referred to in earlier chapters of the book, are presented in Chap.  21 It is possible to change the type of input into the system, or parameter values, and investigate the output very quickly This is as close as one can get to experimentation without the need for expensive equipment Three examination-type papers are listed in Chap.  22 and a complete set of solutions is listed in Chap.  23 Both textbooks and research papers are presented in the list of references The textbooks can be used to gain more background material, and the research papers have been given to encourage further reading and independent study This book is informed by the research interests of the author which are currently nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing Some references include recently published research articles by the author along with an international patent The prerequisites for studying dynamical systems using this book are undergraduate courses in linear algebra, real and complex analysis, calculus, and ordinary differential equations; a knowledge of a computer language such as C or Fortran would be beneficial but not essential Recommended Textbooks M.P Coleman, An Introduction to Partial Differential Equations with MATLAB , 2nd edn Chapman and Hall/CRC Applied Mathematics and Nonlinear Science (Chapman and Hall/CRC, New York, 2013) B Bhattacharya, M Majumdar, Random Dynamical Systems: Theory and Applications (Cambridge University Press, Cambridge, 2007) R Sipahi, T Vyhlídal, S Niculescu, P Pepe (eds.), Time Delay Systems: Methods, Applications and New Trends Lecture Notes in Control and Information Sciences (Springer, New York, 2012) V Volterra, Theory of Functionals and of Integral and Integro-Differential Equations (Dover Publications, New York, 2005) J Mallet-Paret, J Wu, H Zhu, Y Yi, Infinite Dimensional Dynamical Systems (Fields Institute Communications) (Springer, New York, 2013) Instructors may be interested in two new features in MATLAB, namely, Cody Coursework TM and MATLAB apps Instructors can automate grading of MATLAB coursework assignments using Cody Coursework; this is a free resource available to use for faculty who have an active site licence MATLAB apps enable the user to perform computational tasks interactively and many of the toolboxes come with apps More details can be found on the relevant MathWorks web pages I would like to express my sincere thanks to MathWorks for supplying me with the latest versions of MATLAB and its toolboxes Thanks also go to all of the reviewers from the editions of the Maple and Mathematica books Special thanks go to Birkhäuser and Springer publishers Finally, thanks to my family, especially my wife, Gaynor, and our children, Sebastian and Thalia, for their continuing love, inspiration, and support Stephen Lynch Manchester, UK Contents A Tutorial Introduction to MATLAB 1.​1 Tutorial One:​ The Basics and the Symbolic Math Toolbox (1 h) 1.​2 Tutorial Two:​ Plots and Differential Equations (1 h) 1.​3 MATLAB Program Files or M-Files 1.​4 Hints for Programming 1.​5 MATLAB Exercises References Linear Discrete Dynamical Systems 2.​1 Recurrence Relations 2.​2 The Leslie Model 2.​3 Harvesting and Culling Policies 2.​4 MATLAB Commands 2.​5 Exercises References Nonlinear Discrete Dynamical Systems 3.​1 The Tent Map and Graphical Iterations 3.​2 Fixed Points and Periodic Orbits 3.​3 The Logistic Map, Bifurcation Diagram, and Feigenbaum Number 3.​4 Gaussian and Hénon Maps 3.​5 Applications 3.​6 MATLAB Program Files 3.​7 Exercises gain Gauss’s law electricity magnetism Gauss-Newton method Gaussian input pulse Gaussian map Gaussian pulse generalized delta rule generalized fractal dimension generalized fractal dimensions generalized mixed Rayleigh Liénard equations generalized synchronization global bifurcation globally asymptotically stable glucose in blood Gröbner bases gradient gradient vector graph block graphic graphical method gray scale Green’s theorem grey scale Gross National Product (GNP) H Hénon map Hénon-Heiles Hamiltonian Hamiltonian Hamiltonian systems with two degrees of freedom handcrafted patterns hard bifurcation Hartman’s theorem harvesting policy Hausdorff dimension Hausdorff index Hausdorff–Besicovich dimension Heaviside function Hebb’s learning law Hebb’s postulate of learning henry heteroclinic bifurcation orbit heteroclinic orbit heteroclinic tangle heterogeneous hidden layer Hilbert numbers history Hodgkin-Huxley equations Holling-Tanner model homoclinic bifurcation loop orbit tangle homogeneous homogeneous differential equation Hopf bifurcation singularity Hopfield network Hopfield neural network horseshoe dynamics host-parasite system human population hyperbolic attracting critical point fixed point repelling stable limit cycle unstable limit cycle hyperbolic iterated function system hyperpolarize hyperpolarized hysteresis I ideal Ikeda map image analysis image compression incident index inductance infected population infectives inflation unemployment model information dimension inhibitory initial value problem input vector insect population instability instant physician integrable integrate and fire neuron integrating factor integrator block intensity interacting species intermittency route to chaos invariant axes inverse discrete Fourier transform inverted Koch snowflake inverted Koch square isoclines isolated periodic solution isothermal chemical reaction iterated function system (IFS) iteration J Jacobian Jacobian matrix Jordan curve jth point of period i Julia set K KAM theorem tori kernel machines Kerr effect Kerr type kinetic energy Kirchhoff’s current law laws voltage law Koch snowflake Koch curve Koch square L ladybirds and aphids laminarize Landolt clock Laplace transform large-amplitude limit cycle bifurcation laser law of mass action learning process learning rate least mean squared (LMS) algorithm Legendre transformation Leslie matrix model lexicographical order Liénard equation Liénard plane Liénard system large parameter local results Liénard’s theorem lie detector limit cycle 3-D hyperbolic neuron nonexistence Lindstedt-Poincaré technique linear differential equation linear phase shift linear stability analysis linear transformation linearization linearized system Lipschitz continuous Lipschitz condition load local bifurcation log-log plot logic gates logistic equation logistic growth logistic map Lorenz attractor equations Lorenz equations loss in the fiber Lotka-Volterra model low pass filter low-gain saturation function lowest common multiple Lyapunov quantity stability Lyapunov domain of stability Lyapunov exponent Lyapunov function, 254, 257, 290, 336, 367 Hopfield network Lyapunov quantities Lyapunov stability theorem lynx and snowshoe hares M M-file magnetic field vector magnetic flux magnetostrictive ribbon Mandelbrot Mandelbrot set manifold MATLAB based exam MATLAB apps maximal interval of existence Maxwell’s equations Maxwell-Bloch equations Maxwell-Debye equations McCulloch-Pitts neuron mean infectivity period mean latency period mechanical oscillator mechanical system Melnikov function integral memory devices memristance memristor meteorology method of multiple scales method of steepest descent micro-parasite—zooplankton—fish system minimal chaotic neuromodule minimal Gröbner basis mixed fundamental memories mixing modulo monomial ordering mortgage assessment motif multidegree multifractal formalism Hénon map Sierpiński triangle spectra multistability multistable murder mutual exclusion mylein sheath N national income negative limit set negative semiorbit negatively invariant net reproduction rate network architecture neural network neurodynamics neuromodule neuron module neuronal model neurons neurotransmitters Newton’s law of cooling Newton’s law of motion Newton’s method noise NOLM with feedback nonautonomous system nonconvex closed curve nondegenerate critical point nondeterministic chaos nondeterministic system nonexistence of limit cycles nonhyperbolic critical point fixed point nonlinear center optics nonlinear optics nonlinear phase shift nonlinear refractive index coefficient nonlinearity nonperiodic behavior nonsimple canonical system normal form normalized eigenvector not robust numeric::solve Numerical solutions O occasional proportional feedback (OPF) OGY method ohm Ohm’s law optical bistability computer fiber fiber double ring memories resonator sensor optimal sustainable orbit ordinary differential equation oscillation of a violin string output vector ozone production P parasitic infection partial differential equations partition function Pascal’s triangle circuit Peixoto’s theorem in the plane pendulum perceptron period bubblings limit cycle of limit cycle undoublings period-doubling period-doubling bifurcations to chaos period-n cycle period-one behavior period-three behavior period-two period-two behavior periodic orbit windows periodic behavior periodicity permittivity of free space perturbation methods phase portrait phase shift physiology piecewise linear function pinched hysteresis pitchfork bifurcation pixels planar manifold plastics Poincaré section Poincaré compactification Poincaré map Poincaré-Bendixson theorem, 229, 321, 364 Poisson brackets polar coordinates pole placement technique pollution polymer population population model population of rabbits positive limit set positive semiorbit positively invariant potato man potential difference potential energy potential function power of a waterwheel spectra power law power-splitting ratio Prandtl number preallocate predation rate predator-prey models system probe vector propagation psychological profiling Pyragas’s method Q qth moment qualitative behavior qualitatively equivalent quasi-periodicity quasiperiodic route to chaos quasiperiodic forcing R Rössler attractor Rössler system radioactive decay random behavior rate constant rate-determining steps rationally independent Rayleigh number Rayleigh system rate equation real distinct eigenvalues recurrence relation recurrent neural network red and grey squirrels red blood cells reduced reduced Gröbner basis reflected refractive index refractive nonlinearity refuge regulator poles relative permeabilities relative permittivities repeated real eigenvalues repolarization resistance resonance terms resonant response system restoring coefficient restoring force restrictions in programming return map constant reversed fundamental memories ring ringing RLC circuit robust rubbers S S-polynomial saddle point saddle-node bifurcation safe bifurcation scaling scope script file sea lions and penguins seasonal effects seasonality second iterative method second order linear difference equation second part of Hilbert’s sixteenth problem differential equation secular term sedimentary rocks self-similar self-similar fractal self-similarity semistable limit cycle semistable critical point sensitivity to initial conditions, 34, 39, 294 separable differential equation separation of variables separatrix cycle series solutions SFR resonator sharks and fish Sierpiński triangle sigmoid function signal processing simple canonical system simple nonlinear pendulum simply connected domain simulation singlet singular node Smale horseshoe map Smale-Birkhoff theorem perturbation small perturbation small-amplitude limit cycle soft bifurcation solar system solution curves soma spatial vector spectrum of Lyapunov exponents speed of light spike train spin-glass states spirals spurious steady state SR flip-flop stability diagram stable critical point fixed point focus limit cycle manifold node stable critical point staircases stationary point steady state stiff system stiffness stochastic methods stock market analysis stoichiometric equations Stokes’s theorem strange attractor stretching and folding strictly dominant structurally stable unstable subcritical Hopf bifurcation subharmonic oscillations summing junction supercritical Hopf bifurcation supervised learning susceptible population susceptibles sustainable switches synaptic cleft synaptic gap synaptic vesicles synaptic weights synchronization synchronization of chaos synchronous updating T target vector targeting τ( q ) Taylor series expansion tent map three-dimensional system threshold threshold value time series chaos detection plot Toda Hamiltonian tolerance topological dimension topologically equivalent torus total degree totally connected totally disconnected training trajectory transcritical bifurcation transfer function transient transmitted transversal transversely travelling salesman problem triangular input pulse triangular pulse trivial fixed point turbulence two-neuron module U unconstrained optimization problem uncoupled uniform asymptotic expansion uniform harvesting unipolar activation function uniqueness theorem universality Unix unstable critical point fixed point focus limit cycle manifold node unstable critical point unsupervised learning vacuum value of homes in Boston van der Pol equation van der Pol system vector field plot V vectorize velocity of light Verhulst’s equation viscosity viscous fingering volt voltage drop W wave equations wave vector wavelength light WC Windows wing rock WS WU X X-ray spectroscopy XOR gate Y You Tube youngest class harvesting Z Zq ... Publishing Switzerland 2014 Stephen Lynch, Dynamical Systems with Applications using MATLAB , DOI 10.1007/978-3-319-06820-6_1 A Tutorial Introduction to MATLAB Stephen Lynch1 (1) Department of Computing... Equations with MATLAB , 2nd edn Chapman and Hall/CRC Applied Mathematics and Nonlinear Science (Chapman and Hall/CRC, New York, 2013) B Bhattacharya, M Majumdar, Random Dynamical Systems: Theory and Applications. ..Stephen Lynch Dynamical Systems with Applications using MATLAB Stephen Lynch Department of Computing and Mathematics, Manchester Metropolitan