Nonlinear Dynamics. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna.
|
|
- Randall Mason
- 5 years ago
- Views:
Transcription
1 Nonlinear Dynamics Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna
2 2
3 Introduction: Dynamics of Simple Maps 3
4 Dynamical systems A dynamical system can be informally defined as any system where some fixed rule describes the time dependence of the position and velocity of a point in geometric space Examples Planet orbiting around a star Oscillating pendulum Chemical reaction Cellular automata (more about these later) 4
5 Fixed points There are many different types of motion For example, a moving object may reach a fixed point Fixed point (e.g., a pendulum coming to a complete stop due to friction). Limit cycle (the system state eventually repeats itself; e.g., planet orbiting around a star) Quasiperiodi orbit (the system is periodic, but its state does not precisely repeat; e.g., multiple planets orbiting a star with non-resonant orbits) 5
6 Chaos For a long time it was believed that every dynamical system had either a fixed point, a periodic orbit or a quasiperiodic orbit Now, we understand that there are plenty of examples of systems that do not fall in any of the above classes Turbulence in water or air Wobble of planets following complicate orbits Weather pattern Electric activity of the brain Double rod pendulum 6
7 Example 7
8 Logistic map x n+1=r x n (1 x n ), r [0, 4], x n [0,1] Simple model of population growth when the population size is small, the population will increase at a rate proportional to the current population. the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population Note: the book uses the slightly different formulation x n+1=4 r x n (1 x n ) we will not use this: we adopt the standard formulation at the top of this slide, as commonly used 8
9 Logistic map The equation is fully deterministic: apparently, nothing surprising can happen there x n+1= f ( x n )=r x n (1 x n ) y r/4 0 1/2 1 x 9
10 Fixed points We want to study the steady-state dynamics of the logistic map, for every value of r We start from a given x0 and iterate the recurrence xn+1 = f(xn) = rxn( 1 xn ) What are the fixed points of f? Those values x for which f(x) = x 10
11 Logistic map There are two fixed points: x = 0 and x = (r - 1) / r The second one is valid only if r 1 x n+1= f ( x n )=r x n (1 x n ) y y=x r/4 0 1/2 (r-1) / r 1 x 11
12 Iterates of the logistic map (r = 2.8) 12
13 Iterates of the logistic map (r = 3.2) 13
14 Iterates of the logistic map (r = 3.52) 14
15 Iterates of the logistic map (r = 4) 15
16 Classification of fixed points Let xf be a fixed point for function f If f '(xf) = 0 super-stable If f '(xf) < 1 attracting and stable If f '(xf) = 1 neutral If f '(xf) > 1 repelling and unstable 16
17 The logistic map r 1 1<r 3 Iterations eventually converge to the fixed point 0 (stable) Unstable fixed point 0 Stable fixed point (1 - r) / r r>3 Chaotic behavior, period-doubling bifurcations 17
18 18
19 For those mathematically inclined Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd edition, Westview Press, 2003, ISBN
20 Higher Dimensions 20
21 Higher dimensions Chaotic behavior can be observed by iterating some simple maps in higher dimensions Example: Arnold's cat map Γ :( x, y) ( (2x + y) mod 1,( x+ y) mod 1 ) 21
22 Arnold's cat map This (and similar) map is usually shown to illustrate the Poincaré recurrence theorem Certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state Iteration of the cat map eventually produces the initial image 22
23 Baker's map 23
24 Baker's map 24
25 Iterating the baker's map 25
26 Invariant image 26
27 Strange attractors Strange attractors are attractors with a fractal structure Let us consider the Hénon map (introduced by the French astronomer Michel Hénon) that maps two points (xt, yt) into a new pair of points (xt+1, yt+1), as follows: 2 t x t +1 =a x +b y t y t +1 =x t where a, b are two constants Again, this is a fully deterministic map 27
28 The Hénon map for a=1.29, b=0.3 (Only xt is shown, random initial values) xt t 28
29 The Hénon map for a=1.29, b=0.3 (Only xt is shown, random initial values) xt t 29
30 The Hénon map for a=1.29, b=0.3 (Only xt is shown, random initial values) xt x = is an unstable attractor t 30
31 The Hénon attractor viewed at different scales (plots of yt versus xt) 31
32 Hénon map attractor The Hénon map attractor is made of those points that map into the attractor In other words, the attractor is invariant in the Hénon map The Hénon map attractor can be computed by warping a square according to the Hénon map 32
33 Producer-Consumer Dynamics 33
34 Predator-Prey Model Volterra and Lotka df = F (a bs ) dt F = small fish population S = shark population a = reproduction rate of small fish b = number of small fish that a shark can eat c = amount of energy that a small fish supplies to a shark ds =S (cf d ) dt If c is large, csf will be large, meaning that the shark population increases d = death rate of sharks 34
35 Population Dynamics Fixed point at F = d/c, S = a/b 35
36 Generalization The Volterra-Lotka model can be extended to an arbitrary number n of species n dx i = x i Aij (1 x j ) dt j=1 where xi represents the i-th species and Aij represents the effect that species j has on species i 36
37 Example Sample Models Biology Wolf Sheep Predation 37
Introduction to Dynamical Systems Basic Concepts of Dynamics
Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic
More informationUnit Ten Summary Introduction to Dynamical Systems and Chaos
Unit Ten Summary Introduction to Dynamical Systems Dynamical Systems A dynamical system is a system that evolves in time according to a well-defined, unchanging rule. The study of dynamical systems is
More informationxt+1 = 1 ax 2 t + y t y t+1 = bx t (1)
Exercise 2.2: Hénon map In Numerical study of quadratic area-preserving mappings (Commun. Math. Phys. 50, 69-77, 1976), the French astronomer Michel Hénon proposed the following map as a model of the Poincaré
More informationPHY411 Lecture notes Part 5
PHY411 Lecture notes Part 5 Alice Quillen January 27, 2016 Contents 0.1 Introduction.................................... 1 1 Symbolic Dynamics 2 1.1 The Shift map.................................. 3 1.2
More information... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré
Chapter 2 Dynamical Systems... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré One of the exciting new fields to arise out
More informationDynamical Systems and Chaos Part I: Theoretical Techniques. Lecture 4: Discrete systems + Chaos. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part I: Theoretical Techniques Lecture 4: Discrete systems + Chaos Ilya Potapov Mathematics Department, TUT Room TD325 Discrete maps x n+1 = f(x n ) Discrete time steps. x 0
More informationNonlinear dynamics & chaos BECS
Nonlinear dynamics & chaos BECS-114.7151 Phase portraits Focus: nonlinear systems in two dimensions General form of a vector field on the phase plane: Vector notation: Phase portraits Solution x(t) describes
More informationFundamentals of Dynamical Systems / Discrete-Time Models. Dr. Dylan McNamara people.uncw.edu/ mcnamarad
Fundamentals of Dynamical Systems / Discrete-Time Models Dr. Dylan McNamara people.uncw.edu/ mcnamarad Dynamical systems theory Considers how systems autonomously change along time Ranges from Newtonian
More informationLesson 4: Non-fading Memory Nonlinearities
Lesson 4: Non-fading Memory Nonlinearities Nonlinear Signal Processing SS 2017 Christian Knoll Signal Processing and Speech Communication Laboratory Graz University of Technology June 22, 2017 NLSP SS
More informationMechanisms of Chaos: Stable Instability
Mechanisms of Chaos: Stable Instability Reading for this lecture: NDAC, Sec. 2.-2.3, 9.3, and.5. Unpredictability: Orbit complicated: difficult to follow Repeatedly convergent and divergent Net amplification
More informationDynamical Systems: Lecture 1 Naima Hammoud
Dynamical Systems: Lecture 1 Naima Hammoud Feb 21, 2017 What is dynamics? Dynamics is the study of systems that evolve in time What is dynamics? Dynamics is the study of systems that evolve in time a system
More informationONE DIMENSIONAL CHAOTIC DYNAMICAL SYSTEMS
Journal of Pure and Applied Mathematics: Advances and Applications Volume 0 Number 0 Pages 69-0 ONE DIMENSIONAL CHAOTIC DYNAMICAL SYSTEMS HENA RANI BISWAS Department of Mathematics University of Barisal
More informationLab 5: Nonlinear Systems
Lab 5: Nonlinear Systems Goals In this lab you will use the pplane6 program to study two nonlinear systems by direct numerical simulation. The first model, from population biology, displays interesting
More informationDynamics of a Population Model Controlling the Spread of Plague in Prairie Dogs
Dynamics of a opulation Model Controlling the Spread of lague in rairie Dogs Catalin Georgescu The University of South Dakota Department of Mathematical Sciences 414 East Clark Street, Vermillion, SD USA
More informationLesson 9: Predator-Prey and ode45
Lesson 9: Predator-Prey and ode45 9.1 Applied Problem. In this lesson we will allow for more than one population where they depend on each other. One population could be the predator such as a fox, and
More informationMotivation and Goals. Modelling with ODEs. Continuous Processes. Ordinary Differential Equations. dy = dt
Motivation and Goals Modelling with ODEs 24.10.01 Motivation: Ordinary Differential Equations (ODEs) are very important in all branches of Science and Engineering ODEs form the basis for the simulation
More informationMore Details Fixed point of mapping is point that maps into itself, i.e., x n+1 = x n.
More Details Fixed point of mapping is point that maps into itself, i.e., x n+1 = x n. If there are points which, after many iterations of map then fixed point called an attractor. fixed point, If λ
More informationLokta-Volterra predator-prey equation dx = ax bxy dt dy = cx + dxy dt
Periodic solutions A periodic solution is a solution (x(t), y(t)) of dx = f(x, y) dt dy = g(x, y) dt such that x(t + T ) = x(t) and y(t + T ) = y(t) for any t, where T is a fixed number which is a period
More informationPhysics: spring-mass system, planet motion, pendulum. Biology: ecology problem, neural conduction, epidemics
Applications of nonlinear ODE systems: Physics: spring-mass system, planet motion, pendulum Chemistry: mixing problems, chemical reactions Biology: ecology problem, neural conduction, epidemics Economy:
More informationChaos & Recursive. Ehsan Tahami. (Properties, Dynamics, and Applications ) PHD student of biomedical engineering
Chaos & Recursive Equations (Properties, Dynamics, and Applications ) Ehsan Tahami PHD student of biomedical engineering Tahami@mshdiau.a.ir Index What is Chaos theory? History of Chaos Introduction of
More information2 Discrete growth models, logistic map (Murray, Chapter 2)
2 Discrete growth models, logistic map (Murray, Chapter 2) As argued in Lecture 1 the population of non-overlapping generations can be modelled as a discrete dynamical system. This is an example of an
More informationvii Contents 7.5 Mathematica Commands in Text Format 7.6 Exercises
Preface 0. A Tutorial Introduction to Mathematica 0.1 A Quick Tour of Mathematica 0.2 Tutorial 1: The Basics (One Hour) 0.3 Tutorial 2: Plots and Differential Equations (One Hour) 0.4 Mathematica Programs
More informationEen vlinder in de wiskunde: over chaos en structuur
Een vlinder in de wiskunde: over chaos en structuur Bernard J. Geurts Enschede, November 10, 2016 Tuin der Lusten (Garden of Earthly Delights) In all chaos there is a cosmos, in all disorder a secret
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationI NONLINEAR EWORKBOOK
I NONLINEAR EWORKBOOK Chaos, Fractals, Cellular Automata, Neural Networks, Genetic Algorithms, Gene Expression Programming, Wavelets, Fuzzy Logic with C++, Java and SymbolicC++ Programs Willi-Hans Steeb
More informationFractals, Dynamical Systems and Chaos. MATH225 - Field 2008
Fractals, Dynamical Systems and Chaos MATH225 - Field 2008 Outline Introduction Fractals Dynamical Systems and Chaos Conclusions Introduction When beauty is abstracted then ugliness is implied. When good
More informationDiscrete Time Coupled Logistic Equations with Symmetric Dispersal
Discrete Time Coupled Logistic Equations with Symmetric Dispersal Tasia Raymer Department of Mathematics araymer@math.ucdavis.edu Abstract: A simple two patch logistic model with symmetric dispersal between
More informationInfinity Unit 2: Chaos! Dynamical Systems
Infinity Unit 2: Chaos! Dynamical Systems Iterating Linear Functions These questions are about iterating f(x) = mx + b. Seed: x 1. Orbit: x 1, x 2, x 3, For each question, give examples and a symbolic
More informationFrom Last Time. Gravitational forces are apparent at a wide range of scales. Obeys
From Last Time Gravitational forces are apparent at a wide range of scales. Obeys F gravity (Mass of object 1) (Mass of object 2) square of distance between them F = 6.7 10-11 m 1 m 2 d 2 Gravitational
More informationDIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS
DIFFERENTIAL EQUATIONS, DYNAMICAL SYSTEMS, AND AN INTRODUCTION TO CHAOS Morris W. Hirsch University of California, Berkeley Stephen Smale University of California, Berkeley Robert L. Devaney Boston University
More informationContents Dynamical Systems Stability of Dynamical Systems: Linear Approach
Contents 1 Dynamical Systems... 1 1.1 Introduction... 1 1.2 DynamicalSystems andmathematical Models... 1 1.3 Kinematic Interpretation of a System of Differential Equations... 3 1.4 Definition of a Dynamical
More informationPopulation Dynamics II
Population Dynamics II In this class, we shall analyze behavioral patterns of ecosystems, in which more than two species interact with each other. Such systems frequently exhibit chaotic behavior. Chaotic
More information6.2 Brief review of fundamental concepts about chaotic systems
6.2 Brief review of fundamental concepts about chaotic systems Lorenz (1963) introduced a 3-variable model that is a prototypical example of chaos theory. These equations were derived as a simplification
More information1 Simple one-dimensional dynamical systems birth/death and migration processes, logistic
NOTES ON A Short Course and Introduction to Dynamical Systems in Biomathematics Urszula Foryś Institute of Applied Math. & Mech. Dept. of Math., Inf. & Mech. Warsaw University 1 Simple one-dimensional
More informationTWO DIMENSIONAL FLOWS. Lecture 5: Limit Cycles and Bifurcations
TWO DIMENSIONAL FLOWS Lecture 5: Limit Cycles and Bifurcations 5. Limit cycles A limit cycle is an isolated closed trajectory [ isolated means that neighbouring trajectories are not closed] Fig. 5.1.1
More informationDynamical Systems with Applications
Stephen Lynch Dynamical Systems with Applications using MATLAB Birkhauser Boston Basel Berlin Preface xi 0 A Tutorial Introduction to MATLAB and the Symbolic Math Toolbox 1 0.1 Tutorial One: The Basics
More informationTHREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations
THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a
More informationMATH 415, WEEKS 14 & 15: 1 Recurrence Relations / Difference Equations
MATH 415, WEEKS 14 & 15: Recurrence Relations / Difference Equations 1 Recurrence Relations / Difference Equations In many applications, the systems are updated in discrete jumps rather than continuous
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic angular frequency
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. small angle approximation. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ small angle approximation θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic
More informationLECTURE 8: DYNAMICAL SYSTEMS 7
15-382 COLLECTIVE INTELLIGENCE S18 LECTURE 8: DYNAMICAL SYSTEMS 7 INSTRUCTOR: GIANNI A. DI CARO GEOMETRIES IN THE PHASE SPACE Damped pendulum One cp in the region between two separatrix Separatrix Basin
More informationChapter 23. Predicting Chaos The Shift Map and Symbolic Dynamics
Chapter 23 Predicting Chaos We have discussed methods for diagnosing chaos, but what about predicting the existence of chaos in a dynamical system. This is a much harder problem, and it seems that the
More informationDynamical Systems. Dennis Pixton
Dynamical Systems Version 0.2 Dennis Pixton E-mail address: dennis@math.binghamton.edu Department of Mathematical Sciences Binghamton University Copyright 2009 2010 by the author. All rights reserved.
More informationThe logistic difference equation and the route to chaotic behaviour
The logistic difference equation and the route to chaotic behaviour Level 1 module in Modelling course in population and evolutionary biology (701-1418-00) Module author: Sebastian Bonhoeffer Course director:
More informationMathematical Foundations of Neuroscience - Lecture 7. Bifurcations II.
Mathematical Foundations of Neuroscience - Lecture 7. Bifurcations II. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010 Filip
More informationDynamical Systems with Applications using Mathematica
Stephen Lynch Dynamical Systems with Applications using Mathematica Birkhäuser Boston Basel Berlin Contents Preface xi 0 A Tutorial Introduction to Mathematica 1 0.1 A Quick Tour of Mathematica 2 0.2 Tutorial
More informationLogistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations
Logistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations S. Y. Ha and J. Park Department of Mathematical Sciences Seoul National University Sep 23, 2013 Contents 1 Logistic Map 2 Euler and
More informationLectures on Periodic Orbits
Lectures on Periodic Orbits 11 February 2009 Most of the contents of these notes can found in any typical text on dynamical systems, most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationDynamical Systems: Ecological Modeling
Dynamical Systems: Ecological Modeling G Söderbacka Abstract Ecological modeling is becoming increasingly more important for modern engineers. The mathematical language of dynamical systems has been applied
More informationChaos in Dynamical Systems. LIACS Natural Computing Group Leiden University
Chaos in Dynamical Systems Overview Introduction: Modeling Nature! Example: Logistic Growth Fixed Points Bifurcation Diagrams Application Examples 2 INTRODUCTION 3 Linear and Non-linear dynamic systems
More informationLecture 1: A Preliminary to Nonlinear Dynamics and Chaos
Lecture 1: A Preliminary to Nonlinear Dynamics and Chaos Autonomous Systems A set of coupled autonomous 1st-order ODEs. Here "autonomous" means that the right hand side of the equations does not explicitly
More informationLecture 1: Introduction, history, dynamics, nonlinearity, 1-D problem, phase portrait
Lecture 1: Introduction, history, dynamics, nonlinearity, 1-D problem, phase portrait Dmitri Kartofelev, PhD Tallinn University of Technology, School of Science, Department of Cybernetics, Laboratory of
More informationNonlinear Dynamics and Chaos Summer 2011
67-717 Nonlinear Dynamics and Chaos Summer 2011 Instructor: Zoubir Benzaid Phone: 424-7354 Office: Swart 238 Office Hours: MTWR: 8:30-9:00; MTWR: 12:00-1:00 and by appointment. Course Content: This course
More informationDIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS
WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Series Editor: Leon O. Chua Series A Vol. 66 DIFFERENTIAL GEOMETRY APPLIED TO DYNAMICAL SYSTEMS Jean-Marc Ginoux Université du Sud, France World Scientific
More informationEdward Lorenz. Professor of Meteorology at the Massachusetts Institute of Technology
The Lorenz system Edward Lorenz Professor of Meteorology at the Massachusetts Institute of Technology In 1963 derived a three dimensional system in efforts to model long range predictions for the weather
More informationChaos. Dr. Dylan McNamara people.uncw.edu/mcnamarad
Chaos Dr. Dylan McNamara people.uncw.edu/mcnamarad Discovery of chaos Discovered in early 1960 s by Edward N. Lorenz (in a 3-D continuous-time model) Popularized in 1976 by Sir Robert M. May as an example
More informationChapter 4. Transition towards chaos. 4.1 One-dimensional maps
Chapter 4 Transition towards chaos In this chapter we will study how successive bifurcations can lead to chaos when a parameter is tuned. It is not an extensive review : there exists a lot of different
More informationDynamics and Chaos. Copyright by Melanie Mitchell
Dynamics and Chaos Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015 Dynamics: The general study of how systems change over time Copyright by Melanie Mitchell Conference on Complex
More informationAn Introduction to Evolutionary Game Theory: Lecture 2
An Introduction to Evolutionary Game Theory: Lecture 2 Mauro Mobilia Lectures delivered at the Graduate School on Nonlinear and Stochastic Systems in Biology held in the Department of Applied Mathematics,
More information11 Chaos in Continuous Dynamical Systems.
11 CHAOS IN CONTINUOUS DYNAMICAL SYSTEMS. 47 11 Chaos in Continuous Dynamical Systems. Let s consider a system of differential equations given by where x(t) : R R and f : R R. ẋ = f(x), The linearization
More informationPredator-Prey Population Dynamics
Predator-Prey Population Dynamics Gonzalo Mateos Dept. of ECE and Goergen Institute for Data Science University of Rochester gmateosb@ece.rochester.edu http://www.ece.rochester.edu/~gmateosb/ October 2,
More informationDynamical Systems and Chaos Part II: Biology Applications. Lecture 6: Population dynamics. Ilya Potapov Mathematics Department, TUT Room TD325
Dynamical Systems and Chaos Part II: Biology Applications Lecture 6: Population dynamics Ilya Potapov Mathematics Department, TUT Room TD325 Living things are dynamical systems Dynamical systems theory
More informationsecond order Runge-Kutta time scheme is a good compromise for solving ODEs unstable for oscillators
ODE Examples We have seen so far that the second order Runge-Kutta time scheme is a good compromise for solving ODEs with a good precision, without making the calculations too heavy! It is unstable for
More informationExample Chaotic Maps (that you can analyze)
Example Chaotic Maps (that you can analyze) Reading for this lecture: NDAC, Sections.5-.7. Lecture 7: Natural Computation & Self-Organization, Physics 256A (Winter 24); Jim Crutchfield Monday, January
More informationReconstruction Deconstruction:
Reconstruction Deconstruction: A Brief History of Building Models of Nonlinear Dynamical Systems Jim Crutchfield Center for Computational Science & Engineering Physics Department University of California,
More informationCS 365 Introduction to Scientific Modeling Fall Semester, 2011 Review
CS 365 Introduction to Scientific Modeling Fall Semester, 2011 Review Topics" What is a model?" Styles of modeling" How do we evaluate models?" Aggregate models vs. individual models." Cellular automata"
More informationSPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE. Itishree Priyadarshini. Prof. Biplab Ganguli
SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE By Itishree Priyadarshini Under the Guidance of Prof. Biplab Ganguli Department of Physics National Institute of Technology, Rourkela CERTIFICATE This is to
More informationLecture 20/Lab 21: Systems of Nonlinear ODEs
Lecture 20/Lab 21: Systems of Nonlinear ODEs MAR514 Geoffrey Cowles Department of Fisheries Oceanography School for Marine Science and Technology University of Massachusetts-Dartmouth Coupled ODEs: Species
More information1.Introduction: 2. The Model. Key words: Prey, Predator, Seasonality, Stability, Bifurcations, Chaos.
Dynamical behavior of a prey predator model with seasonally varying parameters Sunita Gakkhar, BrhamPal Singh, R K Naji Department of Mathematics I I T Roorkee,47667 INDIA Abstract : A dynamic model based
More informationLecture 6. Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of:
Lecture 6 Chaos Lorenz equations and Malkus' waterwheel Some properties of the Lorenz Eq.'s Lorenz Map Towards definitions of: Chaos, Attractors and strange attractors Transient chaos Lorenz Equations
More informationChapitre 4. Transition to chaos. 4.1 One-dimensional maps
Chapitre 4 Transition to chaos In this chapter we will study how successive bifurcations can lead to chaos when a parameter is tuned. It is not an extensive review : there exists a lot of different manners
More informationScenarios for the transition to chaos
Scenarios for the transition to chaos Alessandro Torcini alessandro.torcini@cnr.it Istituto dei Sistemi Complessi - CNR - Firenze Istituto Nazionale di Fisica Nucleare - Sezione di Firenze Centro interdipartimentale
More informationThe Pattern Recognition System Using the Fractal Dimension of Chaos Theory
Original Article International Journal of Fuzzy Logic and Intelligent Systems Vol. 15, No. 2, June 2015, pp. 121-125 http://dx.doi.org/10.5391/ijfis.2015.15.2.121 ISSN(Print) 1598-2645 ISSN(Online) 2093-744X
More informationWe have two possible solutions (intersections of null-clines. dt = bv + muv = g(u, v). du = au nuv = f (u, v),
Let us apply the approach presented above to the analysis of population dynamics models. 9. Lotka-Volterra predator-prey model: phase plane analysis. Earlier we introduced the system of equations for prey
More informationDO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.
EE 533 Homeworks Spring 07 Updated: Saturday, April 08, 07 DO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED. Some homework assignments refer to the textbooks: Slotine
More informationStability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations
Stability and Hopf bifurcation analysis of the Mackey-Glass and Lasota equations Sreelakshmi Manjunath Department of Electrical Engineering Indian Institute of Technology Madras (IITM), India JTG Summer
More informationChaotic Modelling and Simulation
Chaotic Modelling and Simulation Analysis of Chaotic Models, Attractors and Forms Christos H. Skiadas Charilaos Skiadas @ CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint
More informationChaos in the Dynamics of the Family of Mappings f c (x) = x 2 x + c
IOSR Journal of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 10, Issue 4 Ver. IV (Jul-Aug. 014), PP 108-116 Chaos in the Dynamics of the Family of Mappings f c (x) = x x + c Mr. Kulkarni
More informationIntroduction to Classical Chaos
Introduction to Classical Chaos WeiHan Hsiao a a Department of Physics, The University of Chicago E-mail: weihanhsiao@uchicago.edu ABSTRACT: This note is a contribution to Kadanoff Center for Theoretical
More informationLecture 3. Dynamical Systems in Continuous Time
Lecture 3. Dynamical Systems in Continuous Time University of British Columbia, Vancouver Yue-Xian Li November 2, 2017 1 3.1 Exponential growth and decay A Population With Generation Overlap Consider a
More information2D-Volterra-Lotka Modeling For 2 Species
Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More informationIs chaos possible in 1d? - yes - no - I don t know. What is the long term behavior for the following system if x(0) = π/2?
Is chaos possible in 1d? - yes - no - I don t know What is the long term behavior for the following system if x(0) = π/2? In the insect outbreak problem, what kind of bifurcation occurs at fixed value
More informationDelay Coordinate Embedding
Chapter 7 Delay Coordinate Embedding Up to this point, we have known our state space explicitly. But what if we do not know it? How can we then study the dynamics is phase space? A typical case is when
More informationThe Structure of Hyperbolic Sets
The Structure of Hyperbolic Sets p. 1/35 The Structure of Hyperbolic Sets Todd Fisher tfisher@math.umd.edu Department of Mathematics University of Maryland, College Park The Structure of Hyperbolic Sets
More informationComputers, Lies and the Fishing Season
1/47 Computers, Lies and the Fishing Season Liz Arnold May 21, 23 Introduction Computers, lies and the fishing season takes a look at computer software programs. As mathematicians, we depend on computers
More informationA MINIMAL 2-D QUADRATIC MAP WITH QUASI-PERIODIC ROUTE TO CHAOS
International Journal of Bifurcation and Chaos, Vol. 18, No. 5 (2008) 1567 1577 c World Scientific Publishing Company A MINIMAL 2-D QUADRATIC MAP WITH QUASI-PERIODIC ROUTE TO CHAOS ZERAOULIA ELHADJ Department
More informationAsynchronous and Synchronous Dispersals in Spatially Discrete Population Models
SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 7, No. 2, pp. 284 310 c 2008 Society for Industrial and Applied Mathematics Asynchronous and Synchronous Dispersals in Spatially Discrete Population Models Abdul-Aziz
More information2 One-dimensional models in discrete time
2 One-dimensional models in discrete time So far, we have assumed that demographic events happen continuously over time and can thus be written as rates. For many biological species with overlapping generations
More informationẋ = f(x, y), ẏ = g(x, y), (x, y) D, can only have periodic solutions if (f,g) changes sign in D or if (f,g)=0in D.
4 Periodic Solutions We have shown that in the case of an autonomous equation the periodic solutions correspond with closed orbits in phase-space. Autonomous two-dimensional systems with phase-space R
More informationModelling biological oscillations
Modelling biological oscillations Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Outline Outline of Topics Van der Pol equation Van
More informationLecture3 The logistic family.
Lecture3 The logistic family. 1 The logistic family. The scenario for 0 < µ 1. The scenario for 1 < µ 3. Period doubling bifurcations in the logistic family. 2 The period doubling bifurcation. The logistic
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationMath 232, Final Test, 20 March 2007
Math 232, Final Test, 20 March 2007 Name: Instructions. Do any five of the first six questions, and any five of the last six questions. Please do your best, and show all appropriate details in your solutions.
More informationPOPULATION DYNAMICS: TWO SPECIES MODELS; Susceptible Infected Recovered (SIR) MODEL. If they co-exist in the same environment:
POPULATION DYNAMICS: TWO SPECIES MODELS; Susceptible Infected Recovered (SIR) MODEL Next logical step: consider dynamics of more than one species. We start with models of 2 interacting species. We consider,
More informationDiscrete time dynamical systems (Review of rst part of Math 361, Winter 2001)
Discrete time dynamical systems (Review of rst part of Math 36, Winter 2) Basic problem: x (t);; dynamic variables (e.g. population size of age class i at time t); dynamics given by a set of n equations
More information3.5 Competition Models: Principle of Competitive Exclusion
94 3. Models for Interacting Populations different dimensional parameter changes. For example, doubling the carrying capacity K is exactly equivalent to halving the predator response parameter D. The dimensionless
More informationMath 128A Spring 2003 Week 12 Solutions
Math 128A Spring 2003 Week 12 Solutions Burden & Faires 5.9: 1b, 2b, 3, 5, 6, 7 Burden & Faires 5.10: 4, 5, 8 Burden & Faires 5.11: 1c, 2, 5, 6, 8 Burden & Faires 5.9. Higher-Order Equations and Systems
More informationWhy are Discrete Maps Sufficient?
Why are Discrete Maps Sufficient? Why do dynamical systems specialists study maps of the form x n+ 1 = f ( xn), (time is discrete) when much of the world around us evolves continuously, and is thus well
More informationFish Population Modeling
Fish Population Modeling Thomas Wood 5/11/9 Logistic Model of a Fishing Season A fish population can be modelled using the Logistic Model, P = kp(1 P/a) P is the population at time t k is a growth constant
More information