Lecture 1: Introduction, history, dynamics, nonlinearity, 1-D problem, phase portrait

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1 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 Solid Mechanics Week 1 D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 1 / 31

2 Lecture outline Housekeeping: textbooks, syllabus, course grading, consultation, contact and general info, etc. Demonstration magnetic pendulum in three magnetic potentials Introduction and history of the discipline Definition of dynamics Ordinary differential equations (ODE): order, dimensionality, autonomous, non-autonomous Linearity vs. nonlinearity Phase space Phase portrait Stable and unstable fixed points D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 2 / 31

3 General and contact info (housekeeping) Contact info: Coursework, lecture notes, and other resources: Course syllabus: https: //ois.ttu.ee/portal/page?_pageid=37,674581&_dad=portal& _schema=portal&p_from_tunniplaan=1&p_public=1&p_id=31657 D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 3 / 31

4 General and contact info (housekeeping) Examination: One coursework and one written pass-fail test. Written pass-fail test. Revision questions: ee/~dima/mittelindyn/revision_questions.pdf Two part coursework. Requirements: https: // Coursework is a requisite for taking the test. Course grading criteria: https: // Personal consultation: My office on Wednesdays and Fridays (office hours, schedule the meeting in advance) Visit me at least once to discuss the coursework. Classroom with access to Mathematica software: SOC-408 D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 4 / 31

5 Literature and textbooks S.H. Strogatz, Nonlinear Dynamics and Chaos With Applications to Physics, Biology, Chemistry, and Engineering, Second Edition, Avalon Publishing, K.T. Alligood, T.D. Sauer, J.A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer, H. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals: New Frontiers of Science, New York: Springer-Verlag, Ü. Lepik, J. Engelbrecht, Kaoseraamat, Teaduste Akadeemia Kirjastus, D.W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations, Oxford Univ. Press, H.G. Solari et al., Nonlinear Dynamics, Inst. of Physics, R.C. Hilborn, Chaos and Nonlinear Dynamics, Oxford Univ. Press, D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 5 / 31

6 Literature and textbooks D. Kaplan, L. Glass, Understanding Nonlinear Dynamics, Springer, J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, Encyclopaedia of Nonlinear Sciences, ed. by A. Scott, Routledge, A. Medio, M. Lines, Nonlinear Dynamics: A Primer, Cambridge University Press, During the forthcoming weekend google topics of nonlinear dynamics and chaos. Suggestion: BBC documentary The secret life of chaos. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 6 / 31

7 Introduction to the course Multidisciplinary course (applied), know your sciences. At first surprising and unusual approach (compared to other courses in mathematics and physics). Graphical thinking, intuitive approach. Links between: order and chaos (what happens at the limits?); simplicity and complexity (can they be the same thing?); smoothness and roughness (infinite). By the end of the course you will get a more intimate understanding of differential equations and their solutions. Theory: mechanics (statics, dynamics), linear algebra, calculus (chain rule), multivariable Taylor series expansion, theory of linear and nonlinear systems of ODEs, polar coordinates, etc. Software: Mathematica, pplane (by Rice University), Maple, MATLAB or Octave (open source), ODE packages etc. Nonlinear problems can almost never be solved analytically. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 7 / 31

8 Introduction to the course, main topics Nonlinear dynamic systems, from periodic to chaotic systems. Classification and characteristics of ODEs. Other deterministic chaotic systems (maps, feedback loops). Classification and characteristics of maps. Tools to understand and analyse the above systems. Fractal geometry and fractals. Applications and analysis tools of chaos and fractals. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 8 / 31

9 Introduction to the course, roadmap 1-D systems (homogeneous ODEs). 2-D systems. Classification of fixed points (linear systems). Classification of bifurcations. Quasiperiodisity. 3-D systems and higher order systems. Strange attractors and chaos. Poincaré map. 1-D maps and period doubling. 2-D maps. Classification of fixed points (linear maps). Higher dimensional maps and complex valued maps. Fractal geometry of strange attractors. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 9 / 31

10 Magnetic pendulum in three magnetic potentials Figure: Magnets shown with yellow, red and blue colours. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 10 / 31

11 Magnetic pendulum in three magnetic potentials 2 1 y(t) x(t) Figure: Magnets shown with yellow, red and blue colours, attracting the magnetic pendulum for three neighbouring initial conditions (top view). See Mathematica.nb file uploaded to the course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 11 / 31

12 Magnetic pendulum in three magnetic potentials System is modeled with the following equations of motion: N x i x ẍ + Rẋ ( (xi ) 3 + Cx = 0, i=1 x) 2 + (y i y) 2 + d 2 ÿ + Rẏ N i=1 y i y ( (xi x) 2 + (y i y) 2 + d 2 ) 3 + Cy = 0, where R is proportional to the air resistance and overall attenuation, C is proportional to the effects of gravity, N is the number of magnets, the i-th magnet is positioned at (x i, y i ), d is the distance between the pendulum at rest and the plane of magnets. Additionally we assume that the pendulum length is long compared to the spacing of the magnets. Thus, we may assume for simplicity that the metal ball moves about on a xy-plane. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 12 / 31 (1)

13 Magnetic pendulum Figure: The basins of attraction of the three magnets which are coloured red, blue and yellow. H. Peitgen, et al, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, 2004, pp D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 13 / 31

14 Magnetic pendulum Figure: The basins of attraction of the three magnets which are coloured red, blue and yellow. H. Peitgen, et al, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, 2004, pp D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 14 / 31

15 Magnetic pendulum Figure: Detail of previous slide showing the intertwined structure of the three basins. H. Peitgen, et al, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, 2004, pp D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 15 / 31

16 History of the discipline 1666 Newton Invention of calculus, explanation of planetary motion. Two body problem solved. Problem of the Moon. 1700s 1800s Flowering of calculus and classical mechanics. Analytical studies of planetary motion. Determined chaos (not stochastic), analytical studies. 1890s Poincaré Father of chaos. Geometric approach, 3 body problem explained. Poincaré s work goes unnoticed Nonlinear oscillators in physics and engineering, invention of radio, radar, laser Computer is invented and in use Birkhoff, Komogorov, Arnold, Moser Complex behaviour in Hamiltonian mechanics. KAM theorem Lorenz Strange attractor. Butterfly effect. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 16 / 31

17 History of the discipline 1970s Ruelle and Takens Turbulence and chaos. 1980s 1990s May Chaos in iterative maps. Feigenbaum Universal rout to chaos. Connection between chaos and phase transition. Experimental studies of chaos. Winfree Nonlinear oscillators in biology. Mandelbrot Father of fractal geometry. Fractals. Widespread interest in chaos, fractals, oscillators and their applications. Topic has peaked. Engineering application (encoding communication). Complex systems 2000 present Complex systems, networks (social, economics, internet, biology). D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 17 / 31

18 Introduction to theory: basic definitions Mechanics (solid and fluid) can be roughly divided into statics and dynamics. Statics: the branch of mechanics concerned with the forces acting on stationary bodies. The acting forces are in equilibrium. Dynamics: the branch of mechanics concerned with the motion/changes of bodies/systems under the action of forces. The acting forces are not in equilibrium. The branch of any science in which changes in variables are considered e.g. chemical kinetics, population biology, nonlinear oscillations, econophysics, etc. All these subjects can be placed under a common mathematical framework. Nonlinear dynamics: concerns with dynamical systems or processes that are inherently nonlinear. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 18 / 31

19 Introduction to theory: basic definitions In this course we will be mainly studying systems of ODEs of the form x = f( x), (2) where denotes the time derivative (Lagrange notation for differentiation), x and f are vectors and x R n. The set R n is called (n-dimensional) phase space. We will also consider maps in the form x n+1 = f( x n ), (3) where n is the number of iterates, x is a vector and x R n or x C n. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 19 / 31

20 Introduction to theory: basic definitions The component form of system x = f( x) has the form ẋ 1 = f 1 (x 1, x 2, x 3,..., x n ) ẋ 2 = f 2 (x 1, x 2, x 3,..., x n ) ẋ 3 = f 3 (x 1, x 2, x 3,..., x n ). ẋ n = f n (x 1, x 2, x 3,..., x n ) (4) Linearity: The above system is linear if function f is a linear function. Functions f i are linear compositions of independent variables x i. Variables x i appear in first power only. No products, trigonometric, exponential, etc. functions of x i are present. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 20 / 31

21 Introduction to theory: basic definitions Nonlinearity: Any system that is not linear is nonlinear. Autonomous system: no explicit dependence on time t in f. Non-autonomous system: explicitly dependant on time t in f. ẋ 1 = f 1 (x 1, x 2, x 3,..., x n, t) ẋ 2 = f 2 (x 1, x 2, x 3,..., x n, t) ẋ 3 = f 3 (x 1, x 2, x 3,..., x n, t). ẋ n = f n (x 1, x 2, x 3,..., x n, t) (5) Bulk of the time will be spent on working with autonomous systems. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 21 / 31

22 Introduction to theory: nonlinearity Figure: A nonlinear system is a system in which the change of the output is not proportional to the change of the input. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 22 / 31

23 Introduction to theory Discussed during the lecture. System of 1st order ODEs, n-th order ODEs, phase portrait. Figure: Linearity and nonlinearity in higher order systems. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 23 / 31

24 Linearity vs. nonlinearity Discussed during the lecture. Linear vs. nonlinear systems. Simple harmonic oscillator: mẍ + kx = 0, (6) where m and k are constants and x is the displacement. Mathematical pendulum (normalised, dimensionless): where θ is the angular displacement. θ + sin θ = 0, (7) D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 24 / 31

25 Big picture D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 25 / 31

26 1-D phase portrait Figure: Phase portrait of the 1-D system in the form: ẋ = f(x). Stable and unstable fixed points are shown with filled and unfilled circles, respectively. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 26 / 31

27 1-D examples: phase space, phase portrait Discussed during the lecture. 1-D flow problem: ẋ = sin x. (8) x(t) t See Mathematica.nb file uploaded to the course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 27 / 31

28 1-D examples: phase space, phase portrait Logistic equation (population dynamics in isolation): ( ẋ = rx 1 x ), (9) K where r and K are constants and x is the size of population. 40 x(t) What is the value of carrying capacity K here? t See Mathematica.nb file uploaded to the course webpage. D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 28 / 31

29 Conclusions Magnetic pendulum in three magnetic potentials Introduction and history of the discipline Definition of dynamics Ordinary differential equations (ODE): order, dimensionality, autonomous, non-autonomous Linearity vs. nonlinearity Phase space Phase portrait Stable and unstable fixed points D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 29 / 31

30 Revision questions What is dynamics? Name a dynamical system. Define nonlinearity. Determine if the following equations/systems are linear or nonlinear: ẋ = sin x, (10) ẋ = ln x, { (11) ẋ = y, ẏ = xy, (12) ẍ + ẋ + x = 0. (13) What is ordinary homogeneous differential equation? Define 1-D dynamical system. Name a 1-D problem. What is phase space? D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 30 / 31

31 Revision questions What is phase portrait? What is fixed point? How to find fixed points of differential equations? Find fixed point or points of the following system: ẍ + ẋ + x = 0. (14) Explain fixed point stability. What is linear analysis of a system? D. Kartofelev Nonlinear Dynamics EMR 0060 Week 1 31 / 31

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