NONLINEAR DYNAMICS AND CHAOS. Numerical integration. Stability analysis
|
|
- Chester Smith
- 6 years ago
- Views:
Transcription
1 LECTURE 3: FLOWS NONLINEAR DYNAMICS AND CHAOS Patrick E McSharr Sstems Analsis, Modelling & Prediction Group patrick@mcsharr.net Tel: Numerical integration Stabilit analsis Linear flows in one and two dimensions Harmonic oscillator Damped harmonic oscillator van Der Pol oscillator Rössler sstem Moore-Spiegel sstem Lorenz sstem Trinit Term 7, Weeks 3 and 4 Mondas, Wednesdas & Fridas 9: - : Seminar Room Mathematical Institute Universit of Oford Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSh Numerical integration Stabilit analsis Equation of motion: ẋ = f[(t), t] Man numerical methods for solving ODEs Assume t is a small step (effectivel converting the flow into a map) Z t+ t (t + t) = (t) + f[(t), t]dt t Euler method: (t + t) = (t) + tf[(t), t] + O( t ) Second order Runge-Kutta method (uses trial step): k = tf[(t), t] k = tf [(t) + k /, t + t/] (t + t) = (t) + k + O( t 3 ) Advisable to use fourth order Runge-Kutta method Equation for a flow: Linearising about ields where J((t)) is the Jacobian of f at (t) The perturbation ɛ(t + τ) after a τ is ẋ = f() ɛ(t) = J((t))ɛ(t) ɛ(t + τ) = M((t), τ)ɛ(t) where M((t), τ) is the linear propagator defined b»z t+τ M((t), τ) = ep J((t))dt t A particular perturbation ɛ(t + τ) grows if ɛ(t + τ) / ɛ(t) > There eists some perturbation that will grow if the largest eigenvalue λ of the matri M((t), τ) satisfies λ >. Nonlinear dnamics and chaos c 7 Patrick McSharr p.3 Nonlinear dnamics and chaos c 7 Patrick McSh
2 One-dimensional stabilit analsis Two-dimensional stabilit analsis Equation of motion: ẋ = f((t)) Consider a fied point such that ẋ = f( ) = Linearising about ields ɛ(t) = ep[j( )t]ɛ(t) Consider the growth of a perturbation to the fied point: Stabilit is determined b the sign of λ = J( ) = df/d at Equations of motion: ẋ = f(, ) ẏ = g(, ) Consider a fied point (, ) such that ẋ = ẏ = Calculate the Jacobian matri J at (, ) Define f = f/ etc. The eigenvalues of J are found b solving the characteristic equation: (f λ)(g λ) = f g Solutions are of the form: = ep(λt), = ep(λt) This gives two possible solutions for λ (real or comple conjugates) Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSh Linear motion in one dimension Circular motion Continuous description of linear motion in one dimension e.g. continuous compound interest Equation of motion: ẋ = a Solution: (t) = ep(at) Growth if a > and deca if a < Determinism implies that a trajector cannot intersect itself Poincaré-Bendison theorem: no chaos in two dimensions Require a three-dimensional flow and nonlinearit for chaos Linear motion in two dimensions e.g. Planet orbiting the sun Solution is given b: ẋ = ẏ = (t) = r cos t (t) = r sin t + = r, motion on a circle with constant radius r Sstem has one equilibrium point at (, ) = (, ) (, ) is known as a centre A centre is neutrall stable, it doesn t attract or repel! Nonlinear dnamics and chaos c 7 Patrick McSharr p.7 Nonlinear dnamics and chaos c 7 Patrick McSh
3 Mass and spring (Hooke s law) Simple Pendulum Newton s Law: F = m v where v = ẋ Hooke s Law: F = k, k = spring constant Equation of motion: m v = k Mass undergoes harmonic motion Simplif b setting m = k = : ẋ = v v = Similar to circular motion, but in (, v) state-space Note that F = mv = mẍ is a second order ODE requiring a two-dimensional state space Energ is conserved: E = mv / + k / Again, no attractor or repeller, just a centre An eample of a simple two-dimensional dnamical sstem From Newtons s second law, knowledge of the forces, position and velocit are sufficien determine future motion Pendulum (constrained to move in the plane) Dnamics full specified b the displacement angle θ(t) and the angular velocit θ(t) State vector given b (t) = [θ(t), θ(t)] Let m be the mass of the pendulum g is the acceleration due to gravit l is the length of the pendulum Tangential restoring force due to gravit: mg sin θ Tangential force due to angular acceleration: ml θ In the absence of friction, dnamics are governed b Nonlinear dnamics and chaos c 7 Patrick McSharr p.9 d dt θ = θ d θ dt = g l sin θ Nonlinear dnamics and chaos c 7 Patrick McSha Pendulum dnamics: moment of inertia Pendulum dnamics: Lagrangian formalism Moment of inertia I is defined as: I = ml where m is the mass of the bob and l is the length of the pendulum The tangential restoring force is F = mg sin θ Kinetic energ is Gravitational potential energ is T = ml θ V = mgl cos θ Equations of motion can be derived from the torque τ = Fl: τ = mgl sin θ = Iα = I θ where α is the angular acceleration and g is the acceleration due to gravit Finall, θ = g l sin θ (where the zero potential energ is defined b V = at θ = π The Lagrangian is Lagrangian formula, gives L(θ, θ) = T V = ml θ + mgl cos θ d dt L θ «= L θ θ = g l sin θ Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSha
4 Pendulum dnamics: Hamiltonian formalism Pendulum dnamics (small oscillations) Momentum is The Hamiltonian is given b p = L θ = ml θ H(p, θ) = p θ L(θ, θ) = ml θ mg cos θ For θ, use a Talor series epansion for sin θ: sin θ = θ θ3 3! + θ! +... Assume sin θ = θ giving approimate equation of motion: θ = g l θ The Hamiltonian equations of motion are = p mgl cos θ ml This corresponds to harmonic motion with solution θ(t) = θ ma cos(ω t + φ) θ = H p = p ml () ṗ = H θ = mgl sin θ () Nonlinear dnamics and chaos c 7 Patrick McSharr p.3 where and the natural period is r g ω = l s l T = π g Nonlinear dnamics and chaos c 7 Patrick McSha Pendulum energ (small oscillations) Pendulum energ (large oscillations) Total energ is given b E = T + V = ml θ mgl cos θ For θ, use a Talor series epansion for cos θ: Total energ is given b Eamine specific case of E = mgl E = ml θ mgl cos θ cos θ = θ + θ4 4!... E = ml θ mgl cos θ = mgl Assume cos θ = θ giving approimate equation of the total energ: Dividing across b mgl gives Defining E = E/mgl + mgl, we obtain E = ml θ + mgl θ mgl " # E = θ + θ For small energ (small oscillations), the lines of constant energ are ellipses ω Nonlinear dnamics and chaos c 7 Patrick McSharr p. Lines of constant energ: θ ω = + cos θ = + cos ( θ ) = cos θ θ = ±ω cos θ Nonlinear dnamics and chaos c 7 Patrick McSha
5 Pendulum state space with constant energ contours Damped harmonic oscillator... dθ/dt.. Obtained b adding friction to the harmonic oscillator Assume that friction is proportional to velocit: F = βv Equations of motion: Stable equilibrium at (, v) = (, ) Effect of friction is to lose energ Dissipative sstem ẋ = v v = βv Possible equilibria for damped oscillator: Spiral point (focus) if β < (underdamped) Radial point (node) if β > (overdamped). π π π π θ Nonlinear dnamics and chaos c 7 Patrick McSharr p.7 Nonlinear dnamics and chaos c 7 Patrick McSha van Der Pol oscillator van Der Pol series The van Der Pol Oscillator was the first relaation oscillator and provided a model of the human heartbeat [The Heartbeat considered as a Relaation oscillation, and an Electrical Model of the Heart, Balth. van der Pol and J van der Mark, Phil. Mag. Suppl. 6: (98)] 3 In D: ẍ + + ɛ( )ẋ = ẋ = ẏ = ɛ( ) Unstable equilibrium point (for ɛ > ) at (, ) = (, ). Growth for small values of and Deca for large values of and Sstem has a stable attracting limit ccle All points in the (, )-plane lie in the basis of attraction Top: ɛ =., Bottom: ɛ = Nonlinear dnamics and chaos c 7 Patrick McSharr p.9 Nonlinear dnamics and chaos c 7 Patrick McSha
6 Rössler sstem Rössler series The Rössler equations are ẋ = z, ẏ = + a, ż = b + z( c), Chaotic behaviour for a =., b =., and c = Proposed as a simple eample of chaos in a three-dimensional flow z z Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSha Moore-Spiegel Moore-Spiegel series The three-dimensional Moore-Spiegel flow: ẋ =, ẏ = z, ż = z (a b + b ) a, 3 Values giving chaotic behaviour are a = 6 and b = This provides a model for a parcel of ionised gas in the atmosphere of a star z is the height of the parcel is the velocit and z is the acceleration 3 Nonlinear dnamics and chaos c 7 Patrick McSharr p.3 Nonlinear dnamics and chaos c 7 Patrick McSha
7 The Lorenz model Lorenz series The Lorenz equations are ẋ = σ + σ, ẏ = z + r, ż = bz, Chaotic behaviour for σ =, b = 8/3, and r = 8 Raleigh-Bernard convection: flow of fluid between two rigid horizontal plates subject to gravit, with temperature gradient T between them, the top tpicall being cooler The fluid near the lower plate epands, and buoanc causes the fluid to rise, while the cooler more dense fluid near the top plate falls For some ranges of temperature gradient between the plates, T, stead convective cellular flow occurs As T increases, the flow becomes chaotic The variable is proportional to the circulator fluid flow velocit, characterises the temperature difference between rising and falling fluid regions, and z characterises vertical temperature variation: σ and r are proportional to the Prandtl number and Raleigh number respectivel z 4 3 Nonlinear dnamics and chaos c 7 Patrick McSharr p. Nonlinear dnamics and chaos c 7 Patrick McSha Lorenz return map Stabilit analsis Lorenz equations of motion: Fied points at ẋ = σ + σ ẏ = z + r ż = bz (,, ) 4 z ma (i+) Jacobian is given b (± p b(r ), ± p b(r ), r ) 6 J() = 4 σ σ r b z ma (i) Nonlinear dnamics and chaos c 7 Patrick McSharr p.7 Nonlinear dnamics and chaos c 7 Patrick McSha
8 Stabilit analsis of (,,) Stabilit analsis for pair of fied points Stabilit matri at (,, ) Eigenvalues are (,, ) is stable for r < 6 4 λ, = σ + (,, ) becomes a saddle point for r > σ σ r b ± 3 7 q (σ + ) + 4(r )σ λ 3 = b Stabilit matri for other fied points 6 4 Eigenvalues are roots of σ σ ± p b(r ) ± p b(r ) ± p b(r ) b P(λ) = λ 3 + (σ + b + )λ + b(σ + r)λ + bσ(r ) = r = λ = λ = b λ 3 = (σ + ) Define Hopf boundar: r H = σ σ + b + 3 σ b For < r < r H, smmetric pair of fied points are stable For r > r H real parts of λ, > and this pair becomes unstable 3 7 Nonlinear dnamics and chaos c 7 Patrick McSharr p.9 Nonlinear dnamics and chaos c 7 Patrick McSha Lorenz stabilit graph Power spectra 4 4 CHAOS 3 3 r STEADY b+ σ Nonlinear dnamics and chaos c 7 Patrick McSharr p.3 Nonlinear dnamics and chaos c 7 Patrick McSha
PHYS2330 Intermediate Mechanics Fall Final Exam Tuesday, 21 Dec 2010
Name: PHYS2330 Intermediate Mechanics Fall 2010 Final Exam Tuesday, 21 Dec 2010 This exam has two parts. Part I has 20 multiple choice questions, worth two points each. Part II consists of six relatively
More informationMechanics Departmental Exam Last updated November 2013
Mechanics Departmental Eam Last updated November 213 1. Two satellites are moving about each other in circular orbits under the influence of their mutual gravitational attractions. The satellites have
More informationClassical Mechanics Comprehensive Exam Solution
Classical Mechanics Comprehensive Exam Solution January 31, 011, 1:00 pm 5:pm Solve the following six problems. In the following problems, e x, e y, and e z are unit vectors in the x, y, and z directions,
More informationPhysics Mechanics. Lecture 32 Oscillations II
Physics 170 - Mechanics Lecture 32 Oscillations II Gravitational Potential Energy A plot of the gravitational potential energy U g looks like this: Energy Conservation Total mechanical energy of an object
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 informationP321(b), Assignement 1
P31(b), Assignement 1 1 Exercise 3.1 (Fetter and Walecka) a) The problem is that of a point mass rotating along a circle of radius a, rotating with a constant angular velocity Ω. Generally, 3 coordinates
More informationTheoretical physics. Deterministic chaos in classical physics. Martin Scholtz
Theoretical physics Deterministic chaos in classical physics Martin Scholtz scholtzzz@gmail.com Fundamental physical theories and role of classical mechanics. Intuitive characteristics of chaos. Newton
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 informationChapter 13 Lecture. Essential University Physics Richard Wolfson 2 nd Edition. Oscillatory Motion Pearson Education, Inc.
Chapter 13 Lecture Essential University Physics Richard Wolfson nd Edition Oscillatory Motion Slide 13-1 In this lecture you ll learn To describe the conditions under which oscillatory motion occurs To
More informationChapter 14 Periodic Motion
Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.
More informationPeriodic Motion. Periodic motion is motion of an object that. regularly repeats
Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A special kind of periodic motion occurs in mechanical systems
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 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 informationTorque and Simple Harmonic Motion
Torque and Simple Harmonic Motion Recall: Fixed Axis Rotation Angle variable Angular velocity Angular acceleration Mass element Radius of orbit Kinematics!! " d# / dt! " d 2 # / dt 2!m i Moment of inertia
More informationM2A2 Problem Sheet 3 - Hamiltonian Mechanics
MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,
More informationOscillatory Motion SHM
Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A
More informationChapter 14. Oscillations. Oscillations Introductory Terminology Simple Harmonic Motion:
Chapter 14 Oscillations Oscillations Introductory Terminology Simple Harmonic Motion: Kinematics Energy Examples of Simple Harmonic Oscillators Damped and Forced Oscillations. Resonance. Periodic Motion
More informationCHAPTER 12 OSCILLATORY MOTION
CHAPTER 1 OSCILLATORY MOTION Before starting the discussion of the chapter s concepts it is worth to define some terms we will use frequently in this chapter: 1. The period of the motion, T, is the time
More informationOscillations. Oscillations and Simple Harmonic Motion
Oscillations AP Physics C Oscillations and Simple Harmonic Motion 1 Equilibrium and Oscillations A marble that is free to roll inside a spherical bowl has an equilibrium position at the bottom of the bowl
More informationOscillations. Phys101 Lectures 28, 29. Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum
Phys101 Lectures 8, 9 Oscillations Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum Ref: 11-1,,3,4. Page 1 Oscillations of a Spring If an object oscillates
More informationEE222 - Spring 16 - Lecture 2 Notes 1
EE222 - Spring 16 - Lecture 2 Notes 1 Murat Arcak January 21 2016 1 Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Essentially Nonlinear Phenomena Continued
More informationIn the presence of viscous damping, a more generalized form of the Lagrange s equation of motion can be written as
2 MODELING Once the control target is identified, which includes the state variable to be controlled (ex. speed, position, temperature, flow rate, etc), and once the system drives are identified (ex. force,
More informationThe... of a particle is defined as its change in position in some time interval.
Distance is the. of a path followed by a particle. Distance is a quantity. The... of a particle is defined as its change in position in some time interval. Displacement is a.. quantity. The... of a particle
More informationA plane autonomous system is a pair of simultaneous first-order differential equations,
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A plane autonomous system is a pair of simultaneous first-order differential equations, ẋ = f(x, y), ẏ = g(x, y). This system has an equilibrium
More informationNonlinear Systems Examples Sheet: Solutions
Nonlinear Sstems Eamples Sheet: Solutions Mark Cannon, Michaelmas Term 7 Equilibrium points. (a). Solving ẋ =sin 4 3 =for gives =as an equilibrium point. This is the onl equilibrium because there is onl
More informationn n. ( t) ( ) = = Ay ( ) a y
Sstems of ODE Example of sstem with ODE: = a+ a = a + a In general for a sstem of n ODE = a + a + + a n = a + a + + a n = a + a + + a n n n nn n n n Differentiation of matrix: ( t) ( t) t t = = t t ( t)
More informationThe Pendulum. The purpose of this tab is to predict the motion of various pendulums and compare these predictions with experimental observations.
The Pendulum Introduction: The purpose of this tab is to predict the motion of various pendulums and compare these predictions with experimental observations. Equipment: Simple pendulum made from string
More informationPhysics 106a, Caltech 4 December, Lecture 18: Examples on Rigid Body Dynamics. Rotating rectangle. Heavy symmetric top
Physics 106a, Caltech 4 December, 2018 Lecture 18: Examples on Rigid Body Dynamics I go through a number of examples illustrating the methods of solving rigid body dynamics. In most cases, the problem
More informationFirst Year Physics: Prelims CP1 Classical Mechanics: DR. Ghassan Yassin
First Year Physics: Prelims CP1 Classical Mechanics: DR. Ghassan Yassin MT 2007 Problems I The problems are divided into two sections: (A) Standard and (B) Harder. The topics are covered in lectures 1
More informationMathematical Model of Forced Van Der Pol s Equation
Mathematical Model of Forced Van Der Pol s Equation TO Tsz Lok Wallace LEE Tsz Hong Homer December 9, Abstract This work is going to analyze the Forced Van Der Pol s Equation which is used to analyze the
More informationCP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017
CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and
More informationSecond-Order Linear Differential Equations C 2
C8 APPENDIX C Additional Topics in Differential Equations APPENDIX C. Second-Order Homogeneous Linear Equations Second-Order Linear Differential Equations Higher-Order Linear Differential Equations Application
More informationChapter 15. Oscillatory Motion
Chapter 15 Oscillatory Motion Part 2 Oscillations and Mechanical Waves Periodic motion is the repeating motion of an object in which it continues to return to a given position after a fixed time interval.
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 informationDaba Meshesha Gusu and O.Chandra Sekhara Reddy 1
International Journal of Basic and Applied Sciences Vol. 4. No. 1 2015. Pp.22-27 Copyright by CRDEEP. All Rights Reserved. Full Length Research Paper Solutions of Non Linear Ordinary Differential Equations
More information4452 Mathematical Modeling Lecture 13: Chaos and Fractals
Math Modeling Lecture 13: Chaos and Fractals Page 1 442 Mathematical Modeling Lecture 13: Chaos and Fractals Introduction In our tetbook, the discussion on chaos and fractals covers less than 2 pages.
More informationChapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx
Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull
More informationGeneral Physics I. Lecture 12: Applications of Oscillatory Motion. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 1: Applications of Oscillatory Motion Prof. WAN, Xin ( 万歆 ) inwan@zju.edu.cn http://zimp.zju.edu.cn/~inwan/ Outline The pendulum Comparing simple harmonic motion and uniform circular
More informationClassical Mechanics. FIG. 1. Figure for (a), (b) and (c). FIG. 2. Figure for (d) and (e).
Classical Mechanics 1. Consider a cylindrically symmetric object with a total mass M and a finite radius R from the axis of symmetry as in the FIG. 1. FIG. 1. Figure for (a), (b) and (c). (a) Show that
More informationNonlinear Dynamic Systems Homework 1
Nonlinear Dynamic Systems Homework 1 1. A particle of mass m is constrained to travel along the path shown in Figure 1, which is described by the following function yx = 5x + 1x 4, 1 where x is defined
More informationLimit Cycles II. Prof. Ned Wingreen MOL 410/510. How to prove a closed orbit exists?
Limit Cycles II Prof. Ned Wingreen MOL 410/510 How to prove a closed orbit eists? numerically Poincaré-Bendison Theorem: If 1. R is a closed, bounded subset of the plane. = f( ) is a continuously differentiable
More informationSimple and Physical Pendulums Challenge Problem Solutions
Simple and Physical Pendulums Challenge Problem Solutions Problem 1 Solutions: For this problem, the answers to parts a) through d) will rely on an analysis of the pendulum motion. There are two conventional
More informationChapter 15 - Oscillations
The pendulum of the mind oscillates between sense and nonsense, not between right and wrong. -Carl Gustav Jung David J. Starling Penn State Hazleton PHYS 211 Oscillatory motion is motion that is periodic
More informationFundamentals Physics. Chapter 15 Oscillations
Fundamentals Physics Tenth Edition Halliday Chapter 15 Oscillations 15-1 Simple Harmonic Motion (1 of 20) Learning Objectives 15.01 Distinguish simple harmonic motion from other types of periodic motion.
More information28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod)
28. Pendulum phase portrait Draw the phase portrait for the pendulum (supported by an inextensible rod) θ + ω 2 sin θ = 0. Indicate the stable equilibrium points as well as the unstable equilibrium points.
More informationChaos. Lendert Gelens. KU Leuven - Vrije Universiteit Brussel Nonlinear dynamics course - VUB
Chaos Lendert Gelens KU Leuven - Vrije Universiteit Brussel www.gelenslab.org Nonlinear dynamics course - VUB Examples of chaotic systems: the double pendulum? θ 1 θ θ 2 Examples of chaotic systems: the
More informationReview: control, feedback, etc. Today s topic: state-space models of systems; linearization
Plan of the Lecture Review: control, feedback, etc Today s topic: state-space models of systems; linearization Goal: a general framework that encompasses all examples of interest Once we have mastered
More informationOscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is
Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring
More informationDynamics of multiple pendula without gravity
Chaotic Modeling and Simulation (CMSIM) 1: 57 67, 014 Dnamics of multiple pendula without gravit Wojciech Szumiński Institute of Phsics, Universit of Zielona Góra, Poland (E-mail: uz88szuminski@gmail.com)
More informationNonlinear Oscillations and Chaos
CHAPTER 4 Nonlinear Oscillations and Chaos 4-. l l = l + d s d d l l = l + d m θ m (a) (b) (c) The unetended length of each spring is, as shown in (a). In order to attach the mass m, each spring must be
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 informationPhysics 106b: Lecture 7 25 January, 2018
Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with
More informationMCE 366 System Dynamics, Spring Problem Set 2. Solutions to Set 2
MCE 366 System Dynamics, Spring 2012 Problem Set 2 Reading: Chapter 2, Sections 2.3 and 2.4, Chapter 3, Sections 3.1 and 3.2 Problems: 2.22, 2.24, 2.26, 2.31, 3.4(a, b, d), 3.5 Solutions to Set 2 2.22
More informationLocal Phase Portrait of Nonlinear Systems Near Equilibria
Local Phase Portrait of Nonlinear Sstems Near Equilibria [1] Consider 1 = 6 1 1 3 1, = 3 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating
More informationFor a rigid body that is constrained to rotate about a fixed axis, the gravitational torque about the axis is
Experiment 14 The Physical Pendulum The period of oscillation of a physical pendulum is found to a high degree of accuracy by two methods: theory and experiment. The values are then compared. Theory For
More information* τσ σκ. Supporting Text. A. Stability Analysis of System 2
Supporting Tet A. Stabilit Analsis of Sstem In this Appendi, we stud the stabilit of the equilibria of sstem. If we redefine the sstem as, T when -, then there are at most three equilibria: E,, E κ -,,
More informationTHE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A 2 2 SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS
THE SEPARATRIX FOR A SECOND ORDER ORDINARY DIFFERENTIAL EQUATION OR A SYSTEM OF FIRST ORDER ODE WHICH ALLOWS A PHASE PLANE QUANTITATIVE ANALYSIS Maria P. Skhosana and Stephan V. Joubert, Tshwane University
More informationTransitioning to Chaos in a Simple Mechanical Oscillator
Transitioning to Chaos in a Simple Mechanical Oscillator Hwan Bae Physics Department, The College of Wooster, Wooster, Ohio 69, USA (Dated: May 9, 8) We vary the magnetic damping, driver frequency, and
More informationENGI 9420 Lecture Notes 4 - Stability Analysis Page Stability Analysis for Non-linear Ordinary Differential Equations
ENGI 940 Lecture Notes 4 - Stability Analysis Page 4.01 4. Stability Analysis for Non-linear Ordinary Differential Equations A pair of simultaneous first order homogeneous linear ordinary differential
More information本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權
本教材內容主要取自課本 Physics for Scientists and Engineers with Modern Physics 7th Edition. Jewett & Serway. 注意 本教材僅供教學使用, 勿做其他用途, 以維護智慧財產權 教材網址 : https://sites.google.com/site/ndhugp1 1 Chapter 15 Oscillatory Motion
More informationRotational motion problems
Rotational motion problems. (Massive pulley) Masses m and m 2 are connected by a string that runs over a pulley of radius R and moment of inertia I. Find the acceleration of the two masses, as well as
More informationOSCILLATIONS ABOUT EQUILIBRIUM
OSCILLATIONS ABOUT EQUILIBRIUM Chapter 13 Units of Chapter 13 Periodic Motion Simple Harmonic Motion Connections between Uniform Circular Motion and Simple Harmonic Motion The Period of a Mass on a Spring
More informationHÉNON HEILES HAMILTONIAN CHAOS IN 2 D
ABSTRACT HÉNON HEILES HAMILTONIAN CHAOS IN D MODELING CHAOS & COMPLEXITY 008 YOUVAL DAR PHYSICS DAR@PHYSICS.UCDAVIS.EDU Chaos in two degrees of freedom, demonstrated b using the Hénon Heiles Hamiltonian
More informationStatic Equilibrium, Gravitation, Periodic Motion
This test covers static equilibrium, universal gravitation, and simple harmonic motion, with some problems requiring a knowledge of basic calculus. Part I. Multiple Choice 1. 60 A B 10 kg A mass of 10
More informationMassachusetts Institute of Technology Department of Physics. Final Examination December 17, 2004
Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 004 Final Examination December 17, 004 Instructions Do not start until you are told to do so. Solve
More informationPHYSICS. Chapter 15 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.
PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 15 Lecture RANDALL D. KNIGHT Chapter 15 Oscillations IN THIS CHAPTER, you will learn about systems that oscillate in simple harmonic
More informationAP Pd 3 Rotational Dynamics.notebook. May 08, 2014
1 Rotational Dynamics Why do objects spin? Objects can travel in different ways: Translation all points on the body travel in parallel paths Rotation all points on the body move around a fixed point An
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 informationSTABILITY. Phase portraits and local stability
MAS271 Methods for differential equations Dr. R. Jain STABILITY Phase portraits and local stability We are interested in system of ordinary differential equations of the form ẋ = f(x, y), ẏ = g(x, y),
More informationS13 PHY321: Final May 1, NOTE: Show all your work. No credit for unsupported answers. Turn the front page only when advised by the instructor!
Name: Student ID: S13 PHY321: Final May 1, 2013 NOTE: Show all your work. No credit for unsupported answers. Turn the front page only when advised by the instructor! The exam consists of 6 problems (60
More informationPhysics 8, Fall 2011, equation sheet work in progress
1 year 3.16 10 7 s Physics 8, Fall 2011, equation sheet work in progress circumference of earth 40 10 6 m speed of light c = 2.9979 10 8 m/s mass of proton or neutron 1 amu ( atomic mass unit ) = 1 1.66
More informationChapter 14 Oscillations. Copyright 2009 Pearson Education, Inc.
Chapter 14 Oscillations Oscillations of a Spring Simple Harmonic Motion Energy in the Simple Harmonic Oscillator Simple Harmonic Motion Related to Uniform Circular Motion The Simple Pendulum The Physical
More informationMAS212 Assignment #2: The damped driven pendulum
MAS Assignment #: The damped driven pendulum Sam Dolan (January 8 Introduction In this assignment we study the motion of a rigid pendulum of length l and mass m, shown in Fig., using both analytical and
More informationPHYSICS 110A : CLASSICAL MECHANICS
PHYSICS 110A : CLASSICAL MECHANICS 1. Introduction to Dynamics motion of a mechanical system equations of motion : Newton s second law ordinary differential equations (ODEs) dynamical systems simple 2.
More informationNewton's Laws You should be able to state these laws using both words and equations.
Review before first test Physical Mechanics Fall 000 Newton's Laws You should be able to state these laws using both words and equations. The nd law most important for meteorology. Second law: net force
More informationChapter 15 Periodic Motion
Chapter 15 Periodic Motion Slide 1-1 Chapter 15 Periodic Motion Concepts Slide 1-2 Section 15.1: Periodic motion and energy Section Goals You will learn to Define the concepts of periodic motion, vibration,
More informationChapter 14 Oscillations
Chapter 14 Oscillations Chapter Goal: To understand systems that oscillate with simple harmonic motion. Slide 14-2 Chapter 14 Preview Slide 14-3 Chapter 14 Preview Slide 14-4 Chapter 14 Preview Slide 14-5
More informationChapter 14. PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman. Lectures by Wayne Anderson
Chapter 14 Periodic Motion PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 14 To describe oscillations in
More informationHomework 1. Due Thursday, January 21
Homework 1. Due Thursday, January 21 Problem 1. Rising Snake A snake of length L and linear mass density ρ rises from the table. It s head is moving straight up with the constant velocity v. What force
More informationLecture 41: Highlights
Lecture 41: Highlights The goal of this lecture is to remind you of some of the key points that we ve covered this semester Note that this is not the complete set of topics that may appear on the final
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 informationPHYS 1114, Lecture 33, April 10 Contents:
PHYS 1114, Lecture 33, April 10 Contents: 1 This class is o cially cancelled, and has been replaced by the common exam Tuesday, April 11, 5:30 PM. A review and Q&A session is scheduled instead during class
More information520 Chapter 9. Nonlinear Differential Equations and Stability. dt =
5 Chapter 9. Nonlinear Differential Equations and Stabilit dt L dθ. g cos θ cos α Wh was the negative square root chosen in the last equation? (b) If T is the natural period of oscillation, derive the
More informationPhys 7221, Fall 2006: Midterm exam
Phys 7221, Fall 2006: Midterm exam October 20, 2006 Problem 1 (40 pts) Consider a spherical pendulum, a mass m attached to a rod of length l, as a constrained system with r = l, as shown in the figure.
More informationMidterm EXAM PHYS 401 (Spring 2012), 03/20/12
Midterm EXAM PHYS 401 (Spring 2012), 03/20/12 Name: Signature: Duration: 75 minutes Show all your work for full/partial credit! In taking this exam you confirm to adhere to the Aggie Honor Code: An Aggie
More informationPHYSICS 221, FALL 2011 EXAM #2 SOLUTIONS WEDNESDAY, NOVEMBER 2, 2011
PHYSICS 1, FALL 011 EXAM SOLUTIONS WEDNESDAY, NOVEMBER, 011 Note: The unit vectors in the +x, +y, and +z directions of a right-handed Cartesian coordinate system are î, ĵ, and ˆk, respectively. In this
More informationChapter 5 Oscillatory Motion
Chapter 5 Oscillatory Motion Simple Harmonic Motion An object moves with simple harmonic motion whenever its acceleration is proportional to its displacement from some equilibrium position and is oppositely
More informationUnforced Oscillations
Unforced Oscillations Simple Harmonic Motion Hooke s Law Newton s Second Law Method of Force Competition Visualization of Harmonic Motion Phase-Amplitude Conversion The Simple Pendulum and The Linearized
More informationSystem Control Engineering 0
System Control Engineering 0 Koichi Hashimoto Graduate School of Information Sciences Text: Nonlinear Control Systems Analysis and Design, Wiley Author: Horacio J. Marquez Web: http://www.ic.is.tohoku.ac.jp/~koichi/system_control/
More informationRaymond A. Serway Chris Vuille. Chapter Thirteen. Vibrations and Waves
Raymond A. Serway Chris Vuille Chapter Thirteen Vibrations and Waves Periodic Motion and Waves Periodic motion is one of the most important kinds of physical behavior Will include a closer look at Hooke
More information10 Back to planar nonlinear systems
10 Back to planar nonlinear sstems 10.1 Near the equilibria Recall that I started talking about the Lotka Volterra model as a motivation to stud sstems of two first order autonomous equations of the form
More informationHarmonic Oscillator - Model Systems
3_Model Systems HarmonicOscillators.nb Chapter 3 Harmonic Oscillator - Model Systems 3.1 Mass on a spring in a gravitation field a 0.5 3.1.1 Force Method The two forces on the mass are due to the spring,
More informationAnalytical Mechanics ( AM )
Analytical Mechanics ( AM ) Olaf Scholten KVI, kamer v8; tel nr 6-55; email: scholten@kvinl Web page: http://wwwkvinl/ scholten Book: Classical Dynamics of Particles and Systems, Stephen T Thornton & Jerry
More informationEE Homework 3 Due Date: 03 / 30 / Spring 2015
EE 476 - Homework 3 Due Date: 03 / 30 / 2015 Spring 2015 Exercise 1 (10 points). Consider the problem of two pulleys and a mass discussed in class. We solved a version of the problem where the mass was
More informationPHYSICS 1 Simple Harmonic Motion
Advanced Placement PHYSICS 1 Simple Harmonic Motion Student 014-015 What I Absolutely Have to Know to Survive the AP* Exam Whenever the acceleration of an object is proportional to its displacement and
More informationPHYSICS PART II SECTION- I. Straight objective Type
PHYSICS PAT II SECTION- I Straight objective Tpe This section contains 9 multiple choice questions numbered to 1. Each question has choices,, (C) and, out of which ONLY ONE is correct.. A parallel plate
More informationModelling in Biology
Modelling in Biology Dr Guy-Bart Stan Department of Bioengineering 17th October 2017 Dr Guy-Bart Stan (Dept. of Bioeng.) Modelling in Biology 17th October 2017 1 / 77 1 Introduction 2 Linear models of
More informationis conserved, calculating E both at θ = 0 and θ = π/2 we find that this happens for a value ω = ω given by: 2g
UNIVERSITETET I STAVANGER Institutt for matematikk og naturvitenskap Suggested solutions, FYS 500 Classical Mechanics Theory 2016 fall Set 5 for 23. September 2016 Problem 27: A string can only support
More information= F ( x; µ) (1) where x is a 2-dimensional vector, µ is a parameter, and F :
1 Bifurcations Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 A bifurcation is a qualitative change
More informationGravitational potential energy
Gravitational potential energ m1 Consider a rigid bod of arbitrar shape. We want to obtain a value for its gravitational potential energ. O r1 1 x The gravitational potential energ of an assembl of N point-like
More information