Course: Nonlinear Dynamics. Laurette TUCKERMAN Maps, Period Doubling and Floquet Theory
|
|
- Abel James
- 5 years ago
- Views:
Transcription
1 Course: Nonlinear Dynamics Laurette TUCKERMAN Maps, Period Doubling and Floquet Theory
2 Discrete Dynamical Systems or Mappings y n+1 = g(y n ) y,g R N Fixed point: ȳ = g(ȳ) 1D linear stability ofȳ: Sety n = ȳ + ǫ n ȳ + ǫ n+1 = g(ȳ + ǫ n ) = g(ȳ) + g (ȳ)ǫ n g (ȳ)ǫ 2 n ǫ n+1 g (ȳ)ǫ n g (ȳ) <> 1 ǫ ȳ stable / unstable Superstability: g (ȳ) = 0 = ǫ n g (ȳ)ǫ 2 n
3 Multidimensional system g (ȳ) = Dg(ȳ) (Jacobian) ȳ stable all eigenvaluesµof Dg(ȳ) inside unit circle µ exits at ( 1,0) µ exits ate ±iθ µ exits at (1,0)
4 Illustrate exit at (±1,0) via graphical construction (+1, 0) ( 1, 0)
5 Change of stability Bifurcations Saddle-node bifurcation: ẋ = µ x 2 x n+1 x n = µ x 2 n x n+1 = f(x n ) x n + µ x 2 n f (x n ) = 1 2x n µ < 0 µ > 0 no fixed point x = ± µ f (± µ) = 1 2 µ 1
6 Supercritical pitchfork bifurcation: ẋ = µx x 3 x n+1 x n = µx n x 3 n x n+1 = f(x n ) x n + µx n x 3 n f (x n ) = 1 + µ 3x 2 n µ < 0 µ > 0 x = 0 x = 0,± µ f (0) = 1 + µ < 1 f (0) = 1 + µ > 1 f (± µ) = 1 2µ < 1
7 Subcritical pitchfork bifurcation: ẋ = µx + x 3 x n+1 x n = µx n + x 3 n x n+1 = f(x n ) x n + µx n + x 3 n f (x n ) = 1 + µ + 3x 2 n µ < 0 µ > 0 x = 0,± µ x = 0 f (0) = 1 + µ < 1 f (0) = 1 + µ > 1 f (± µ) = 1 2µ < 1
8 Eig at Bifurcations Leads to (+1,0) saddle-node, pitchfork, transcritical other steady states e ±iθ secondary Hopf or Neimark-Sacker torus (next chapter) ( 1,0) flip or period-doubling two-cycle Period-doubling: impossible for continuous dynamical systems Illustrate via logistic map: x n+1 = ax n (1 x n ) { multiple of x n for x n small, but Next value x n+1 is reduced whenx n too large Mentioned in 1800s, popularized in 1970s: models population Famous period-doubling cascade discovered in 1970s by Feigenbaum in Los Alamos, U.S., and by Coullet and Tresser in Nice, France
9 Logistic Map x n+1 = f(x n ) ax n (1 x n ) for x n [0,1], 0 < a < 4 Fixed points: x = a x(1 x) { x = 0 or = 1 = a(1 x) = 1 x = 1/a = x = 1 1/a f(x) = ax(1 x) for x = 0 and x = 1 1/a a = 0.4, 1.2, 2.0, 2.8, 3.6 as function of a
10 Stability f(x) = ax(1 x) f (x) = a(1 x) ax = a(1 2x) f (0) = a = f (0) < 1 for a < 1 ( f 1 1 ) ( ( = a )) = a + 2 a ( a 1 < f 1 1 ) < 1 a 1 < a + 2 < 1 1 < a 2 < 1 transcrit: 1 < a < 3???
11 Graphical construction a = 0.8: x n 0 a = 2.0: x n 1 1/a a = 2.6: x n 1 1/a a = 3.04: x n 2 cycle
12 Seek two-cycle formed ata = 3, wheref ( x) = 1 f(x 1 ) = x 2 and f(x 2 ) = x 1 = f 2 (x 1 ) f(f(x 1 )) = x 1 f 2 (x) = af(x)[1 f(x)] = a(ax(1 x)) [1 ax(1 x)] = a 2 x(1 x) [ 1 ax + ax 2] = a 2 x [ 1 ax + ax 2 x + ax 2 ax 3] = a 2 x [ 1 (1 + a)x + 2ax 2 ax 3] Seek fixed points of f 2 : 0 = f 2 (x) x = x [ a 2 (1 (1 + a)x + 2ax 2 ax 3 ) 1 ] contains factors x and (x (1 1/a)) = x(a(x 1) + 1) [ ax 2 (a + 1)x + 4(a + 1) ] x 1,2 = a + 1 ± (a 3)(a + 1) 2a for a > 3
13 f 2 undergoes pitchfork bifurcation a Fixed Points 1.6 x = 0 unstable x = 1 1/a stable 2.3 x = 0 unstable x 1,2 stable x = 1 1/a unstable together comprise two-cycle for f
14 Stability of two-cycle d dx f2 (x 1 ) = f (f(x 1 ))f (x 1 ) = f (x 2 )f (x 1 ) x 1 + x 2 = a = a + 1 x 1 x 2 = a + 1 a a 2 f (x 1 )f (x 2 ) = a(1 2x 1 ) a(1 2x 2 ) = a 2 (1 2(x 1 + x 2 + 4x 1 x 2 ) ( ) ( )) a + 1 a + 1 = a ( a a 2 = a 2 2a(a + 1) + 4(a + 1) = a 2 + 2a = f (x 1 )f (x 2 ) 1 = a 2 + 2a = a 2 + 2a + 3 a = 2 ± = 1 ± 16 2 = 2 ± 4 2 = 3 pitchfork 2-cycle 0 = f (x 1 )f (x 2 ) + 1 = a 2 + 2a = a 2 + 2a + 5 a = 2 ± = = flip bif 4-cycle 2
15 Period-doubling cascade Successive period-doubling bifs occur at successively smaller intervals in a and accumulate at a = n a n n a n a n 1 δ n n 1 / n
16 Renormalization Feigenbaum (1979), Collet & Eckmann (1980), Lanford (1982) f(x) = 1 rx 2 on [ 1,1] T acts on mappingsf: (Tf)(x) 1 α f2 ( αx)
17 Seek fixed point of T : f(x) (Tf)(x) 1 rx 2 f 2 ( αx)/α = f(1 r(αx) 2 )/α = (1 r(1 r(αx) 2 ) 2 )/α = (1 r(1 2r(αx) 2 + r 2 (αx) 4 ))/α = (1 r + 2r 2 (αx) 2 r 3 (αx) 4 )/α Matching constant and quadratic terms: r 1 α = 1 2r 2 α 2 α = r r 1 = α 2rα = 1 = 2r(r 1) = 1 α = r = (1 + 3)/2 = 1.366
18 f(x) (Tf)(x) (T 2 f)(x)... φ(x) 2nd order 4th order 8th order... where limiting function φ(x) = x x x is fixed point oft (mapping-of-mappings), i.e. T(φ) = φ Single unstable direction with multiplierδ = (table) φ, δ, are universal for all map families with quadratic maxima, such asasinπx, 1 rx 2,ax(1 x).
19 If f has a stable 2-cycle, thenf 2 has a stable fixed point. More generally, if f has a stable 2 n cycle, thenf 2 has a stable 2 n 1 cycle. Since T involves taking f 2, then applying T takes maps with 2 n cycles to maps with2 n 1 cycles.
20 Other classes of unimodal maps, characterized by the nature of their extrema, undergo a period-doubling cascade. Each class has its own asymptotic value of δ and α. Examples are functions with a quartic maximum or the tent map { ax for x < 1/2 f(x) = a(1 x) for x > 1/2 a < 1: 0 is stable fixed point a > 1: 0 is unstable fixed point x = a/(1 + a) is stable
21 Periodic Windows f 3 has 3 saddle-node bifurcations at a 3 = = = two 3-cycles, one stable and one unstable ( (f 3 ) 1) Stable and unstablen-cycles created whenf n crosses diagonal
22 Bifurcation Diagram for Logistic Map Each n-cycle undergoes period-doubling cascade
23 Three-cycle: before and after a = a = 3.83 a = intermittency 3-cycle 6-cycle Stable three-cycle for = < r < 3.857
24 Intermittency Slow dynamics near a bifurcation Type I Type III near saddle-node bif near period-doubling bif near pitchfork bif g = x n x 2 n f = 1.2 x n x 3 n off 2 Slow dynamics near ghosts of not-yet-created or unstable fixed points Assume that another mechanism re-injects trajectories back tox 0 Type II intermittency associated with a subcritical Hopf bifurcation
25 Sharkovskii Ordering (odd numbers) (multiples of 2) (multiples of 2 2 ) (powers of 2) Sharkovskii s Theorem: f has a k-cycle = f has an l-cycle for any l k f has a 3-cycle = f has an l-cycle for anyl (Most cycles not stable) Logistic map for any r > r 3 has cycles of all lengths.
26 Period-doubling in Rayleigh-Bénard convection From Libchaber, Fauve & Laroche, Physica D (1983). A: freq f and f/2 = B:f/4 = C:f/8 = D:f/16
27 Rayleigh-Bénard convection in small-aspect-ratio container Type III intermittency Timeseries Poincaré map maxima (I k,i k+2 ) Subharmonic (period-doubled signal) grows til burst, followed by laminar phase From M. Dubois, M.A. Rubio & P. Bergé, Phys. Rev. Lett. 51, 1446 (1983).
28 Rayleigh-Bénard convection in small-aspect-ratio container Type I intermittency non-intermittent timeseries Ra = 280 Ra c intermittent timeseries Ra = 300 Ra c From P. Bergé, M. Dubois, P. Manneville & Y. Pomeau, J. Phys. (Paris) Lett. 41, 341 (1980).
29 Poincaré map of Lorenz system 3D trajectory of the Lorenz system Timeseries of (t) for standard chaotic value of r = 28 Timeseries ofz(t) From P. Manneville, Course notes
30 Successive pairs of maxima ofz resemble tent map: From P. Manneville, Course notes
31 From continuous flows to discrete maps limit cycle after pitchfork after period-doubling
32 Where do discrete dynamical systems come from? Hopf or global bifurcations= limit cycles Study these using Poincaré or first return map: Continuous dynamical system x(t) R d, x d (t) goes throughβ for some set of trajectories y n+1 = g(y n ) x(t) = (y n,β), ẋ d (t) > 0 x(t ) = (y n+1,β), ẋ d (t ) > 0 above not satisfied for any t (t,t ) From P. Manneville class notes, DEA Physique des Liquides From Moehlis, Josic, & Shea-Brown, Scholarpedia
33 Floquet theory Linear equations with constant coefficients: aẍ + bẋ + cx = 0 = x(t) = α 1 e λ1t + α 2 e λ 2t where aλ 2 + bλ + c = 0 ẋ = cx = x(t) = e ct x(0) N N c n x (n) = 0 = x(t) = α n e λ nt n=0 where n=1 N c n λ n = 0 n=0 Generalize to linear equations with periodic coefficients: a(t)ẍ + b(t)ẋ + c(t)x = 0 = x(t) = α 1 (t)e λ1t + α 2 (t)e λ 2t a(t),b(t),c(t) have period T = α 1 (t),α 2 (t) have period T
34 Floquet theory continued a(t)ẍ + b(t)ẋ + c(t)x = 0 = x(t) = α 1 (t)e λ 1t + α 2 (t)e λ 2t α 1 (t),α 2 (t) Floquet functions λ 1, λ 2 e λ1t, e λ 2T Floquet exponents Floquet multipliers λ 1,λ 2 not roots of polynomial = must calculate numerically or asymptotically ẋ = c(t)x = x(t) = e λt α(t) N N c n (t)x (n) = 0 = x(t) = e λnt α n (t) n=0 n=1
35 Floquet theory and linear stability analysis Dynamical system: Limit cycle solution: Stability of x(t) : ẋ = f(x) x(t + T) = x(t) with x(t) = f( x(t)) x(t) = x(t) + ǫ(t) x + ǫ = f( x(t)) + f ( x(t))ǫ(t) +... ǫ = f ( x(t))ǫ(t) Floquet form! ǫ(t) = e λt α(t) with α(t) of period T Re(λ) > 0 = x(t) unstable Re(λ) < 0 = x(t) stable λ complex = most unstable or least stable pert has period different from x(t)
36 Region of stability for exponent λ for multiplier e λt Imaginary part non-unique = choose Im(λ) ( πi/t, πi/t] In R N, x(t) stable iff real parts of all λ j are negative Monodromy matrix: Ṁ = Df( x(t))m withm(t = 0) = I Floquet multipliers/functions = eigenvalues/vectors ofm(t)
37 Faraday instability Faraday (1831): Vertical vibration of fluid layer = stripes, squares, hexagons In 1990s: first fluid-dynamical quasicrystals: Edwards & Fauve Kudrolli, Pier & Gollub J. Fluid Mech. (1994) Physica D (1998)
38 Oscillating frame of reference = oscillating gravity G(t) = g(1 acos(ωt)) G(t) = g(1 a[cos(mωt) + δcos(nωt + φ 0 )]) Flat surface becomes linearly unstable for sufficiently higha Consider domain to be horizontally infinite (homogeneous) = solutions exponential/trigonometric in x = (x,y) Seek bounded solutions = trigonometric: exp(ik x) Height ζ(x,y,t) = k e ik xˆζk (t) Oscillating gravity = temporal Floquet problem, T = 2π/ω ˆζ k (t) = j e λj k t f j k (t)
39 Height ζ(x,y,t) = k e ik xˆζ k (t) Ideal fluids (no viscosity), sinusoidal forcing= Mathieu eq. ω t ˆζ k + ω 2 0 [1 acos(ωt)] ˆζ k = 0 combinesg, densities, surface tension, wavenumber k 2 ˆζ k (t) = e λj k t f j k (t) j=1 Re(λ j k ) > 0 for somej,k= ˆζ k ր= flat surface unstable = Faraday waves with wavelength 2π/k Im(λ j k ) eλt waves period 0 1 harmonic same as forcing ω/2 1 subharmonic twice forcing period
40
41 Floquet functions λ (ζ-< ζ >)/ t / T λ (ζ-< ζ >)/ t / T (ζ-< ζ >)/ t / T within tongue 1 /2 within tongue 2 /2 within tongue 3 /2 subharmonic harmonic subharmonic µ = 1 µ = +1 µ = 1 λ From Périnet, Juric & Tuckerman, J. Fluid Mech. (2009)
42 Hexagonal patterns in Faraday instability From Périnet, Juric & Tuckerman, J. Fluid Mech. (2009)
43 Ideal flow Cylinder wake with downstream recirculation zone Von Kármán vortex street (Re 46) Laboratory experiment (Taneda, 1982) Off Chilean coast past Juan Fernandez islands
44 von Kárman vortex street: Re = U d/ν 46 spatially: temporally: two-dimensional (x,y) periodic,st = fd/u (homogeneous inz) appears spontaneously U 2D (x,y,t mod T)
45 Stability analysis of von Kármán vortex street 2D limit cycle U 2D (x,y,tmod T) obeys: t U 2D = (U 2D )U 2D P 2D + 1 Re U 2D Perturbation obeys: t u 3D = (U 2D (t) )u 3D (u 3D )U 2D (t) p 3D + 1 Re u 3D Equation homogeneous inz, periodic in t= u 3D (x,y,z,t) e iβz e λβt f β (x,y,t mod T) Fixβ, calculate largestµ = e λβt via linearized Navier-Stokes
46 From Barkley & Henderson, J. Fluid Mech. (1996) mode A:Re c = β c = = λ c 4 mode B:Re c = 259 β c = 7.64 = λ c 1 Temporally: µ = 1 = steady bifurcation to limit cycle with same periodicity asu 2D Spatially: circle pitchfork (any phase in z)
47 mode A at Re = 210 mode B at Re = 250 From M.C. Thompson, Monash University, Australia ( mct/mct/docs/cylinder.html)
48 transition to mode A initially faster than exp = subcritical transition to mode B initially slower than exp = supercritical
49 Describe via complex amplitudesa n,b n amplitude and phase of mode A, B at fixed instant of cycle A n+1 = ( µ A + α A A n 2 A n β A A n 4) A n B n+1 = ( µ B α B B n 2) B n
50 Numerical and experimental frequencies From Barkley & Henderson, JFM (1996) Solution to minimal model From Barkley, Tuckerman & Golubitsky, PRE (2000) Interaction between A and B: A n+1 = ( µ A + α A A n 2 + γ A B n 2 β A A n 4) A n B n+1 = ( µ B α B B n 2 + γ B A n 2) B n As Re is increased: mode A = A + B = modeb
Hydrodynamics. Class 11. Laurette TUCKERMAN
Hydrodynamics Class 11 Laurette TUCKERMAN laurette@pmmh.espci.fr Faraday instability Faraday (1831): Vertical vibration of fluid layer = stripes, squares, hexagons In 1990s: first fluid-dynamical quasicrystals:
More informationNonlinear Dynamics. Laurette TUCKERMAN Maps, Period Doubling and Floquet Theory
Nonlinear Dynamics Laurette TUCKERMAN laurette@pmmh.espci.fr Maps, Period Doubling and Floquet Theory 1 1 Discrete Dynamical Systems or Mappings A discrete dynamical system is of the form y n+1 = g(y n
More informationBoulder School for Condensed Matter and Materials Physics. Laurette Tuckerman PMMH-ESPCI-CNRS
Boulder School for Condensed Matter and Materials Physics Laurette Tuckerman PMMH-ESPCI-CNRS laurette@pmmh.espci.fr Dynamical Systems: A Basic Primer 1 1 Basic bifurcations 1.1 Fixed points and linear
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 informationLaurette TUCKERMAN Codimension-two points
Laurette TUCKERMAN laurette@pmmh.espci.fr Codimension-two points The 1:2 mode interaction System with O(2) symmetry with competing wavenumbersm = 1 andm = 2 Solutions approximated as: u(θ,t) = z 1 (t)e
More informationPart II. Dynamical Systems. Year
Part II Year 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2017 34 Paper 1, Section II 30A Consider the dynamical system where β > 1 is a constant. ẋ = x + x 3 + βxy 2, ẏ = y + βx 2
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 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 informationPHY411 Lecture notes Part 4
PHY411 Lecture notes Part 4 Alice Quillen February 1, 2016 Contents 0.1 Introduction.................................... 2 1 Bifurcations of one-dimensional dynamical systems 2 1.1 Saddle-node bifurcation.............................
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 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 information16 Period doubling route to chaos
16 Period doubling route to chaos We now study the routes or scenarios towards chaos. We ask: How does the transition from periodic to strange attractor occur? The question is analogous to the study of
More informationLaurette TUCKERMAN Rayleigh-Bénard Convection and Lorenz Model
Laurette TUCKERMAN laurette@pmmh.espci.fr Rayleigh-Bénard Convection and Lorenz Model Rayleigh-Bénard Convection Rayleigh-Bénard Convection Boussinesq Approximation Calculation and subtraction of the basic
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 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 informationProblem Set Number 2, j/2.036j MIT (Fall 2014)
Problem Set Number 2, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Mon., September 29, 2014. 1 Inverse function problem #01. Statement: Inverse function
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,
More informationNonlinear Dynamics and Chaos
Ian Eisenman eisenman@fas.harvard.edu Geological Museum 101, 6-6352 Nonlinear Dynamics and Chaos Review of some of the topics covered in homework problems, based on section notes. December, 2005 Contents
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 informationQuasipatterns in surface wave experiments
Quasipatterns in surface wave experiments Alastair Rucklidge Department of Applied Mathematics University of Leeds, Leeds LS2 9JT, UK With support from EPSRC A.M. Rucklidge and W.J. Rucklidge, Convergence
More informationUnstable Periodic Orbits as a Unifying Principle in the Presentation of Dynamical Systems in the Undergraduate Physics Curriculum
Unstable Periodic Orbits as a Unifying Principle in the Presentation of Dynamical Systems in the Undergraduate Physics Curriculum Bruce M. Boghosian 1 Hui Tang 1 Aaron Brown 1 Spencer Smith 2 Luis Fazendeiro
More informationPeriod doubling cascade in mercury, a quantitative measurement
Period doubling cascade in mercury, a quantitative measurement A. Libchaber, C. Laroche, S. Fauve To cite this version: A. Libchaber, C. Laroche, S. Fauve. Period doubling cascade in mercury, a quantitative
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 08, 2018, at 08 30 12 30 Johanneberg Kristian
More informationLaurette TUCKERMAN Numerical Methods for Differential Equations in Physics
Laurette TUCKERMAN laurette@pmmh.espci.fr Numerical Methods for Differential Equations in Physics Time stepping: Steady state solving: 0 = F(U) t U = LU + N(U) 0 = LU + N(U) Newton s method 0 = F(U u)
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 4. Bifurcations Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Local bifurcations for vector fields 1.1 The problem 1.2 The extended centre
More information7 Two-dimensional bifurcations
7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed
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 informationBIFURCATION TO TRAVELING WAVES IN THE CUBIC-QUINTIC COMPLEX GINZBURG LANDAU EQUATION
BIFURCATION TO TRAVELING WAVES IN THE CUBIC-QUINTIC COMPLEX GINZBURG LANDAU EQUATION JUNGHO PARK AND PHILIP STRZELECKI Abstract. We consider the 1-dimensional complex Ginzburg Landau equation(cgle) which
More informationOne Dimensional Dynamical Systems
16 CHAPTER 2 One Dimensional Dynamical Systems We begin by analyzing some dynamical systems with one-dimensional phase spaces, and in particular their bifurcations. All equations in this Chapter are scalar
More informationSpatio-Temporal Chaos in Pattern-Forming Systems: Defects and Bursts
Spatio-Temporal Chaos in Pattern-Forming Systems: Defects and Bursts with Santiago Madruga, MPIPKS Dresden Werner Pesch, U. Bayreuth Yuan-Nan Young, New Jersey Inst. Techn. DPG Frühjahrstagung 31.3.2006
More informationStability of flow past a confined cylinder
Stability of flow past a confined cylinder Andrew Cliffe and Simon Tavener University of Nottingham and Colorado State University Stability of flow past a confined cylinder p. 1/60 Flow past a cylinder
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 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 informationThe Big, Big Picture (Bifurcations II)
The Big, Big Picture (Bifurcations II) Reading for this lecture: NDAC, Chapter 8 and Sec. 10.0-10.4. 1 Beyond fixed points: Bifurcation: Qualitative change in behavior as a control parameter is (slowly)
More informationSolutions to homework assignment #7 Math 119B UC Davis, Spring for 1 r 4. Furthermore, the derivative of the logistic map is. L r(x) = r(1 2x).
Solutions to homework assignment #7 Math 9B UC Davis, Spring 0. A fixed point x of an interval map T is called superstable if T (x ) = 0. Find the value of 0 < r 4 for which the logistic map L r has a
More informationBifurcation of Fixed Points
Bifurcation of Fixed Points CDS140B Lecturer: Wang Sang Koon Winter, 2005 1 Introduction ẏ = g(y, λ). where y R n, λ R p. Suppose it has a fixed point at (y 0, λ 0 ), i.e., g(y 0, λ 0 ) = 0. Two Questions:
More informationCan weakly nonlinear theory explain Faraday wave patterns near onset?
Under consideration for publication in J. Fluid Mech. 1 Can weakly nonlinear theory explain Faraday wave patterns near onset? A. C. S K E L D O N 1, and A. M. R U C K L I D G E 2 1 Department of Mathematics,
More informationPoincaré Map, Floquet Theory, and Stability of Periodic Orbits
Poincaré Map, Floquet Theory, and Stability of Periodic Orbits CDS140A Lecturer: W.S. Koon Fall, 2006 1 Poincaré Maps Definition (Poincaré Map): Consider ẋ = f(x) with periodic solution x(t). Construct
More information6 Linear Equation. 6.1 Equation with constant coefficients
6 Linear Equation 6.1 Equation with constant coefficients Consider the equation ẋ = Ax, x R n. This equating has n independent solutions. If the eigenvalues are distinct then the solutions are c k e λ
More informationEntrance Exam, Differential Equations April, (Solve exactly 6 out of the 8 problems) y + 2y + y cos(x 2 y) = 0, y(0) = 2, y (0) = 4.
Entrance Exam, Differential Equations April, 7 (Solve exactly 6 out of the 8 problems). Consider the following initial value problem: { y + y + y cos(x y) =, y() = y. Find all the values y such that the
More informationarxiv:patt-sol/ v1 7 Mar 1997
Hexagonal patterns in finite domains P.C. Matthews Department of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, United Kingdom arxiv:patt-sol/9703002v1 7 Mar 1997 Abstract In many
More informationVIII. Nonlinear Tourism
Université Pierre et Marie Curie Master Sciences et Technologie (M2) Spécialité : Concepts fondamentaux de la physique Parcours : Physique des Liquides et Matière Molle Cours : Dynamique Non-Linéaire Laurette
More information2 Qualitative theory of non-smooth dynamical systems
2 Qualitative theory of non-smooth dynamical systems In this chapter, we give an overview of the basic theory of both smooth and non-smooth dynamical systems, to be expanded upon in later chapters. In
More informationSimplest Chaotic Flows with Involutional Symmetries
International Journal of Bifurcation and Chaos, Vol. 24, No. 1 (2014) 1450009 (9 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414500096 Simplest Chaotic Flows with Involutional Symmetries
More informationSolutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10)
Solutions for B8b (Nonlinear Systems) Fake Past Exam (TT 10) Mason A. Porter 15/05/2010 1 Question 1 i. (6 points) Define a saddle-node bifurcation and show that the first order system dx dt = r x e x
More informationLecture 5. Outline: Limit Cycles. Definition and examples How to rule out limit cycles. Poincare-Bendixson theorem Hopf bifurcations Poincare maps
Lecture 5 Outline: Limit Cycles Definition and examples How to rule out limit cycles Gradient systems Liapunov functions Dulacs criterion Poincare-Bendixson theorem Hopf bifurcations Poincare maps Limit
More informationMathematics for Engineers II. lectures. Differential Equations
Differential Equations Examples for differential equations Newton s second law for a point mass Consider a particle of mass m subject to net force a F. Newton s second law states that the vector acceleration
More informationGeorgia Institute of Technology. Nonlinear Dynamics & Chaos Physics 4267/6268. Faraday Waves. One Dimensional Study
Georgia Institute of Technology Nonlinear Dynamics & Chaos Physics 4267/6268 Faraday Waves One Dimensional Study Juan Orphee, Paul Cardenas, Michael Lane, Dec 8, 2011 Presentation Outline 1) Introduction
More informationLotka Volterra Predator-Prey Model with a Predating Scavenger
Lotka Volterra Predator-Prey Model with a Predating Scavenger Monica Pescitelli Georgia College December 13, 2013 Abstract The classic Lotka Volterra equations are used to model the population dynamics
More informationBifurcations in the Quadratic Map
Chapter 14 Bifurcations in the Quadratic Map We will approach the study of the universal period doubling route to chaos by first investigating the details of the quadratic map. This investigation suggests
More informationComplex Dynamic Systems: Qualitative vs Quantitative analysis
Complex Dynamic Systems: Qualitative vs Quantitative analysis Complex Dynamic Systems Chiara Mocenni Department of Information Engineering and Mathematics University of Siena (mocenni@diism.unisi.it) Dynamic
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 informationThree-dimensional Floquet stability analysis of the wake in cylinder arrays
J. Fluid Mech. (7), vol. 59, pp. 79 88. c 7 Cambridge University Press doi:.7/s78798 Printed in the United Kingdom 79 Three-dimensional Floquet stability analysis of the wake in cylinder arrays N. K.-R.
More informationSolution to Homework #5 Roy Malka 1. Questions 2,3,4 of Homework #5 of M. Cross class. dv (x) dx
Solution to Homework #5 Roy Malka. Questions 2,,4 of Homework #5 of M. Cross class. Bifurcations: (M. Cross) Consider the bifurcations of the stationary solutions of a particle undergoing damped one dimensional
More informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Systems of Differential Equations Phase Plane Analysis Hanz Richter Mechanical Engineering Department Cleveland State University Systems of Nonlinear
More information1. < 0: the eigenvalues are real and have opposite signs; the fixed point is a saddle point
Solving a Linear System τ = trace(a) = a + d = λ 1 + λ 2 λ 1,2 = τ± = det(a) = ad bc = λ 1 λ 2 Classification of Fixed Points τ 2 4 1. < 0: the eigenvalues are real and have opposite signs; the fixed point
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 14, 2019, at 08 30 12 30 Johanneberg Kristian
More informationPolynomial Functions
Polynomial Functions Equations and Graphs Characteristics The Factor Theorem The Remainder Theorem http://www.purplemath.com/modules/polyends5.htm 1 A cross-section of a honeycomb has a pattern with one
More informationBifurcations in systems with Z 2 spatio-temporal and O(2) spatial symmetry
Physica D 189 (2004) 247 276 Bifurcations in systems with Z 2 spatio-temporal and O(2) spatial symmetry F. Marques a,, J.M. Lopez b, H.M. Blackburn c a Departament de Física Aplicada, Universitat Politècnica
More information6. Well-Stirred Reactors III
6. Well-Stirred Reactors III Reactors reaction rate or reaction velocity defined for a closed system of uniform pressure, temperature, and composition situation in a real reactor is usually quite different
More informationRoute to chaos for a two-dimensional externally driven flow
PHYSICAL REVIEW E VOLUME 58, NUMBER 2 AUGUST 1998 Route to chaos for a two-dimensional externally driven flow R. Braun, 1 F. Feudel, 1,2 and P. Guzdar 2 1 Institut für Physik, Universität Potsdam, PF 601553,
More information4 Problem Set 4 Bifurcations
4 PROBLEM SET 4 BIFURCATIONS 4 Problem Set 4 Bifurcations 1. Each of the following functions undergoes a bifurcation at the given parameter value. In each case use analytic or graphical techniques to identify
More informationSummary of topics relevant for the final. p. 1
Summary of topics relevant for the final p. 1 Outline Scalar difference equations General theory of ODEs Linear ODEs Linear maps Analysis near fixed points (linearization) Bifurcations How to analyze a
More informationTo link to this article : DOI: /S URL :
Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationLectures on Dynamical Systems. Anatoly Neishtadt
Lectures on Dynamical Systems Anatoly Neishtadt Lectures for Mathematics Access Grid Instruction and Collaboration (MAGIC) consortium, Loughborough University, 2007 Part 3 LECTURE 14 NORMAL FORMS Resonances
More informationLecture 3 : Bifurcation Analysis
Lecture 3 : Bifurcation Analysis D. Sumpter & S.C. Nicolis October - December 2008 D. Sumpter & S.C. Nicolis General settings 4 basic bifurcations (as long as there is only one unstable mode!) steady state
More informationNumerical simulation of Faraday waves
J. Fluid Mech. (29), vol. 63, pp. 1 26. c Cambridge University Press 29 doi:1.117/s22112971 1 Numerical simulation of Faraday waves NICOLAS PÉRINET 1, DAMIR JURIC 2 AND LAURETTE S. TUCKERMAN 1 1 Laboratoire
More informationChapter 7. Nonlinear Systems. 7.1 Introduction
Nonlinear Systems Chapter 7 The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. - Jules Henri Poincaré (1854-1912)
More informationQUASIPERIODIC RESPONSE TO PARAMETRIC EXCITATIONS
QUASIPERIODIC RESPONSE TO PARAMETRIC EXCITATIONS J. M. LOPEZ AND F. MARQUES Abstract. Parametric excitations are capable of stabilizing an unstable state, but they can also destabilize modes that are otherwise
More informationTitle. Author(s)Ito, Kentaro; Nishiura, Yasumasa. Issue Date Doc URL. Rights. Type. File Information
Title Intermittent switching for three repulsively coupled Author(s)Ito, Kentaro; Nishiura, Yasumasa CitationPhysical Review. E, statistical, nonlinear, and soft Issue Date 8- Doc URL http://hdl.handle.net/5/9896
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 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 informationDifferential equations, comprehensive exam topics and sample questions
Differential equations, comprehensive exam topics and sample questions Topics covered ODE s: Chapters -5, 7, from Elementary Differential Equations by Edwards and Penney, 6th edition.. Exact solutions
More informationMATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation
MATH 415, WEEK 11: Bifurcations in Multiple Dimensions, Hopf Bifurcation 1 Bifurcations in Multiple Dimensions When we were considering one-dimensional systems, we saw that subtle changes in parameter
More informationVortex wake and energy transitions of an oscillating cylinder at low Reynolds number
ANZIAM J. 46 (E) ppc181 C195, 2005 C181 Vortex wake and energy transitions of an oscillating cylinder at low Reynolds number B. Stewart J. Leontini K. Hourigan M. C. Thompson (Received 25 October 2004,
More informationSpatial and temporal resonances in a periodically forced hydrodynamic system
Physica D 136 (2) 34 352 Spatial and temporal resonances in a periodically forced hydrodynamic system F. Marques a,, J.M. Lopez b a Departament de Física Aplicada, Universitat Politècnica de Catalunya,
More information2 Lecture 2: Amplitude equations and Hopf bifurcations
Lecture : Amplitude equations and Hopf bifurcations This lecture completes the brief discussion of steady-state bifurcations by discussing vector fields that describe the dynamics near a bifurcation. From
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 informationSymmetry breaking of two-dimensional time-periodic wakes
Under consideration for publication in J. Fluid Mech. 1 Symmetry breaking of two-dimensional time-periodic wakes By H. M. BLACKBURN, 1 F. MARQUES 2 AND J. M. LOPEZ 3 1 CSIRO Manufacturing and Infrastructure
More informationBand to band hopping in one-dimensional maps
Home Search Collections Journals About Contact us My IOPscience Band to band hopping in one-dimensional maps This content has been downloaded from IOPscience. Please scroll down to see the full text. View
More informationCapillary-gravity waves: The effect of viscosity on the wave resistance
arxiv:cond-mat/9909148v1 [cond-mat.soft] 10 Sep 1999 Capillary-gravity waves: The effect of viscosity on the wave resistance D. Richard, E. Raphaël Collège de France Physique de la Matière Condensée URA
More informationONE DIMENSIONAL FLOWS. Lecture 3: Bifurcations
ONE DIMENSIONAL FLOWS Lecture 3: Bifurcations 3. Bifurcations Here we show that, although the dynamics of one-dimensional systems is very limited [all solutions either settle down to a steady equilibrium
More informationRésonance et contrôle en cavité ouverte
Résonance et contrôle en cavité ouverte Jérôme Hœpffner KTH, Sweden Avec Espen Åkervik, Uwe Ehrenstein, Dan Henningson Outline The flow case Investigation tools resonance Reduced dynamic model for feedback
More informationExamples include: (a) the Lorenz system for climate and weather modeling (b) the Hodgkin-Huxley system for neuron modeling
1 Introduction Many natural processes can be viewed as dynamical systems, where the system is represented by a set of state variables and its evolution governed by a set of differential equations. Examples
More informationPAPER 331 HYDRODYNAMIC STABILITY
MATHEMATICAL TRIPOS Part III Thursday, 6 May, 016 1:30 pm to 4:30 pm PAPER 331 HYDRODYNAMIC STABILITY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal
More informationProf. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait
Prof. Krstic Nonlinear Systems MAE28A Homework set Linearization & phase portrait. For each of the following systems, find all equilibrium points and determine the type of each isolated equilibrium. Use
More informationSolution to Homework #4 Roy Malka
1. Show that the map: Solution to Homework #4 Roy Malka F µ : x n+1 = x n + µ x 2 n x n R (1) undergoes a saddle-node bifurcation at (x, µ) = (0, 0); show that for µ < 0 it has no fixed points whereas
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 informationA Model of Evolutionary Dynamics with Quasiperiodic Forcing
paper presented at Society for Experimental Mechanics (SEM) IMAC XXXIII Conference on Structural Dynamics February 2-5, 205, Orlando FL A Model of Evolutionary Dynamics with Quasiperiodic Forcing Elizabeth
More informationDepartment of Mathematics IIT Guwahati
Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,
More informationGeneral introduction to Hydrodynamic Instabilities
KTH ROYAL INSTITUTE OF TECHNOLOGY General introduction to Hydrodynamic Instabilities L. Brandt & J.-Ch. Loiseau KTH Mechanics, November 2015 Luca Brandt Professor at KTH Mechanics Email: luca@mech.kth.se
More informationClearly the passage of an eigenvalue through to the positive real half plane leads to a qualitative change in the phase portrait, i.e.
Bifurcations We have already seen how the loss of stiffness in a linear oscillator leads to instability. In a practical situation the stiffness may not degrade in a linear fashion, and instability may
More informationDynamical scaling behavior of the Swift-Hohenberg equation following a quench to the modulated state. Q. Hou*, S. Sasa, N.
ELSEVIER Physica A 239 (1997) 219-226 PHYSICA Dynamical scaling behavior of the Swift-Hohenberg equation following a quench to the modulated state Q. Hou*, S. Sasa, N. Goldenfeld Physics Department, 1110
More informationLocalized structures as spatial hosts for unstable modes
April 2007 EPL, 78 (2007 14002 doi: 10.1209/0295-5075/78/14002 www.epljournal.org A. Lampert 1 and E. Meron 1,2 1 Department of Physics, Ben-Gurion University - Beer-Sheva 84105, Israel 2 Department of
More informationApplication demonstration. BifTools. Maple Package for Bifurcation Analysis in Dynamical Systems
Application demonstration BifTools Maple Package for Bifurcation Analysis in Dynamical Systems Introduction Milen Borisov, Neli Dimitrova Department of Biomathematics Institute of Mathematics and Informatics
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 informationDynamical Systems in Neuroscience: Elementary Bifurcations
Dynamical Systems in Neuroscience: Elementary Bifurcations Foris Kuang May 2017 1 Contents 1 Introduction 3 2 Definitions 3 3 Hodgkin-Huxley Model 3 4 Morris-Lecar Model 4 5 Stability 5 5.1 Linear ODE..............................................
More informationAdditive resonances of a controlled van der Pol-Duffing oscillator
Additive resonances of a controlled van der Pol-Duffing oscillator This paper has been published in Journal of Sound and Vibration vol. 5 issue - 8 pp.-. J.C. Ji N. Zhang Faculty of Engineering University
More information