VIII. Nonlinear Tourism

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1 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 TUCKERMAN laurette@pmmh.espci.fr VIII. Nonlinear Tourism English-language version 1

2 Nonlinear Tourism In this chapter, we will survey the model equations which are most commonly used to study nonlinear dynamical systems. We have already seen a number of these; we will review them and present the remaining most important model equations. You can find articles, films, and, best of all, interactive programs on most of these systems in bprotas/gallery/wake animations mcc/chaos Course masax/research/spiral spectra model model shawn/lcs-tutorial/motivation.html alievr/rubin.html nonlin/ There are even videos on YouTube. Try searching for Reaction-Diffusion, Turing, Brussellator, Belousov-Zhabotinsky, Spatio-temporal chaos, Softology. 1 Systems of ordinary differential equations These are continuous-time dynamical systems (also called flows) with 1, 2 or 3 degrees of freedom. 1.1 Lorenz Model The Lorenz model is: Ẋ = σ(y X) (1a) Ẏ = XZ + rx Y (1b) Ż = XY bz We recall the origin of these equations. For 2D Rayleigh-Bénard convection with boundary conditions that are free-slip in the vertical direction and periodic in the horizontal direction the first instability is a supercritical pitchfork bifurcation to a trigonometric eigenmode: a pair of rolls. X, Y, and Z (1c) 2

3 Figure 1: 3D view of Lorenz attractor at r = 28. are amplitudes of the streamfunction, the temperature, and the second harmonic of temperature. The parameter σ is the Prandtl number, usually set to 10, its value for water; b is usually set to 8/3; and r is the reduced Rayleigh number, i.e. Ra/Ra c, which is varied. The supercritical pitchfork bifurcation modeling onset of convection occurs at r = 1, and a subcritical Hopf bifurcation leads to chaos at r = The Lorenz model was proposed by Edward Lorenz (MIT) in 1963 and was the first strange attractor studied. Trajectories converge towards a strange attractor, but they diverge within the attractor. A classic view of the Lorenz attractor is shown in figure 1. In addition to the articles on Wikipedia, Scholarpedia and Mathworld, see and eng.html. 1.2 Rössler system The Rössler system ẋ = y z ẏ = x + ay ż = b + z(x c) (2a) (2b) (2c) was proposed in 1976 by Otto Rössler (University of Tubingen, Germany) There are two fixed points, both with a complex conjugate pair of eigenvalues and one real eigenvalue. For one fixed point the c.c. pair has positive real part and the real eigenvalue is negative (and vice versa for the other one). Trajectories spiral outwards (c.c. pair) from the first fixed point and eventually return to it (negative real eigenvalue). This system also undergoes a period-doubling cascade, when two of the parameters a, b, c are fixed and the third is varied. 1.3 Van der Pol oscillator ẍ ǫ(1 x 2 )ẋ + x = F cos(ω f t) (3) 3

4 Figure 2: Rössler attractor. Left: from attractor. Right: from This is a harmonic oscillator to which nonlinear damping and forcing have been added. When the forcing is zero, (3) can be written as ẋ = y ẏ = ǫ(1 x 2 )y x (4a) (4b) Scholarpedia has an extensive website on the van der Pol oscillator (with animations) by T. Kanamaru. 1.4 Duffing oscillator The Duffing oscillator is governed by the equation ẍ + δẋ x + x 3 = γ cos(ωt) (5) In contrast to the van der Pol oscillator, here the damping (δẋ) is linear, but there is a nonlinear potential V (x 4 /4 x 2 /2). Scholarpedia has an extensive website (with animations) on the Duffing oscillator by T. Kanamaru. 4

5 Figure 3: Van der Pol oscillator. Left: Nullclines in the (x, y) = (x, ẋ) plane. Right: timeseries showing relaxation oscillations. From T. Kanamura, Van der Pol oscillator, Scholarpedia 2(1):2202 Figure 4: The Duffing oscillator describes the behavior of the magneto-elastic beam. 5

6 2 Discrete-time maps in one or two variables 2.1 Logistic Map Figure 5: Logistic map. Left: logistic map f(x) for r = 0.76 showing convergence to a two-cycle under iteration of the map. Right: Members x of cycles as a function of r, including period-doubling cascade and periodic windows. The logistic mapping: x n+1 = f(x n ) = 4rx n (1 x n ) for x n [0, 1], 0 < r < 1 (6) was studied by Robert May, in the context of population growth, by Mitchell Feigenbaum (then at Los Alamos) and by Pierre Coullet and Charles Tresser (then at University of Nice), as a universal perioddoubling system in the 1970s. It is for this system that the period-doubling cascade was first discovered. A fixed point created at r 1 = 1/4 = 0.25 is succeeded by a two-cycle at r 2 = 3/4 = 0.75, a four-cycle at r 4 = (1 + 6)/4 = 0.862, and so on until the accumulation point at r = This can be explained by considering the successive scaled iterates of the mapping, a process which converges to a universal mapping and which is an example of the technique of renormalization. Other periodic cycles are created via saddle-node bifurcations for r > r. 2.2 Sine circle map The sine circle map of Arnol d: θ n+1 = f Ω,K (θ n ) [ θ n + Ω K ] 2π sin(2πθ n) mod 1 (7) models the first return map of flow on a torus. For K = 0, trajectories have winding number Ω. For K > 0, saddle-node bifurcations generate Arnold tongues, which are intervals of Ω over which the winding number remains at the same rational value and which widen with K. This is called frequency locking; see figure 6. 6

7 Figure 6: Schematic representation of frequency locking tongues in the (Ω, K) plane. From M. Cross, CalTech, mcc/chaos Course/Lesson19/Circle.pdf 2.3 Hénon map x n+1 = y n + 1 ax 2 n (8a) y n+1 = bx n (8b) The Hénon map was introduced by Michel Hénon (Observatoire de Nice, France) in The website by Evgeny Demidov has an extensive treatment of the Hénon map. Its actions of stretching and folding are an essential part of chaos and are shared by other prototypical maps, such as the Smale Horseshoe and the Baker s Map. See figure 7. 7

8 Figure 7: Above: Hénon map. Left: Hénon map explained as a decomposition of three operations: stretching, folding, and rotation. From E. Demidov, Right: Attractor of Hénon map for a = 1.4 and b = 0.3. From map. Below: Smale horseshoe map composes stretching and folding. From 8

9 3 Reaction-Diffusion Systems This is a large category of systems that overlaps with both the previous and with the next sections. A reaction-diffusion system is: u t = u xx + f(u, v) (9a) v t = v xx + g(u, v) (9b) The number of spatial dimensions can be increased to two or to three. We have already seen several examples, in particular the FitzHugh-Nagumo equations: and the Barkley model: The Brusselator: f(u, v) = u u 3 /3 v + I g(u, v) = 0.08 (u v) f(u, v) = 1 ( ǫ u(1 u) u v + b ) a g(u, v) = u v f(u, v) = a + u 2 v bu u g(u, v) = bu u 2 v (10a) (10b) (11a) (11b) (12a) (12b) was formulated by the group of I. Prigogine in Belgium to describe autocatalytic reactions. Prigogine was interested in irreversibility in quantum mechanics and was one of the pioneers of the study of oscillating chemical reactions. He was awarded the Nobel Prize in Chemistry in The Gray-Scott model: f(u, v) = uv 2 + a(1 u) g(u, v) = uv 2 (1 + k)v (13a) (13b) forms self-replicating spots, as seen in figure 8. 9

10 Figure 8: Laboratory chemical experiment compared with simulations of the Grey-Scott model. From K. Lee, W.D. McCormick, J.E. Pearson & H.L. Swinney, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature 369, 215 (1994). 10

11 4 Partial Differential Equations 4.1 Burgers equation Burgers equation: u t + uu x = νu xx (14) was formulated by J.M. Burgers and has a clear correspondence to hydrodynamics. If ν = 0, then shocks can appear and the equation is used to model gas dynamics and traffic flow If ν 0, the Cole-Hopf transformation u = 2νφ x /φ (15) transforms (14) into the diffusion equation. φ t = νφ xx (16) 4.2 Kuramoto-Sivashinsky equation Figure 9: Growth rate σ as a function of spatial wavenumber k for the Kuramoto-Sivashinsky equation. The trivial u = 0 state is unstable to periodic perturbations with wavenumbers 0 < k < 1. The Kuramoto-Sivashinsky equation: u t + uu x = u xx u xxxx x [ L/2, L/2] (17) also has a clear correspondence to hydrodynamics. However, the second derivative term u xx has a negative sign and so is destabilizing instead of stabilizing: substituting u sinkx leads to +k 2 u on the right-hand side. In the K-S equation, it is the fourth derivative term u xxxx which is stabilizing: substituting u sinkx leads to k 4 u on the right-hand side. Boundary conditions can be periodic, or 11

12 Dirichlet or Neumann (four Dirichlet or Neumann conditions are needed for this fourth-order equation). The K-S equation was originally formulated to describe flame fronts. For periodic boundary conditions, linear instability of the trivial u = 0 solution to sinkx is governed by σ = k 2 k 4 (18) shown in figure 9, where it can be seen that the unstable wavenumbers are 0 < k < 1. Allowed values of the wavenumber k, i.e. those which can fit into the box of length L, are multiplies of 2π/L. If L < 2π, then k min = 2π/L > 1 and so u = 0 is stable. As the size L of the box is increased, an increasing number of unstable wavenumbers can fit, leading to more bifurcations from the trivial state. There exists a large number of solutions to the Kuramoto-Sivashinsky equation, even for fixed L. Some solutions are shown in figure Swift-Hohenberg (SH) equation The Swift-Hohenberg equation was formulated by Jack Swift and Pierre Hohenberg in 1977 to describe pattern-forming systems. We can begin to motivate this equation as follows. Consider a trivial state u = 0 which loses stability to perturbations of the form u e ±ik x. If the physical configuration is isotropic, then the growth rate σ must depend on the magnitude but not the orientation (which includes the sign) of k. To be differentiable it must be a function of k 2 rather than of k. In order for perturbations with infinitely large wavenumbers to be damped, σ must be negative for large k 2, and in order for some perturbations to grow and patterns to be formed, σ must be positive for some range of k 2. Then the growth rate is of the form: σ(k) = a 0 + a 2 k 2 k 4 (19) where we have set a 4 = 1 by scaling time. Setting k 2 c a 2 /2 and µ a 0 + k 4 c, (19) can be rewritten as: σ(k) = µ (k 2 c k 2 ) 2 (20) The curve σ(k) resembles that in figure 9, for the values k c = 1/ 2 and µ = k 4 c = 1/4. Subsituting σ t and k 2 in (20) leads to the partial differential equation u t = µu (k 2 c + ) 2 u (21) In order to halt the exponential growth due to linear instability, one must also include a nonlinear saturating term. The nonlinear term chosen for the Swift-Hohenberg equation is usually (but not always) u 3. Thus, the SH equation is: u t = [ µ ( + k 2 c) 2] u u 3 (22) The Swift-Hohenberg equation reproduces many of the well-known instabilities of striped (roll) patterns: the Eckhaus (E) instability: change in wavelength the zigzag (Z) instability: sinusoidal in-phase oscillations along roll axes the skew-varicose (SV) instability: sinusoidal out-of-phase oscillations along roll axes the cross-roll (CR) instability: appearance of perpendicular rolls the oscillatory (OS) instability: time-dependent oscillations along roll axes These were discovered numerically and explored extensively in a series of papers by Friedrich Busse and R. Clever in the 1970s on Rayleigh-Bénard convection. The occurrence of these instabilities depends on 12

13 Figure 10: Some solutions to the Kuramoto-Sivashinsky equation for lengths L = 10, 12, 16, 22, 24. Horizontal axis is t, vertical axis is x. From P. Cvitanović et al., 13

14 Figure 11: Patterns produced by Swift-Hohenberg equation and modifications of the SH equation. Simulations from Java applets of M. Cross, Caltech, three parameters: Rayleigh number Ra, Prandtl number P r and wavenumber k of the underlying striped (roll) pattern. The volume in (Ra, P r, k) space within which straight roll patterns are stable to these instabilities is called the Busse balloon. When the nonlinearity includes a quadratic term, then hexagons can be obtained; see figure 11. Lifshitz and Petrich further modified the SH equation by including two different critical wavenumbers and were able to simulate quasipatterns; see figure 11. The fact that these patterns and instabilities also occur in the Swift-Hohenberg equation shows that they are not particular to Rayleigh-Bénard convection. Figures 12 and 13 shows the manifestation of some of these in experiments and simulations of a granular layer subjected to vertical oscillation, together with an adaptation of the Busse balloon to this case; the amplitude Γ of the vibrations acts analogously to the Rayleigh number. 4.4 Ginzburg-Landau (GL), Newell-Whitehead (NW), and Newell-Whitehead-Segel (NWS) equations The Ginzburg-Landau equation A t = µa A 3 + A xx (23) 14

15 Figure 12: Squares, stripes, hexagons in a granular layer. From C. Bizon, M.D. Shattuck, J.B. Swift, W.D. McCormick & H.L. Swinney, Patterns in 3D vertically oscillated granular layers: simulation and experiment, Phys. Rev. Lett. 80, 57 (1998). Figure 13: Instabilities of a striped pattern in a vertically-vibrated granular layer. Left top: skew-varicose instability. Left bottom: cross-roll instability. Right: stability boundaries in the (< k >, Γ) plane, where 15 < k > is the mean wavenumber and Γ is the amplitude of the acceleration. From J. de. Bruyn, C. Bizon, M.D. Shattuck, D. Goldman, J.B. Swift & H.L. Swinney, Continuum-type stability balloon in oscillated granulated layers, Phys. Rev. Lett. 81, 1421 (1998).

16 was originally formulated by Ginzburg and Landau in the Soviet Union in 1950 to model superconductivity. It was rederived by Newell and Whitehead (then at UCLA) to model Rayleigh-Bénard convection: A t = µa A 2 A + A xx (24) Unlike the Lorenz model, which completely fixes the spatial structure, this equation allows modulation of the rolls: in particular, they may change wavelength. That is, (24) describes the evolution of the long-wavelength envelope of a field via: w(x, t) = A(X, T)e iqcx + c.c. (25) where X ǫx, T ǫ 2 t and can be derived via a multiple-scale expansion from the SH equation. The Newell-Whitehead-Segel equation also includes variations along the direction y of the roll axes: where Y ǫ/2 y. ( A t = µa A 2 A + x i 2 2 Y 2 ) 2 A (26) 4.5 Complex Ginzburg-Landau (CGL) equation The complex Ginzburg-Landau equation t A = µa (1 + iβ) A 2 A + (1 + iα) A (27) is used when the first instability is a Hopf bifurcation. 4.6 Cross-Newell equations Cross and Newell have considered the spatial phase θ(x, Y, T) and its gradient k = θ of a system of convection rolls or a more general striped pattern and formulated evolution equations for these fields. 16

17 5 Bifurcations in Hamiltonian systems Having seen the crucial role played in Hamiltonian systems (and thus in spatial analysis) by centers (elliptic fixed points) and saddles (hyperbolic fixed points), we now examine bifurcations of these points. 5.1 Hamiltonian saddle-node bifurcation A B p Q Q + - p Q + Q C q q D ø p q Q Figure 14: Phase portraits of the Hamiltonian saddle-node normal form (28), with p = q. A: δ = 0.2, B: δ = 0.1, C: δ = 0. D: Corresponding potential Φ associated to each phase portrait A, B or C, with ṗ = Φ/ q. An elliptic region bounded by the separatrix that starts and ends on the fixed point Q + (homoclinic orbit) is present on A and B. Phase portrait C displays the critical merging of fixed points Q + and Q +, and the disappearance of the elliptic region. A B C Q - q The normal form for the Hamiltonian saddle-node bifurcation is q = δ q 2 { q = p = H p ṗ = δ q 2 = H q (28) where H = p2 p2 + Φ(q) = q3 δq (29) 3 Figure 14D shows the potential Φ for various values of δ. The fixed points are extrema of Φ: p = 0 q = ± δ for δ > 0 (30a) (30b) 17

18 Their stability is determined by the eigenvalues of [ ] 0 1 λ 2 = 2q (31) 2q 0 We have q = + δ λ 2 < 0 λ imaginary q a center q = δ λ 2 > 0 λ real q a saddle (32a) (32b) So at δ = 0, a saddle and center are created simultaneously, as is shown in figure 14A,B,C. 5.2 Hamiltonian pitchfork bifurcation Figure 15: Behavior of the Ginzburg-Landau equation for µ = 1 and for µ = +1. Top row: potential Φ(u) with fixed points ū = 0 for µ = 1 and ū = ± µ = ±1 for µ = +1. For µ = 1, a portion of a trajectory is shown which originates at u = and goes to u = +. For µ = +1, the trajectory shown goes from ū = 1 to ū = +1. Bottom row: phase portraits in the (u, u) plane. (0, 0) is a hyperbolic fixed point for µ = 1 and an elliptic fixed point for µ = +1. ( µ,0) are hyperbolic fixed points for µ = +1 and are connected by a heteroclinic orbit. In figure 15, already presented in a previous section, we show how a saddle is transformed into a center surrounded by two saddles. The corresponding normal form is: { ü = u 3 u = v = H/ v µu ṗ = u 3 (33) µu = H/ u where H = v2 2 v2 Φ(u) = 2 u4 4 + µu2 2 (34) 18

19 6 Some mathematical techniques Here we present two important mathematical constructions used in bifurcation problems. 6.1 Center manifold reduction Consider a system undergoing a bifurcation whose Jacobian has mostly negative eigenvalues and one or more eigenvalues that are zero (for complex eigenvalues, we are referring to the real part). We diagonalize the Jacobian and use coordinates corresponding to the eigenvectors. We write the directions corresponding to the zero eigenvalues as the vector x and those corresponding to the negative eigenvalues as vector y. ẋ = Ax + f(x,y) ẏ = By + g(x, y) (35a) (35b) where A and B are the Jacobian reduced to x and y respectively. We assume that the negative eigenvalues are bounded away from zero, and so y evolves very quickly compared to x. After a short initial transient, we say that y is slaved to x and write: 0 = By + g(x,y) (36) relating y and x which implicitly gives y as a function of x, so that in principle: y = h(x) with y = O( x 2 ) (37) In practice, (36) is solved approximately by expanding h(x) as a series, whose elements are monomials in the components of x and y. We then substitute the exact or approximate h(x) into (35a) to obtain: ẋ = Ax + f(x,h(x)) (38a) The Reduction Principle states that system (35) is locally topologically equivalent near the origin to (38a) with ẏ = By (38b) in which x and y are decoupled. 6.2 Fredholm alternative We begin with an evolution equation for a field u with a term L which is linear in u and which depends on a control parameter r, and a term N which is quadratic in u: t u = L(r)u + N(u, u) (39) The trivial solution u = 0 undergoes a steady bifurcation at r = r c so that 0 = L(r c )u c (40) and we are interested in the behavior of solutions near r c. We write ǫ (r r c ) γ and expand u = ǫ (u 0 + u 1 ǫ + u 2 ǫ 2 ) (41a) L = L 0 + L 1 ǫ + L 2 ǫ (41b) 19

20 where u contains no O(1) term because we are considering a solution bifurcating from u = 0 near the bifurcation point. Subsituting the expansions (41) into (39) leads to: t u k ǫ k+1 = L j u k ǫ j+k+1 + N(u j, u k )ǫ j+k+2 (42) k 0 j,k 0 j,k 0 Separating the terms of (42) in powers of ǫ and setting t u to zero yields 0 = L 0 u 0 (43a) 0 = L 0 u 1 + L 1 u 0 + N(u 0, u 0 ) (43b) 0 = L 0 u 2 + L 1 u 1 + L 2 u 0 + N(u 0, u 1 ) + N(u 1, u 0 ) (43c) Equation (43a) is a restatement of (40), that is, we identify u 0 with u c and L(r c ) with L 0. Specifically, u 0 is a null eigenvector of the singular operator L 0. Equation (43b) relates L 0 u 1 to u 0, and, in general each succeeding equation relates L 0 u k to the previously calculated u k 1, u k 2,...,u 1. These equations are treated using the Fredholm alternative for singular operators. The statement of the Fredholm alternative for self-adjoint operators is as follows. Suppose A is a linear operator and f a vector such that Af = 0 (44) Then the equation has a solution if and only if If A is not self adjoint, then its adjoint A also has a null vector and (46) is generalized to Ax = b (45) f,b = 0 (46) A f = 0 (47) f,b = 0 (48) where f is the null eigenvector of the adjoint operator A or, equivalently, the left eigenvector of A. The solution to (45) is non-unique since any multiple of the null vector f can be added to x. One direction is easy to show. Taking the inner product of (45) with the null vector f leads to: f,b = f, Ax = A f,x = 0 (49) Thus, (45) implies (48). The other direction is more difficult to show in the abstract, but is also simple in finite dimensions. If A is a singular matrix and f its null vector, then we can carry out a change of basis such that the new A has zeroes in its first row. Equation (48) then merely says that any vector of the form Ax must have zero as its first component. This is because the new A has zeroes in its first column, the new f is the first unit vector, and the inner product of a vector with f is the first component. Thus, equation (45) becomes X X X X X X X X X X X X 20 Ax = b (50) 0 = b c d x y z w

21 where X designates entries of A that are not necessarily zero. Equation (47) becomes 0 X X X 0 X X X 0 X X X 0 X X X A f = 0 (51) = and equation (48) becomes < f,b >= [ ] a b c d = a = 0 (52) The condition (52) suffices for (50) to have a solution if there are no remaining null vectors, since then the remaining lower 3 3 matrix of A is non-singular. If there are other null vectors, then the right-hand-side b must be orthogonal to these as well. Returning to the bifurcation problem, equation (43a) tells us that u 0 is a null vector of the singular operator L 0 : L 0 u 0 = 0 (53) whose amplitude α remains to be determined: u 0 = αf where L 0 f = 0 and f = 1 (54) Equation (43b) states that u 1 is a solution to a linear problem involving the singular L 0 : L 0 u 1 = L 1 u 0 N(u 0, u 0 ) = αl 1 f α 2 N(f, f) (55) where we have used (54). Let f be the normalized null vector of the adjoint L 0. The Fredholm alternative states that (55) has a solution if its right-hand-side is orthogonal to f : 0 = α f, L 1 f α 2 f, N(f, f)) = 0 = f, L 1 f + α f, N(f, f)) (56) where we have used u 0 0 = α 0 to divide through by α. If then α is determined by (56) to be: f, N(f, f)) 0 (57) α = f L 1 f f, N(f, f) In the case of a pitchfork bifurcation, (57) is not satisfied and higher-order equations must be used to determine α. (58) 21