Dynamical Systems Generated by ODEs and Maps: Final Examination Project

Transcription

1 Dynamical Systems Generated by ODEs and Maps: Final Examination Project Jasmine Nirody 26 January, 2010

2 Contents 1 Introduction and Background 1 2 Equilibria 1 3 Attracting Domain 2 4 Overview of Behavior Under Changes in r 4 5 Bifurcations of Equilibria 5 6 The Strange Invariant Set 11 7 The Strange Attractor 14 8 Geometric Models 18 9 Period Doubling and Intermittent Chaos Summary 23

3 1 Introduction and Background In the following we consider the Lorenz system: ẋ = σx + σy ẏ = rx y xz ż = bz + xy Unless otherwise mentioned, the parameter values considered will be σ = 10, b = 8 3, r > 0. The system arose from a model proposed by Lord Rayleigh for thermal convection: δ 2 ψ δt = δ(ψ, 2 ψ) δ(ζ, η) + ν 4 ψ + gα δθ δζ δθ δt = δ(ψ, Θ δ(ζ,η) + δt H + δψ δζ + κ 2 Θ. In this model, ψ is a stream function for two dimensional motion, Θ is the function describing the temperature variation when there is no convection, ζ, η are spatial coordinates, T the imposed temperature difference, g is the acceleration due to gravity, α bouyancy, κ thermal diffusivity, ν viscosity, and t is time [7]. Using a Fourier series in the spatial coordinates to expand ψ and Θ, with functions of t as coefficients, and then reducing the infinite system by setting all modes but three to zero [10], we obtain a finite system of ordinary differential equations, which are the basis for the system shown above [7]. It is important to note that the coordinates in the Lorenz system are not spatial coordinates. Rather, x is proportional to the speed of motion of the air due to convection, y is a measure of temperature difference between the rising (warm) and falling (cool) air, and z is the vertical temperature difference as we move through the system. The parameters also have physical meaning: σ is called the Prandtl number which is related to the nature of the air involved, b is representative of the size of the area considered, and r is called the Rayleigh number which says at which point convection will begin. 2 Equilibria We now wish to solve for the stationary points or equilibrium points of the system, x = f( x), where f( x) : R 3 R 3 is defined as: ẋ = σx + σy f( x) = ẏ = rx y xz ż = bz + xy. 1

4 We define stationary points as those values at which f( x) = 0. It is easy to see that the origin, (x,y,z) = (0, 0, 0), is such a point, but the system also has two other equilibria, (x,y,z) = (±b r 1, ±b r 1,r 1) which we refer to from now on as C 1 (+) and C 2 ( ). Note that C 1,C 2 are real-valued for r > 1. At this point, let us quickly note that there is a natural symmetry defined by the transformation: (x,y,z) ( x, y,z). This property is seen in the equilibria C 1 and C 2, but we wish to verify this symmetry. In order to show symmetry, we must find a 3 3 matrix, R, which commutes with the system. That is: Rf( x) = f(r x). We choose R based on the transformation mentioned above: R = We check the commutative relationship: σ(x y) Rf( x) = x(r z) y, xy bz f(r x) = f x y z = σ(x y) x(r z) y xy bz = Rf( x). Thus, we have shown that this relationship corresponds to a natural symmetry in the system. 3 Attracting Domain Let us consider a general 3D system. The time evolution of a volume, V (t), enclosed by a surface, S(t), is given by: V = f NdA, S where N is the normal vector to S. Then, by Green s Theorem, this is equivalent to: V = fdv. V 2

5 The divergence of the Lorenz system is given as: And so: f = d d [σ(y x)] + dx dy (rx y xz) + d (xy bz), dz f = (σ b). V = (σ b)v V (t) = V (0)e (σ+1+b)t. We see that the exponent is negative, and therefore volumes under the action of the Lorenz system shrink exponentially fast, showing that there exists an attracting set of zero volume. A set is considered to be a positively-invariant domain of a system if, for any starting point x 0 S, under the action of the system, x(t, x 0 ) S for all t > 0. We now show that there is a bounded sphere, S, into which all trajectories enter and upon entering, never leave (i.e. S is a positively invariant domain of the Lorenz system). First, let us define a Lyapunov function. A scalar function, V, is considered a Lyapunov function on a region D if it is continuous, positive definite, and has continuous first order derivitives on all of D. Now, we consider a specific such function: V (x,y,z) = rx 2 + σy 2 + σ(z 2r) 2. The derivitive of a Lyapunov function, with respect to the ODE system x = f( x), is given to be: V = V ( x) f( x). So, considering the Lorenz system and Lyapunov function we have introduced: V = 2σ(rx 2 + y 2 + bz 2 2brz). Let us call D the bounded region in which V > 0, and let c be the maximum of V in D. Specifically, we can define D as the ellipsoid which consists of the points for which: rx 2 + y 2 + bz 2 = 2brz. Consider now a bounded region, S consisting of the points which satisfy: x 2 + y 2 + (z r σ) 2 = R 2, and let us choose R so that D is completely contained within S. Then, if a point x lies outside of S, then it must also lie outside of D, and by definition, V (x) 0. Rather, we can also say that V (x) δ for some δ > 0. Now, if a trajectory begins with initial condition x outside of S, then V (x) will decrease with time and will eventually (within finite time for δ > 0) enter S. Additionally, since on the surface of S, V 0, trajectories may only cross inwards. Therefore, once trajectories pass inwards through the boundary of S, they will never leave it, and S is therefore a positvely-invariant domain into which all trajectories will enter. 3

6 4 Overview of Behavior Under Changes in r In the following sections, we wish to observe changes in the dynamics of the Lorenz system as we vary one parameter, namely r. In this section, we provide a quick overview of the changes we see with this varying, with only a few proofs. However, most of the statements made in this section will be discussed in detail in subsequent sections. In the interval 0 < r < 1, the origin is globally stable and all trajectories are attracted to it. We now show the global stability of the origin. Consider the Lyapunov function: with: V (x,y,z) = 1 σ x2 + y 2 + z 2 V = 2[(r + 1)xy x 2 y 2 bz 2 ] = 2[x r y]2 2[1 ( r )2 ]y 2 2bz 2 From this we see that for 0 < r < 1, V < 0 if (x,y,z) (0, 0, 0), and we can conclude that as t, V 0. So we can conclude that any point (x,y,z) (0, 0, 0); that is, all points will tend to the origin. For 1 < r, the origin is unstable. In the interval 1 < r < , C 1,2 are stable. The first statement can be shown by a simple analysis of the eigenvalues of the system evaluated at the origin. To analyse stability of equilibria, we first consider the Jacobian: J = σ σ 0 r z 1 x y x b. At the origin, this reduces to: J (0,0,0) = σ σ 0 r b. The eigenvalues of this matrix are: λ 1,2 = 1 2 ( 1 σ ± 1 2σ + 4rσ + σ 2 ), 4

7 λ 3 = b. For r > 1, λ 1 > 0 and the origin becomes unstable. A discussion of the stability of C 1,2 will be given in a later section. For < r, both C 1 and C 2 are unstable. At this point, there are no stable fixed points in the system. A discussion of this loss of stability will be provided in a later section. Our interest lies in what happens for r > We know there are no more stable fixed points, and we will see in later sections that there do not exist any limit cycles around C 1,2 either. We also know that volumes contract exponentially fast, and so trajectories cannot escape to infinity. We have shown that there exists an attracting set to which all trajectories tend. In this region, however, the attractor does not contain any critical points (as they are not stable) and is unlikely to contain any limit cycles (as we will see, the Hopf bifurcation which occurs nearby is subcritical, so the limit cycles caused by it will be unstable as well). A discussion about the shape of the attractor will be provided later, as will a general discussion of techniques used to analyse such a question. However, before we consider this region, we first quickly discuss the region r < , which also has some interesting events. 5 Bifurcations of Equilibria A bifurcation occurs when a small change in parameters causes a qualitative change in the behavior of the system. We will first be concerned with local bifurcations, which can be analysed by observing changes in stability of equilibria, periodic orbits, or other invariant sets as parameters cross what we deem to be critical values. In a later section, we discuss one kind of global bifurcation, the homoclinic bifurcation. In this section, we focus on bifurcations concerned with equilibria, primarily the pitchfork and Hopf bifurcations. We consider again the Jacobian of the system: σ σ 0 J = r z 1 x, y x b and evaluated at the origin: 5

8 J (0,0,0) = σ σ 0 r b. A one parameter system x = f( x,λ) with a pitchfork bifurcation is defined as one has having the following necessary conditions: f x (0, 0) = 0 (bifurcation condition), f(0, λ) = 0 (bifurcation condition), f x x (0, 0) = 0 (suggests local symmetry). We wish to find such a point in the Lorenz system. We know the second bifurcation condition must hold for any equilibrium point with any parameter values. At a pitchfork point, we know that rank(j) is not maximal as per the first condition, that is, rank(j) < 3. So the determinant of this matrix, J, must be zero. The determinant is given by: det(j) = 0 = r r BP = 1. Note that we have considered the parameter values for σ and b given in the first section. Alternatively, we can also show that J (0,0,0) does not have 3 linearly independent columns. Because of the 0 in the right upper column, we will consider the first two columns. A( σ) = σ A(r) = 1 A(y) = x. From this, it is apparent that these columns are linearly dependent when x = y = 0,A = 1,r BP = 1. The local symmetry condition holds (as expected, given the natural symmetry of the system), apparent from the fact that there are no quadratic terms in the system. We wish to analyse the direction of this bifurcation; that is, whether it is considered subcritical or supercritical. In order to do this, we begin by introducing the paramter ρ = r 1 so that the system now becomes: 6

9 ẋ = σx + σy ẏ = ρx + x y xz ż = bz + xy Now, we realise that the pitchfork bifurcation will occur at ρ = 0 and so we analyse the Jacobian at this state: σ σ 0 J = b We see that J has eigenvalues λ 1,2,3 = 0, (σ + 1), b. We now rewrite our system in the form: where: f(ū) = T 1 JT[ū] + T 1 R(T(ū)), and R( x) = f( x) J ( A T 1 c JT = A s ), where A c and A s are block matrices whose diagonals contain the eigenvalues with Re(λ) = 0 and Re(λ) < 0 respectively. Note that we have accounted for the fact that Re(λ) 0 for all eigenvalues in this system. Using T as the matrix consisting of the eigenvectors: 1 σ 0 T = 1 1 0, we arrive at the extended system which takes into account the derivitive of ρ: u = σ (ρ w)(u + σv) 1+σ f(ū) = ẇ = bw + 1 ρ = 0. v = (1 + σ)v σ 1+σ The central manifold will be of the form: ( (ρ w)(u + σv)) (1 + σ)(u + σv)(u v) W c = {(u,v,w,ρ) : v = h 1 (u,ρ),w = h 2 (u,ρ),h i (0, 0) = 0,J h (0, 0) = 0}. 7

10 We substitute these values for v and w and consider the power series: h 1 (u,ρ) = a 1 u 2 + a 2 uρ + a 3 ρ 2 + higher order terms h 2 (u,ρ) = b 1 u 2 + b 2 uρ + b 3 ρ 2 + higher order terms. After comparing coefficients for u 2 and uρ, we find: leaving us with: 1 v = (1 + σ) 2 w = 1 b u2, u = 1 u3 (ρu 1 + σ b ) ρ = 0. Therfore, the equilibrium is stable for ρ 0 and unstable for ρ > 0, corresponding to r 1 and r > 1, respectively, as we found previously. Furthermore, since the sign in front of the cubic term is negative, we say we have a supercritical pitchfork bifurcation, which we already suspect from the fact that the bifurcation results in the appearance of two stable fixed points C 1,2. We also find this bifurcation numerically: XBP r Figure 1: Numerical continuation in MATCONT showing branching point and resulting pitchfork appearance. A Hopf bifurcation occurs when a pair of complex conjugate eigenvalues cross the imaginary axis, that is when Re(λ x,y ) = 0 and the eigenvalues are purely imaginary. Specifically, the necessary conditions for a Hopf bifurcation are: 8

11 f x (0, 0) = 0 (bifurcation condition), Re(µ 1,2 ) σ(j) = 0 and Im(µ 1,2 ) σ(j) 0, where sigma is the spectrum of J. That is, the Jacobian has a pair of purely imaginary eigenvalues. Genericity conditions usually seen are: There are no other eigenvalues σ(j) on the imaginary axis d dλ Re(µ 1,2) 0 for all µ σ(j). The genericity conditions assure that the bifurcation is topologically equivalent to the normal form. We wish to find the value at which a Hopf bifurcation may occur in the system. For a Hopf point in this system, we realise that there are three eigenvalues, two of which will be purely imaginary conjugates. Therefore, at the Hopf point, tr(j) will be equal to the third eigenvalue. We see that tr(j) = σ 1 b = λ 3 (which is not a critical value, as per the first genericity condition), where λ 1,2 are purely imaginary. Remembering that det(j λ 3 I) = 0 (because λ 3 is an eigenvalue of the system), we call A = J λ 3 I: 1 + b σ 0 A = r z σ + b x. y x σ + 1 We calculate: det(a) = 0 = x(x + bx σy) + (1 + σ)(b + b 2 + σ + bσ rσ + σz). Using the equations for the stationary points, C 1,2, of the system: we arrive at: (x,y,z) = (±b r 1, ±b r 1,r 1), σ(3 + b + σ) r H = σ b 1. We now check the stability of C 1,2. The Jacobian at C 1 is shown as: σ σ 0 J = 1 1 b(r 1), b(r 1) b(r 1) b and at C 2 : J = σ σ b(r 1) b(r 1) b(r 1) b. 9

12 The eigenvalues of these matrices are the roots of: f(λ) = λ 3 + λ 2 (σ + b + 1) + λb(σ + r) + 2σb(r 1). All three roots are real when r 1, but when r > (with σ,b as usual), we have a pair of complex conjugate roots and one real root. All three roots will have a negative real part for r < r H, while the two complex conjugate roots will have positive real part for r > r H, implying they have non-zero velocity as they cross the imaginary axis, as per the the genericity condition. This means that the equilibrium points C 1,2 are stable for 1 < r < r H , and lose stability after. [8]. We find the Hopf points numerically as well: 10 8 H XBP H r Figure 2: Numerical continuation in MATCONT showing Hopf bifurcation points. From numerical calculations, we find that the first Lyapunov coefficient, l 1 (0), is A Hopf bifurcation is considered subcritical (and will result in a stable equilibrium point losing stability by absorbing an unstable periodic orbit) if l 1 (0) > 0 and supercritical (and will result in a stable equilibrium point losing stability by emitting a stable periodic orbit) if l 1 (0) < 0. Our Hopf bifurcation is subcritical for the values we have chosen. It appears that the bifurcation is subcritical for all values σ and b for which it occurs as long as r > 0 [3]. Because this implies that the Hopf point will absorb periodic orbits rather than emit them, we must investigate where these periodic orbits originate. 10

13 LPC Y z z X r Figure 3: A family of limit cycles going into the Hopf bifurcation point for C 1. 6 The Strange Invariant Set For r > 1, trajectories tend to have a certain behavior. Trajectories that begin on one side of R 3 spiral into C 1 while those that begin on the other side spiral to C 2. These two halves are divided by the stable manifold of the origin [11]. The trajectories which begin on the stable manifold, of course, tend towards the origin. However as r gets larger, and crosses some value r, trajectories on each side of the stable manifold make bigger and bigger spirals until finally, at r > r , trajectories on one side of the stable manifold of the origin cross over and become attracted to the fixed point on the other side. We show this in a MATCONT simulation: x x (a) Trajectory for r = 10 < r (b) Trajectory for r = 15 > r Figure 4: Trajectories beginning from the same point in space (x,y,z) = (2, 3, 0) for different values of r. We know that the stable manifold includes the entire z axis, since it is invariant and all points on it will tend to the origin, and we can explain this change of behavior by expecting a homoclinic orbit at some value r = r, meaning there will be a tangency 11

14 between the stable and unstable manifolds of the origin. This means that, in both forward and backward time, trajectories will tend towards the origin. Formally, an orbit, φ(t), is called homoclinic if φ(t) x 0 as t ±, where x 0 is an equilibrium of the system. We see numerical evidence of a homoclinic orbit if we plot the period of the limit cycles, T, vs. r: Period LPC r Figure 5: Numerics suggest the period of limit cycles tends to as r r. It is difficult to find this orbit numerically, because once approximations jump off the stable manifold, they will move away from it very quickly. However, we can convince ourselves of its existence in another way: if a trajectory beginning at x 1 with a certain r reaches x 2 in a certain time, then a trajectory beginning at a point x 1 using r near r will reach a point near x 2 in the same time. But, if we choose some ǫ > 0, r = r ǫ 2, the trajectory will spiral in towards a fixed point, whereas for r = r + ǫ 2 it will spiral into the other. The existence of a homoclinic orbit at r = r explains this change, since at this time the trajectory will spend infinite time at the origin. Let us choose a surface Σ which contains the equilibria C 1,2 and their 1D stable manifolds such that Σ intersects the 2D stable manifold of the origin along a curve D (shown as the midline in Figure 6). In addition, we choose Σ so that any orbit started on it returns to it after some time, so that we have now defined a Poincaré map, P, on Σ\D. We can put this more intuitively: consider a small box around the origin with two tubes entering the top face and exiting the sides and turning around the branches of the unstable origin. Then, we call F = P(Σ) Σ a return map, which analyses where a trajectory entering in the top face of the box will hit the top face once again (rather, return ) after travelling through the tubes. We show the return map presented by Sparrow in Figure 6. The line D is mapped to two points, denoted by the tips of the shaded triangles, H ±. Let us make note that these points are where the unstable manifold of the origin 12

15 Figure 6: Return maps for the Lorenz system given by Sparrow [11]. hits Σ for the first time. Near the origin, the flow contracts (because λ 1 < 0) in the direction of the eigenvector corresponding to the largest eigenvalue λ 1. Using some coordinates (u,v), where u = 0 defines D, we write F as: where and F(u,v) = (f(u),g(u,v)), 0 < g v (u,v) < c < 1 for u 0, g 0 as u 0. This assumes the existence of a global contracting foliation, a concept which we will revisit when discussing the geometric Lorenz attractor. Now, we wish to see what will happen to F as we increase r. The left panel of Figure 6 shows F for r < r. Here, the fixed points C 1,2 at the left and right sides of the triangles are global attractors. As we increase r to r = r, we see that the points of the triangle (which we remember represent the first intersection of the unstable manifold of the origin with the return surface) hits D, thus suggesting that the unstable manifold of the origin is contained in the stable manifold, and a homoclinic orbit occurs (as we had expected). Beyond r = r, we can only calculate return maps numerically, and so the following results are based on numerical, rather than analytical, evidence. For r > r, F has two new fixed points within the triangles, which we refer to as P ±. These points are from the limit cycles which arise at r from the homoclinic orbits. Now, there exists an invariant set, Λ, of Cantor structure between P + and P. So, points that enter this region stay forever within this region. Furthermore, the dynamics on Λ is conjugate to the shift map, s, of two symbols: [4, 11, 15] where w is a list of symbols s(w) = θ, θ j = w j+1, j Z w = w 1 w 2 w 3..., w i 0, 1 We now show that this implies that there exist within Λ a countable infinity of periodic orbits and an uncountable infinity of aperiodic orbits and orbits that tend to the origin. Any periodic sequence with block length n repeating k Z amount of times (in this case n 2) is called a n-periodic orbit. This set of all periodic sequences 13

16 is countable. Since we can choose any k, the orbits are dense as well. Now suppose that all non-periodic orbits (consisting of the set of aperiodic orbits and orbits that tend to the origin) are listed. Then, we can construct a sequence with w 1 different than in the first sequence, w 2 different from the second sequence, etc, and therefore construct some sequence w not listed, which is a contradiction. Therefore, the set of all non-periodic orbits is uncountable. On this set, the dynamics also show sensitive dependence on initial conditions, meaning that trajectories which begin arbitrarily close to each other move apart with increasing time. Let us consider a map f t with state space M. Then f t shows sensitive dependence for initial conditions if for every point x M, there exists a point y in a neighborhood N of x such that at time t = τ: for some δ > 0. d(f τ (x),f τ (y)) > δ Due to the appearance of this strange invariant set, we arrive at two possible conclusions. We see that the two simplest periodic orbits (one whose period consists of one journey through the right tube, and the other its symmetric counterpart in the left tube) are probable candidates for the periodic orbits which the subcritical Hopf bifurcation at r H will absorb. Other periodic orbits cannot be considered because, as the z axis is invariant, an orbit which winds around the axis will never be able to separate itself from it, since it cannot cross it. Finally, we also see that this invariant set could possibly become the attractor after the three critical points become unstable (since we know there always exists an attractor in the system). 7 The Strange Attractor Formally, we define an attractor as a region in space that is invariant under the evolution of time and attracts most, if not all, nearby trajectories. An attractor is called strange if it has a non-integer dimension or if the trajectories within it display chaotic behavior (that is, display high sensitivity to initial conditions). As we increase r, both P ± and H ± move towards C 1,2, but numerical experiments show that H ± move more slowly. A trajectory here eventually settles to C 1 or C 2 but will wander about the invariant set for some time, τ, before it does so. Finally, at r = r A , P ± hits H ±. This implies that the unstable manifold of the origin intersects the stable manifolds of the periodic orbits. Here, we will see heteroclinic orbits which connect C 1,2 at τ =. A trajectory that passes through the invariant set remains wandering aperiodically in the invariant set forever and it becomes a strange attractor [15]. Figure 7 shows a typical trajectory for r < r = 22 < r A. We show some numerical experiments that highlight the sensitive dependence on initial conditions observed: 14

17 z z z z t Figure 7: A typical trajectory with finite but high τ, r = x (a) Trajectory for z = x (b) Trajectory for z = 9.9 Figure 8: Trajectories beginning from the nearby points in space (x,y,z) = (0, 2,z) for parameter values, σ = 10, b = 8 3, r = t Figure 9: Trajectory shown with respect to time for z = 10 (red) and z = 9.9 (blue). All other values as above. Note that r A < r H Therefore, at r A, C 1,2 are still stable, and some trajectories will be attracted to these even though the strange attractor exists. 15

18 Because return maps can only be calculated numerically past r = r, complete maps cannot be computed since only a finite amount of points can be produced. However, using judgment and selecting a sufficient number of points, certain statements about the properties of the flow can be made. The following are assumptions and observations from computed return maps by Sparrow [11]. We assume that the map stretches almost all distances, implying that we still observe sensitive dependence on initial conditions because (almost) all trajectories will be divergent. The exceptions are those which begin equidistant from the intersection of the smooth manifold and the return plane, which we can call RM. The trajectories of these equidistant points provide the return map with a chaotic appearance. The attractor has a Cantor-like structure. This partially explains why it is called a strange attractor. A strange attractor is one defined as having a noninteger dimension or one on which chaotic dynamics are observed. The Lorenz strange attractor is observed to fit both these qualities. The attractor contains a countable infinity of non-stable periodic orbits and an uncountable infinity of aperiodic orbits and trajectories which terminate at the origin. For r > r A, the simplest periodic orbits (each consisting of one turn around C 1 or C 2 ) are no longer part of the attracting set. This is expected because they are known to have been absorbed by the Hopf bifurcation, but it has been shown that the same is true for r A < r < r H as well. We do not observe any fixed points on the computed return map, which we would expect if there were such periodic orbits. Each periodic orbit is denoted by a series of x s and y s, which are arbitrarily chosen as opposite sides of the return plane, which we call RM. Choosing an initial point R, each application of ψ(r) will bring it closer to RM, until finally it appears on the other side. Therefore, there is a finite i such that there does not exist a sequence with more than i consecutive x s or y s. However, these orbits all existed in the original invariant set, as there was a one-to-one correspondence to each possible sequence. (Note, there is still only at most one orbit corresponding to each sequence.) The explanation for the last point given is that periodic orbits, as well as some aperiodic trajectories, are removed from the attractor in homoclinic explosions. Consider, for example, r = 28, at which value i = 26. Here, ψ 26 (R) lies exactly on RM, and so corresponds to a homoclinic orbit.we must note that i is a very inaccurate measure of the size of the attractor, since we are only considering the first few entries in the symbolic sequence of x s and y s, k(r), which describes the behavior of the right branch of the unstable manifold (an arbitrary choice). 16

19 Each homoclinic explosion creates a new strange invariant set, and so there will be an uncountable number of topologically distinct attractors in any neighborhood of an arbitrary r [14]. Numerical experiments suggest that while the attractor loses orbits (that is, i decreases monotonically) for r A < r < 28, at some value slightly greater than 28, i begins to increase. At r 30, i begins to decrease again, reaching 2 at r 45 and 1 at r For r > 54.6, the original point R lies to the right of RM for the first time. Since we consider the attractor to be monotonic on some interval, each periodic orbit can only be removed once from the attractor. Then, the set of periodic orbits in the original set can be divided into the following disjoint subsets: 1. The simplest periodic orbits absorbed by the Hopf bifurcation. 2. Orbits remaining in the strange attractor throughout. 3. A disjoint union of orbits removed by homoclinic explosions. Using this criterion, we realise that only certain homoclinic explosions can be allowed so as to divide these orbits into completely disjoint seets. This result, which excludes certain homoclinic orbits from the Lorenz system, comes up again important in geometric models. 8 Geometric Models Several intuitive arguments based on numerical experiments and return maps can be made. However, there has also been a considerable amount of work done on model flows or geometric models that contain Lorenz attractors. Many of the observations presented in the last section are based on the concept of a contracting foliation. Let us consider two points, x and y, with distance between them c such that at some n th iterate, d(ψ n (x),ψ n (y)) cλ n, where 0 < λ < 1. Remembering the observation from the previous section, we assume these two points to be equidistant from RM, and so we then can assume there is a whole arc of points between x and y such that the above argument can be applied to any two trajectories started on this arc. If it is possible to fill the return plane with a continuum of such arcs (i.e. that it is possible to begin a trajectory on one such arc that the return map will always hit a point on another such arc), then we can say there exists a contracting foliation [11]. We now wish to present a model flow that is meant to mimic the Lorenz equations. But first, we present some basic concepts related to one-dimensional maps and kneading sequences necessary for analysing these flows. Let us consider a map, f, which satisfies the following conditions [11]: 17

20 f : I I, I = [0, 1], f is continuous and differentiable except at c, 0 < c < 1, f is monotone increasing on [0, c) (c, 1], lim x c f(x) = 1, lim x c f(x) = 0, f(c) = c, For every interval J I, there is some n such that f n (J) = I. This condition we can call locally eventually onto (LEO). Now we define a kneading sequence, k(x), for every point in I. We define this as: { 0 if x < c k(x) = s 0 s 1 s 2 s 3... where s i = 1 if x > c The sequence ends if f i (x) = c. We can see that a sequence k can be for at most one point x I due to the LEO condition: if x 1 x 2 then for some n, f n (x 1 ) and f n (x 2 ) will be on opposite sides of c. The ordering of sequences k is the binary ordering with the additional condition that the empty space is between 0 and 1. So: < 00 < With this ordering we now have a relationship between k(x) and k(f(x)). k(x) = s 0 s 1 s 2 s 3... and k(f(x)) = s 1 s 2 s 3 s 4... So, generally: k(f(x)) = t 0 t 1 t 2... where t i = s i+1. Remember, we call this the shift map. We require that for a sequence to be a kneading sequence for some x I, k(0) s i (k) k(1). These kneading sequences give us a picture of the behavior of f. Aperiodic/periodic points have infinitely nonrepeating/repeating kneading sequences, whereas points that map to c have finite sequences. This condition also implies that k(0) k(1). We define k (x) as being the sequence derived from k(x) where all 0 s have been replaced by 1 s and vice versa. Let us define the concept of topological equivalence for maps using kneading sequences. Two maps f 1 and f 2 are considered topologically equivalent if and only if k f1 (0) = k f2 (1), k f2 (0) = k f1 (1) or if k f 1 (0) = k f2 (0), k f 1 (1) = k f2 (0). Let us call (k(0),k(1)) the kneading invariant for a map f [11]. While these conditions might seem sufficient to define a kneading sequence for x I, we must think of the events that occur within the Lorenz system, specifically the existence of only some certain homoclinic orbits. Some of those which are not allowed will satisfy the above conditions. Let us consider a kneading invariant (000, 111). Then (000111, ) is a sequence which fits the first conditions. The map f given by this kneading sequence will not be LEO (this can be seen by explicitly computing the map [shown in Figure 4] and observing that an interval [A,B] I is mapped to 18

21 itself but does not fill I). We present, then, a final condition: Suppose w 1 and w 2 are two 0-1 sequences such that w 1 begins with 0 and w 2 begins with 1 and both sequences are of length at least two and at least one is finite. Then, k(0) and k(1) cannot be of the forms: k(0) = w 1 (0w 2 ) n 1(1w 1 ) n 2(0w 1 ) n 3... k(1) = w 2 (0w 1 ) p 1(1w 2 ) p 2(0w 1 ) p 3... where, when w 1 and w 2 are finite, the sequences, k(0),k(1), are either finite or infinite with n 1,p 1 > 0 and, when either w 1 or w 2 is infinite, k(0),k(1) are both infinite with either n 1 or p 1 = 1 and the remaining n i,p i = 0 [14]. We can see that the sequences given in our example do not fulfill this condition (n 1 = p 1 = 1,n 2 = p 2 = 0...). It has not yet been proved that these conditions are both necessary and sufficient, though they are generally accepted as such. Note that the maps, f, are not invertible, so before we can construct model flows, we must give a unique past to each point. For this, we construct a map ˆf Î such that every point ˆx Î consists of points...,x 3,x 2,x 1 such that f(x i ) = x i+1. For points ˆx that we choose x i = 0 or 1, we have a problem, since there is no x I such that f(x) = 0 or 1. So we say the history will end if x i = 0 or 1. This is called the pinched inverse limit of f [11]. Now we can construct a 2D return map, F, of the unit square onto itself with the following properties [11]: F(x,y) (f(x),g(x,y)). F is one-to-one everywhere except on a discontinuity c I. dg dy < 1 2 (i.e there is a contraction in the y direction.) As x c ±, g x I tend to constant functions b ±. Since F is contracting, it is logical to assume there is an attracting set Λ. From this, we can construct a flow by placing the unit square with our map over a non-stable fixed point with 2D stable manifold and 1D unstable manifold, and then embedding this cell into a vector field which takes trajectories that leave the area of interest back into the square as determined by the return map. (We can very roughly say that this vector field acts as the tubes in our box and tubes model shown previously.) Finally we present a 3D Lorenz flow. Geometric Lorenz models are vector fields in 3D space (as mentioned above) with the following properties [5]: 19

22 There exists some stationary point O. O has a 1D unstable and a 2D stable manifold. There exists some cross section to this vector field where the return map of the cross section has a contracting foliation. The 1D map induced by the factorization along the leaves of the stable foliation is uniformly expanding. This essentially implies that the 1D map of the return map will fit the conditions we outlined above. It was shown that, not only do such vector fields exist and contain strange attractors, but even more so, under a small perturbation which produces a flow Φ with a return map F and one-dimensional map f, Φ will also have a Lorenz attractor (that is, F and f will have the properties we have given above) [2]. Geometric models prove a useful tool to globally analyse the properties of the Lorenz attractor, but their relevance to the Lorenz equations could be completely invalidated if it was shown that there does not exist a contracting foliation in the map of the Lorenz equations, and therefore, there also does not exist a strange attractor in the equations. Fortunately, it was shown definitively through a computer-assisted proof that there does exist a strange attractor in the Lorenz equations for the parameter values Lorenz originally considered (σ = 10,b = 8,r = 28) [13]. 3 9 Period Doubling and Intermittent Chaos We now move away from geometric models back to the original equations for a look at behavior for values of r > 28. In numerical simulations, stable periodic orbits are observed in some intervals: < r < , < r < , and < r <. In each of these intervals, we see a period doubling cascade, or a series of consecutive period-doubling bifurcations. Numerical simulations outside these intervals show more chaotic behavior [1, 9]. First, let us review some basic terminology and give an overview of three common types of bifurcations (which we say occur at some r = r ) which may occur in periodic orbits [6]: On one side of r there exists a periodic orbit which, after the crossing of r, gives birth to an invariant torus. This is called the Neimark-Sacker bifurcation and occurs when a complex conjugate pair of eigenvalues of the monodromy matrix, M, cross the unit circle in the complex plane at a point, λ 1,2 = 1. (We can also choose to consider the Poincaré return map of the system and look at the eigenvalues of the fixed point corresponding to the periodic cycle.) Note that this bifurcation cannot occur in the Lorenz system, as an invariant torus would not allow for the volume contraction observed in the equations. Suppose on one side of r we have two periodic orbits (these can be nonsymmetric or symmetric). As r r, the orbits move very closely together 20

23 z and after r > r, both orbits disappear. This is referred to as the saddle-node bifurcation. This bifurcation is characterised by an eigenvalue λ = +1 for M. On one side of r, we have a periodic orbit. As r r, its period approaches some value T. On the other side of r, the original orbit continues to exist; however, there is, alongside it, another orbit of period 2T. This is referred to as a period-doubling bifurcation, and corresponds to an eigenvalue λ = 1 for M. Actually, it is not perfectly accurate to say that we observe chaos in between period doubling windows. In fact, at the lower boundaries of the windows, such as < r < , we observe something called noisy periodicity. If on some return plane, we can find n non-overlapping, connected regions U 1,U 2,U 3...U n such that all trajectories pass through these regions in a cyclic order, then we say that the system is semi-periodic with period n. At the top of the in-between regions, we observe intermittent chaos. This is defined as trajectories which are almost periodic, and then they wander off and act chaotically for some time, before returning to almost periodic behavior again, and so on infinitely. We show some simulations of stable periodic orbits in the period doubling windows, as well as some trajectories in the in-between times: t Figure 10: Intermittent chaos at r =

24 z t Figure 11: Noisy periodicity at r = Summary Figure 12: A stable periodic orbit at r = 400. Although we have done so once before, we conclude with a summary of how the behavior of the Lorenz system changes when one parameter, r, is varied: When 0 < r < 1, the origin, (0, 0, 0), is globally stable. All trajectories are attracted to it. When r = 1, a supercritical pitchfork bifurcation occurs. Two new stable equilibrium points C 1 and C 2 appear, while the origin loses stability. When 1 < r < r H , C 1,2 are stable fixed points. When r = r , a homoclinic bifurcation occurs. The simplest unstable limit cycles created here are later absorbed by a Hopf point. At this point a strange invariant set is born. This set contains a countable infinity of 22

25 periodic orbits and an uncountable infinity of aperiodic orbits and trajectories which tend to the origin. When r = r A , the invariant set becomes a strange invariant attractor the Lorenz attractor. When < r < , both stable fixed points (C 1,2 ) and a strange attractor co-exist. Depending on the initial conditions, the solution may tend towards the fixed points (be non-chaotic) or enter the attractor (be chaotic). When r = , a subcritical Hopf bifurcation occurs. C 1,2 lose stability as they absorb unstable limit cycles. When < r < , the Lorenz attractor exists, and behavior is highly sensitive on initial conditions. When < r < , the first period doubling window is observed. Stable periodic behavior occurs within this window. When < r < and < r < , we are between two period doubling windows. Here, again, trajectories are generally aperiodic and wandering. At the lower boundaries of the period doubling windows, noisy periodicity is observed, while at the upper boundaries, intermittent chaos is observed. When < r < and < r <, the second and third period doubling windows occur. 23

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