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 manifold 1.3 The Hopf bifurcation 1.4 Sotomayor s theorem 1.5 Local bifurcation of maps 1
What we learned from Section 1&2. For a system of linear autonomous equations ẋ = Ax, the solutions live on invariant spaces that can be classified according to the eigenvalues of A. The stable (resp. unstable, centre) linear subspace is the span of eigenvectors whose eigenvalues have a negative (resp. positive, null) real part. For nonlinear systems, we define the notion of asymptotic sets (α and ω limit set), the notion of attracting set, and basin of attraction. We define two important notions of stability for a fixed point: (Lyapunov) stability (i.e. solutions remain close ) and exponential stability i.e. ( fixed point is stable AND all nearby solutions converge to the fixed point asymptotically for long time ). Lyapunov functions can be used to test stability. But, finding a Lyapunov function can be difficult. 2
What we learned from Section 3. For a system of autonomous equations ẋ = f(x), we are interested in the trajectories and asymptotic sets in phase space. At a fixed point, we can define local stable, unstable, and centre manifolds based on the corresponding linear subspaces of the linearised system. From the local stable and unstable manifolds, we can define the global stable and unstable manifolds by extending them to all times. If the unstable manifold is non-empty, the fixed point is unstable. If the unstable and centre manifolds are empty, the fixed point is asymptotically stable. If the unstable manifold is empty but the centre manifold is non-empty, we can study the dynamics on the centre manifold by centre manifold reduction. The same notions can be defined for iterative mappings and for periodic orbits. 3
4. Local bifurcations for vector fields
4. Local bifurcations for vector fields 4.1 The problem
The problem Consider the nonlinear, autonomous, first-order system of differential equations ẋ = f(x, µ) where x : E R n, µ : U R p (1) where µ is a vector of parameters. Problem: How does the dynamics change when the parameters are varied? What are the special values where qualitative changes occur? Can the possible changes be classified? Can they be obtained algorithmically? 4
An important example: the transcritical bifurcation Take p = n = 1 and consider the system ẋ = µx x 2 (2) 5
Main idea We consider the problem of determining bifurcations at fixed points. Take µ R (p=1). Then the fixed points are given by the solution of f(x, µ) = 0. (3) For values of µ for which a solution can be found, the solution defines a branch of equilibria x = x(µ). Along this branch, define the matrix D(µ) = D x f (x(µ),µ). (4) 6
Main idea D(µ) = D x f (x(µ),µ). (5) Assume that there is a value µ 0 for which D(µ) has only eigenvalues with non-zero real parts (i.e. the fixed point is hyperbolic for that value). Then, D(µ) is invertible and the local branch of equilibria can be continued locally. We increase (or decrease) µ up to a critical bifurcation value µ c where D(µ) is not invertible. At this point, the branch of equilibria is non-smooth. 7
Definitions For a system ẋ = f(x, µ) where x : E R n, µ : U R p the value µ c is a bifurcation value, if there is a qualitative change of behaviour in any neighborhood V of µ c (and for any small enough pertubations of the vector field). Note 1: The precise definition of bifurcation is related to the the notion of structural stability that implies that small enough perturbations of the vector field do not change locally the phase portrait. That is, there exists a homeomorphism mapping the trajectories of the original system to the trajectories of the perturbed system. A bifurcation occurs when the system fails to be structurally stable. Note 2: Here, as a starting point, we use the working definition for bifurcation at a fixed point: a bifurcation occurs at values where the number or stability of a fixed point change (for small enough perturbations of the vector field or parameters). 8
More definitions For a system ẋ = f(x, µ) where x : E R n, µ : U R p the bifurcation set is the loci in µ-space of bifurcation values. A bifurcation diagram is the loci in (x, µ)-space where f(x, µ) = 0 with at least one bifurcation value. 9
Another important example: the saddle-node bifurcation Consider the system ẋ = µ x 2 (6) 10
Yet another important example: the pitchfork bifurcation Consider the system ẋ = µx x 3 (7) 11
4. Local bifurcations for vector fields 4.2 The extended centre manifold
Bifurcation through reduction We are interested in problems for which a single parameter is responsible for the bifurcation (co-dimension 1 bifurcations). Therefore, we take p = 1 and we have a system of the form ż = F(z, λ), z R n, λ R (8) Assume that for λ = λ c, there is a fixed point z c and the matrix DF(z c ) has eigenvalues with zero real parts. We use the change of variables { z = zc + C z, µ = λ λ c (9) where C is chosen such that C 1 MC = [ A 0 0 B ]. (10) 12
Bifurcation through reduction The matrices A and B of respective dimension n c and n s + n u are s.t.: { Re(λ) = 0 λ Spec(A), Re(λ) 0 λ Spec(B). (11) After the change of variables, the new system in the variable z = (x, y) is {ẋ = Ax + f(x, y, µ), x R n c ẏ = By + g(x, y, µ) y R ns +nu (12) and (x, y) = (0, 0) is a fixed point for µ = 0. 13
Bifurcation through reduction MAIN IDEA: We extend the centre manifold to include the parameter. ẋ = Ax + f(x, y, µ), x R nc ns +nu ẏ = By + g(x, y, µ) y R (13) µ = 0 We can view this system as a dynamical system in the extended phase space in m = n s + n u + n c + 1 dimensions. Note 1: the vector y now denotes variables that are both in the stable and unstable manifolds. Note 2: The important realisation is that the centre manifold has now dimension n c + 1 and is parameterised by the vector (x, µ). 14
Bifurcation through reduction We can proceed as before and look for a center manifold of the form y = h(x, µ), (14) and obtain the extended centre manifold. Once this is known, we can write the dynamics on the extended centre manifold as: ẋ = Ax + f(x, h(x, µ), µ). (15) This equation captures the relevant part of the bifurcation. 15
Example Consider the system {ẋ = µ(x + y) (x + y) 2 ẏ = y µ(x + y) + (x + y) 2 (16) 16
(c). Dynamics on the centre manifold. Locally on the extended Centre Manifold µ =0 is trivial so it is the ẋ equation that is interesting: Example ẋ = µ(x + x 2 µx +...) (x 2 +2x(x 2 µx)+...) = µx x 2 + O(3) Substituting back into the equation for ẋ we get (to leading order) ẋ = µx x }{{} 2 +O(x 3 ). On the center manifold, the Standard dynamics Form for transcritical and bifurcation diagram are: The phase portrait for the reduced dynamics for x is shown in Figure 5.6 and the phase portrait for the original system is in Figure 5.7. µ<0 x µ<0 µ µ<0 Figure 5.6: Phase portraits on the (one-dimensional) centre manifoldandthebifurcation diagram. Remark. If the stable manifold is of higher dimension, then y 1 = h 1 (x, µ),y 2 = h 2 (x, µ) and we need to find h 1,h 2 using the same method. For example, for the system ẋ = µx yz, ẏ = y + x 2, ż = z + x 3. 17
Example In the full space (x, y) the dynamics for µ < 0 (left) and µ > 0 (right) is: MATH64041/44041: Applied Dynamical Systems y Figure 5.7: Full phase portraits of the dynamics in µ<0andµ>0. x Then a 1 =1,a 2 = a 3 = b 1 = b 2 = b 3 =0. Thatisy = x 2 +,butwehavetogoto cubic polynomials to find the stable manifold for z, whichgivesz = x 3 +.Therefore,the reduced dynamics on the stable manifold is ẋ = f µ (x) =µx x 5. 18
Generic bifurcations with one zero eigenvalue Note: If n c = 1, the equation on the center manifold is a single equation and the typical bifurcations are: 1. The saddle-node bifurcation (generic) ẋ = µ ax 2. (17) 2. The transcritical bifurcation (generic, assuming that there is always at least an equilibrium solution) ẋ = µx ax 2. (18) 3. The pitchfork bifurcation (generic, assuming that there is a symmetry in the system that enforces the existence of pair of fixed points after bifurcation) ẋ = µx ax 3. (19) 19
giving two new solutions in whichever sign of µ makes the right hand side positive. There are no other ways of balancing leading order terms (by posing x µ α )sothesearetheonly bifurcating solutions. Since Subcritical vs supercitical Consider the two pitchfork bifurcations x f(x, µ) =f xµµ + 1 2 f xxxx 2 +, (5.10) we see that the solution x ẋ = µx x 3 (20) fµµ 2f xµ µ is stable (locally) if f xµ µ<0andunstableiff xµ µ>0. So the sign of f ẋ = µx + x 3 xµ determines on which side of µ =0thisbranchisstable. (21) The stability of second set of solutions is determined by substituting (5.9) into (5.10) giving In the 2f first xµ µ case, and sothe thenon-trivial stability is the branch opposite is stable of the simple (supercritical branch described bifurcation) above. InThis the is second called case, a pitchfork the non-trivial bifurcation: branch if the non-trivial is unstable branch (subcritical is stablebif.) it is called a supercritical pitchfork bifurcation and if the non-trivial branch is unstable it is called a subcritical pitchfork bifurcation, as shown in Figure 5.8. Supercritical Pitchfork Bifurcation ẋ = µx x 3 Subcritical Pitchfork Bifurcation ẋ = µx + x 3 Figure 5.8: Two types of Pitchfork Bifurcation 20
4. Local bifurcations for vector fields 4.3 The Hopf bifurcation
The Hopf bifurcation There is another generic bifurcation with one parameter. It happens when the eigenvalues at the bifurcation are imaginary. Recall that we can bring a system to its canonical form ẋ = Ax + f(x, y, µ), x R nc ns +nu ẏ = By + g(x, y, µ) y R µ = 0 (22) We have studied the case where A is of dimension 1 and vanishes at the bifurcation. Next, we study the case where [ ] 0 ω A =. (23) ω 0 21
The Hopf bifurcation On the center manifold, at the bifurcation, the dynamics of the linear part (with x = (x, y)) is {ẋ = ωy ẏ = ωx. (24) To obtain the behaviour of the system close to the bifurcation (unfolding), we consider the generic perturbation around the linear system: Close to the bifurcation, the system unfolds to {ẋ = µx ωy + f (x, y, µ) ẏ = ωx + µy + g(x, y, µ). (25) 22
Example Consider the typical example of a Hopf bifurcation {ẋ = µx ωy x(x 2 + y 2 ) ẏ = ωx + µy y(x 2 + y 2 ). (26) In polar coordinates, it reads {ṙ = µr r 3 θ = ω. (27) 23
Example 24
Example 25
The Hopf bifurcation In polar coordinates, the general form of a Hopf bifurcation is {ṙ = dµr + ar 3 θ = ω + cµ + br 2, (28) where a, b, c, d are parameters that depend on the vector field at the bifurcation. The parameters c and d can be found from a linear analysis. Let λ(µ) be the eigenvalue such that λ(0) = iω, then { d = d dµ Re(λ(µ)), c = d dµ Im(λ(µ)). (29) A step-by-step method to compute the coefficients of a Hopf normal form for a planar system is given in the file hopfalgebra.pdf. 26
4. Local bifurcations for vector fields 4.4 Sotomayor s theorem
Sotomayor s theorem (1973) Consider the system ẋ = f(x, µ), x E R n, (30) and assume the general case where = x 0, is a fixed point at µ = µ 0 and there is a single eigenvalue that crosses 0 at µ = µ 0. We can obtain the form of the bifurcation without computing the extended centre manifold every time. 27
Sotomayor s theorem (1973) Define D = Df, (31) (x=x0,µ=µ 0) D has a single zero eigenvalue with the left and right eigenvectors Dv = 0, and wd = 0. (32) Define α = 1 v w w f µ (x=x0,µ=µ 0) β = 1 n ( ) 2 f i w i v j v k v w x j x k (x=x0,µ=µ 0) i,j,k=1 (33) (34) 28
Sotomayor s theorem (1973) Theorem 4.1 (Sotomayor) If α 0 β, then there exists a smooth curve of equilibrium point in R n R passing by (x 0, µ 0 ) and tangent to R n {µ 0 } so that locally: either, there are no fixed points for µ < µ 0 and two for µ > µ 0 ; or, there are no fixed points for µ > µ 0 and two for µ < µ 0 ; This theorem states that if α 0 β, there is a saddle-node bifurcation. We define the centre manifold variable y as the dynamics along v: x(t) = x 0 + y(t)v, (35) λ = µ µ 0, (36) and the dynamics on the extended centre manifold is given by ẏ = αλ + βy 2. (37) 29
Sotomayor s theorem (1973) What happens if α = 0? In this case, we define γ = 2 v w n i,j=1 w i v j ( 2 f i x j µ) (x=x0,µ=µ 0) (38) and in the variable y and λ defined by x(t) = x 0 + y(t)v, (39) λ = µ µ 0, (40) the dynamics on the extended centre manifold is a transcritical bifurcation given by ẏ = γλy + βy 2. (41) Question: what happens if β = 0? 30
Example Consider the bifurcation at the origin for system {ẋ = (1 + µ)x 4y + x 2 2xy ẏ = 2x 4µy y 2 x 2 (42) Compute Df(0, 0) = [ 1 + µ 4 2 4µ ]. (43) There is a bifurcation at µ = 1 with matrix [ ] 2 4 D =, (44) 2 4 and left/right kernels [ ] 2 [ v =, w = 1 1 1 ], v w = 1. (45) 31
Example Since, µ f(0, 0) = (0, 0), we have α = 0 and compute γ = 12, β = 10, (46) Writing [x, y] T = z(t)[2, 1] T, the bifurcation is transcritical ż = 12z(µ 1) + 10z 2. (47) 32
4. Local bifurcations for vector fields 4.5 Local bifurcation of maps
Bifurcation of maps Consider the mapping x n+1 = G(x n ), (48) where x E R m. Assume that x 0 is a fixed point (G(x 0 ) = x 0 ). This fixed point is as. stable if λ < 1 for all λ Spec(DG(x 0 )). The fixed point is unstable if there λ Spec(DG(x 0 )) s.t. λ > 1. So bifurcation will occur when an eigenvalue is on the unit complex circle. 33
Bifurcation of maps We consider a mapping with one parameter µ x n+1 = G(x n, µ), (49) where x E R m. Assume that x 0 = x 0 (µ) is a fixed point (G(x 0, µ) = x 0 ). We are interested in the case where one of the eigenvalue crosses the unit disk. It gives three possibilities at the bifurcation: either λ = 1, λ = 1 or λ λ with λ = 1. 34
Bifurcation of maps: λ = 1 at bifurcation Case I: λ = 1 at bifurcation. This case is similar to the cases obtained for vector fields. Namely, we have the saddle-node bifurcation: The transcritical bifurcation: The pitchfork bifurcation: x x + µ x 2. (50) x x + µx x 2 (51) x x + µx x 3 (52) 35
Bifurcation of periodic orbit Example in polar coordinates: {ṙ = r ( µ (r 2 1) 2) θ = 1 (53) 36
Bifurcation of maps: λ = 1 at bifurcation Case II: λ = 1 at bifurcation. Period-doubling bifurcation x x µx + x 3 (54) 37
The logistic map An important example of period-doubling cascade leading to chaotic dynamics is the logistic map (See file LogisticMap.pdf) x µx(1 x) (55) 38