Introduction to Bifurcation and Normal Form theories

Introduction to Bifurcation and Normal Form theories Romain Veltz / Olivier Faugeras October 9th 2013 ENS - Master MVA / Paris 6 - Master Maths-Bio (2013-2014)

Outline 1 Invariant sets Limit cycles Stable/Unstable manifolds Center manifold Center manifold Center manifold 2 Classification of hyperbolic sets 3 Bifurcation theory Definitions Normal form theory 4 Codimension 1 bifurcations of equilibria of continuous systems

Flow box ẋ = F(x), F smooth Lemma (Flow box) Assume F(x 0 ) 0, then there is a local diffeomorphism h such that h F is constant. F h

Description of limit cycles Poincare Map as a way to transform continous DS to discrete DS Suppose there is limit cycle Γ with x 0 Γ and period T. Consider an hypersurface Σ transverse to Γ en x 0 : F(x 0 ), x x 0 = 0 It is called a cross-section. Definition The Poincare map Π Σ x 0 Π Σ x 0 : is the application { Σ U Σ x φ T +ɛ(x) (x)

Poincare Map Properties Proposition The stability of Γ is equivalent to the stability of x 0 for Π Σ x 0 Theorem The Poincare map Π Σ x 0 is a local diffeomorphism on a neighborhood of x 0 and in a basis with first vector collinear to F(x 0 ), we have [ ] dφ T 1 (x 0 ) = 0 dπ x0 (x 0 ) Corollary The eigenvalues (multipliers) of Π Σ x 0 are independent of x 0 and Σ. Use the fact dφ T (x 0 ) dφ T (y 0 )

Stability of limit cycles Proposition The stability of Γ is equivalent to the stability of x 0 for Π Σ x 0 It is not trivial!

Hyperbolic equilibria Continuous DS ẋ = F(x), F smooth with F(x 0 ) = 0 Let n, n 0, n + be the number of eigenvalues of df(x 0 ) with negative, null, positive real part counted with multiplicity. definitions An equilibrium is: hyperbolic if n 0 = 0 a hyperbolic saddle if n n + 0 Since a generic matrix has no eigenvalues on the imaginary axis, hyperbolicity is a typical property and an equilibrium in a generic system (i.e., one not satisfying certain special conditions) is hyperbolic.

Invariant sets: Local stable manifold Continuous case definitions Consider an equilibrium x 0, its stable and unstable sets are defined by Theorem W s (x 0 ) = {x : lim t φ t x = x 0 } W u (x 0 ) = {x : lim t φt x = x 0 } Let x 0 R n be a hyperbolic equilibrium (i.e., n 0 = 0, n + n + = n ). Then the intersections of W s (x 0 ) and W u (x 0 ) with a sufficiently small neighborhood of x 0 contain smooth submanifolds Wloc s (x 0) and Wloc u (x 0) of dimension n and n +, respectively. Moreover,Wloc s (x 0)(resp. Wloc u (x 0)) is tangent to T s (resp. T u ) where T s (resp. T u ) is the generalized eigenspace corresponding to the eigenvalues of df (x 0 ) of negative (resp. positive) real part.

Invariant sets: Local stable manifold Continuous case Sketch of proof φ 1 (T u ) is a manifold of dimension n + tangent to T u at x 0 Restrict attention to a sufficiently small neighborhood of the equilibrium where the linear part is dominant and repeat the procedure. the iterations converge to a smooth invariant submanifold defined in this neighborhood of x 0 and tangent to T u at x 0. for W s, apply φ 1 to T s

o a smooth invariant submanifold defined in this neighborhood of x 0 and angent to T u at x 0. The limit is the local unstable manifold W u cal stable manifold Wloc s (x0) can be constructed by applying ϕ 1 to T s. loc (x0). The Local stable manifold in the continuous case From Kuznetsov emark: Globally, the invariant sets W s and W u are immersed manifolds ofλ 3 di-ensions n and n +, respectively, and have the same smoothness proper- 2 λ 1 ies as f. Having these properties in mind, we will call the sets W s and u the stable and unstable invariant manifolds of x 0, respectively. Example 2.2 (Saddles and saddle-foci in R 3 ) Figure 2.4 illustrates (a) s W x 0 v 1 u W 1 u v W 2 3 v 2 v 1 u W 1 v 1 u W 1 λ 3 λ 2 λ 1 s W λ2 λ 1 s W x 0 x 0 v 2 λ3 Re v 2 (a) u v W 2 3 (b) Im v 2 W u 2 λ2 λ 1 s W v 1 u W 1 x 0 FIGURE 2.4. (a) Saddle and (b) saddle-focus: The vectors νk are the eigenvecto corresponding to the eigenvalues λk. Figure: a) Saddle b) Saddle-Foci the theorem for the case where n = 3, n = 2, and n + = 1. In th λ3 Re v 2 (b) Im v 2 W u 2

Invariant sets: Local stable manifold Discrete case x F(x), F and F 1 smooth with F(x 0 ) = 0, diffeomorphism Let n, n 0, n + be the number of eigenvalues of df(x 0 ) with modulus < 1, = 1, > 1 counted with multiplicity. definitions An equilibrium is: hyperbolic if n 0 = 0 a hyperbolic saddle if n n + 0

Proof analogous to continuous case if one substitutes φ 1 by F. Local stable manifold in the discrete case definitions Consider an equilibrium x 0, its stable (unstable) set is defined by Theorem W s (x 0 ) = {x : lim k Fk (x) = x 0 } W u (x 0 ) = {x : lim k Fk (x) = x 0 } Let x 0 R n be a hyperbolic equilibrium (n 0 = 0, n + n + = n ). Then the intersections of W s (x 0 ) and W u (x 0 ) with a sufficiently small neighborhood of x 0 contain smooth submanifolds Wloc s (x 0) and Wloc u (x 0) of dimension n and n +, respectively. Moreover,Wloc s (x 0)(resp. Wloc u (x 0)) is tangent to T s (resp. T u ) where T s (resp. T u ) is the generalized eigenspace corresponding to the eigenvalues λ of df(x 0 ) such that λ < 1 ( λ > 1).

Local stable manifold in the discrete case From Kuznetsov 2.2 Classification of equilibria and fixed points 53 Example with positive/negative multiplier W u 1 W s 1 x µ 0 2 µ 1 (a) W u 1 W s 2 W s 1 W u 2 µ 1 µ 2 x 0 (b) W u 2

Hyperbolic limit cycles 2 2 1 2 x =(x 1,x 2 ) T R 2. Let x 0 (t) be a solution corresponding to a limit cycle L 0 of the system, and let T 0 be the (minimal) period of this solution. There Saddle cycles inis3d onlysystems: one multiplier (a) positive of the cycle, multipliers µ 1, which is and positive (b) negative and is givenmultipliers. by From Kuznetsov. { } T0 µ 1 = exp (div f)(x 0 (t)) dt > 0, 0 Invariant manifolds: where div stands W s for the divergence of the vector (L 0 ) = {x : lim φ t field: (x) L 0 } (div f)(x) = f t 1(x) + f 2(x). W u x 1 x 2 (L 0 ) = {x : lim t φt (x) L 0 } If 0 <µ 1 < 1, we have a stable hyperbolic cycle and all nearby orbits converge We have W s,u exponentially to it, while for (x 0, Π) = Σ W s,u µ 1 > 1 we have an unstable hyperbolic cycle with exponentially diverging neighboring (L 0 ) orbits. Σ W u ( L ) W u ( L ) 0 0 W s ( L ) 0 ξ P ( ξ ) L 0 Σ P ( ξ ) ξ P 2 ( ξ ) W s ( L ) 0 L 0

Center manifold Continuous case ẋ = Lx + R(x; µ), L L(R n ), R C k (V x V µ, R m ) (1) R(0; 0) = 0, dr(0; 0) = 0 Theorem 1/2 Write R n = T c T h where T h = T s T u. Then, there is a neighborhood O = O x O µ of (0, 0) in R n R m, a mapping Ψ C q (T c R m ; T h ) with Ψ(0; 0) = 0, dψ(0; 0) = 0 and a manifold M(µ) = {u c + Ψ(u c, µ), u c T c } for µ V µ such that: 1 M(µ) is locally invariant, i.e., x(0) M(µ) O x and x(t) O x for all t [0, T ] implies x(t) M(µ) for all t [0, T ].

Center manifold Continuous case Theorem 2/2 2 M(µ) contains the set of bounded solutions of (1) staying in O x for all t R, i.e. if x is a solution of (1) satisfying for all t R, x(t) O x, then x(0) M(µ). 3 (Parabolic case) if n + = 0, then M(µ) is locally attracting, i.e. if x is a solution of (1) with x(0) O x and x(t) O x for all t > 0, then there exists v(0) M(µ) O c and γ > 0 such that x(t) = v(t) + O(e γt ) as t where v is a solution of (1) with initial condition v(0).

Center manifold From Kuznetsov URE 5.2. One-dimensional center manifold at the fold bifurcat λ 3 λ λ 1 2 v 3 W c x 0 Re v 1 Im v

Center manifold Additional properties 1 The Center manifold is not unique 2 If x c (0) M(µ), then ẋ c = L c x c + P c R(x c + Ψ(x c, µ), µ) f (x c, µ) where P c is the projector on T c. 3 The local coordinates function satisfies dψ(x c, µ) f (x c ) = P h L Ψ(x c, µ) + P h R(x c + Ψ(x c, µ)) 4 There are extensions for non-autonomous systems, with symmetries... 5 Extensions to Banach spaces possible in some cases

Center manifold Discrete case ẋ Lx + R(x, α), L L(R n ), R C k (V x V α, R n ) R(0, 0) = 0, d x R(0, 0) = 0 Write R n = T c T h where T h = T s T u is the hyperbolic part. Then we have the same conclusions as for the continuous case.

Topological equivalence definition A dynamical system {T, R n, φ t } is called locally topologically equivalent near an equilibrium x 0 to a dynamical system {T, R n, ψ t } near an equilibrium y 0 if there exists a homeomorphism h : R n R n that is 1 defined in a small neighborhood U R n of x 0 ; 2 satisfies y 0 = h(x 0 ) 3 maps orbits of the first system in U onto orbits of the second system in V = F(U) R n, preserving the direction of time. In the discrete case x F(x) is equivalent to y G(y) if F = h 1 G h i.e they are conjugate.

Equivalence From Kuznetsov Example: 44 2. Equivalence and Bifurcations (a) (b) Vector fields: { ẋ1 = x 1 { ẋ1 = x, λ = 1 and 1 x 2 FIGURE 2.2. Node-focus equivalence. those of (2.9) are spirals. The phase portraits of the systems are presented in Figure ẋ 2 2.2. = The xequilibrium 2 of the first system is ẋ 2 a node = x (Figure 2 + 2.2(a)), x 1 while in the second system it is a focus (Figure 2.2(b)). The difference in behavior of the systems can also be perceived by saying that perturbations decay near the origin monotonously in the first case and oscillatorily in the second case. The systems are neither orbitally nor smoothly equivalent. The first fact is obvious, while the second follows from the observation that the eigen- There are not smoothly equivalent!, λ = 1 ± i

Topological equivalence of hyperbolic points Continuous case Consider: ẋ = F(x), x R n, F smooth Theorem The phase portraits of the DS near two hyperbolic equilibria, x 0 and y 0, are locally topologically equivalent if and only if these equilibria have the same number n and n + of eigenvalues with Rλ < 0 and with Rλ > 0, respectively.

Topological equivalence of hyperbolic points Continuous case Sketch of the proof: 1 near a hyperbolic equilibrium the system is locally topologically equivalent to its linearization: ξ = df(x 0 )ξ (Grobman-Hartman Theorem). 2 We apply it near x 0 and y 0 3 We prove the topological equivalence of two linear systems having the same numbers of eigenvalues with Rλ < 0 and Rλ > 0 and no eigenvalues on the imaginary axis.

Topological classification of hyperbolic equilibria on R 2 Continuous case Exercise: 2.2 Classification of equilibria and fixed points 49 ( n +, n - ) Eigenvalues Phase portrait Stability (0, 2) node stable focus (1, 1) saddle unstable node (2, 0) unstable focus

Topological equivalence of hyperbolic points Discrete case Consider: Theorem x F(x), x R n, F, F 1 smooth (1) The phase portraits of system (1) near two hyperbolic equilibria, x 0 and y 0, are locally topologically equivalent if and only if these equilibria have the same number n and n + of multipliers with λ < 1 and with λ > 1, respectively, and the signs of the products of all the multipliers with λ < 1 and with λ > 1 are the same for both fixed points. Analogous to the continuous case but one needs to be careful about preserving the direction of time. Recall the conjugacy property...

Invariant sets: Local stable manifold ample 2.4 (Stable fixed points in R 1 ) Suppose x 0 = 0 is a fixed of Discrete a one-dimensional case discrete-time system (n = 1). Let n = 1, meanhat the unique multiplier µ satisfies µ < 1. In this case, according heorem 2.3, all orbits starting in some neighborhood of x 0 converge 0. Example Depending on the sign of the multiplier, we have the two possi- 1-1 µ 1 x 0 x x x 4 3 2 x 1-1 (a) 1-1 µ 1 0 x 5 x 2 4 3 1 x x x x -1 (b) From Kuznetsov RE 2.6. Stable fixed points of one-dimensional systems: (a) 0 <µ<1; (b) µ<0.

orbits converging to x 0 under iterations of f, and the one-dimensional man- Local stable manifold in the discrete case Staircase diagrams for stable fixed points Example 52 2. Equivalence and Bifurcations x ~ = x ~ x = -x x ~ = x ~ x = f ( x ) f ( x ) x f ( x ) x ~ x = f ( x ) (a) (b) FIGURE 2.7. Staircase Fromdiagrams Kuznetsov for stable fixed points.

Topological equivalence and center manifold Continuous case Write the DS in an eigenbasis (1) {ẋc = A c x c + f (x c, x h ) ẋ h = A h x h + g(x c, x h ) proposition The system (1) is locally topologically equivalent to { ẋc = A c x c + f (x c, x c + Ψ(x c )) ẋ h = A h x h the equations discouple { ẏ = y Note that ẋ h = A h x h is equivalent to R n ẏ + = y + R n+

Topological equivalence and center manifold Discrete case Write the DS in an eigenbasis (1) { xc A c x c + f (x c, x h ) x h A h x h + g(x c, x h ) proposition The system (1) is locally topologically equivalent to { xc A c x c + f (x c, x c + Ψ(x c )) x h A h x h the equations discouple

Topological equivalence of DSs Classification of possible phase diagrams of generic systems, at least, locally. Consider two DS: ẋ = F(x, α), ẏ = G(y, β), x, y R n, α, β R m definition The DS are called locally topologically equivalent near the origin if there is a map (x, α) (h α (x), p(α)) defined in a small neighborhood of (x, α) = (0, 0) R n R m such that p is a homeomorphism defined in a small neighborhood of α = 0, p(0) = 0 h : R n R n is a parameter-dependent homeomorphism defined in a small neighborhood U α of x = 0, h 0 (0) = 0, mapping orbits in U α of the first system at parameter values α onto orbits in h α (U α ) of the second system and preserving the direction of time.

Equivalence of DSs We consider parameter dependent dynamical systems x F(x, α) ẋ = F(x, α) x R n, α R m definition The appearance of a topologically nonequivalent phase portrait under variation of parameters is called a bifurcation. definition A bifurcation diagram of the dynamical system is a stratification of its parameter space induced by the topological equivalence, together with representative phase portraits for each stratum. definition The codimension is the number of independent conditions determining the bifurcation.

in the direct product of the phase and parameter spaces, R R with the phase portraits represented by one- or two-dimensional slices α = const. Example 1: Consider, the Pitchfork example, a scalar bifurcation system Local bifurcation, from Kuznetsovẋ = αx x 3,x R 1, α R 1. This system has an equilibrium x 0 = 0 for all α. This equilibrium is stable for α < 0 and unstable for α > 0(α is the eigenvalue of this equilibrium). Can be detected For α > if0, we therefix areany two extra small equilibria neighborhood branching fromofthe the origin equilibrium. (namely, x x 1 x 0 0 x 1,2 = ± α) which are stable. This bifurcation is often called a pitchfork bifurcation, the reason for which becomes ẋ = x(α x 2 immediately clear if one has a look at the bifurcation diagram of the system ) presented in (x, α)-space (see Figure 2.16). Notice that the system demonstrating the pitchfork biα x 2 FIGURE 2.16. Pitchfork bifurcation. Two stratums {α 0} and {α > 0}, codim 1 furcation is invariant under the transformation x x. We will study bifurcations in such symmetric systems in Chapter 7. In the simplest cases, the parametric portrait is composed by a finite

Example 2: the Andronov-Hopf bifurcation Local bifurcation, from Kuznetsov 58 2. Equivalence and Bifurcations Can be detected if we fix any small neighborhood of the equilibrium. In polar coordinates (ρ, θ) it takes the form { { ẋ1 = αx ρ 1 = x ρ(α 2 x ρ 1 (x 2 ), 1 2 + x 2 2) ẋ 2 = x 1 + αx 2 x 2 (x1 2 + x 2 2 (2.17) θ = 1, ) and can be integrated explicitly (see Exercise 6). Since the equations for α < 0 α = 0 α > 0 FIGURE 2.13. Hopf bifurcation. Two stratums {α 0} and {α > 0}, codim 1 ρ and θ are independent in (2.17), we can easily draw phase portraits of the system in a fixed neighborhood of the origin, which is obviously the

Example 3: the Heteroclinic bifurcation Global bifurcation, from Kuznetsov To detect this bifurcation we must fix a region covering both saddle. { ẋ1 = 1 x 2 1 αx 1x 2 ẋ 2 = x 1 x 2 + α(1 x 2 1 ) The system has two saddle equilibria x (1) = ( 1, 0), x (2) = (1, 0). α < 0 α = 0 α > 0 Three stratums {α < 0}, {α = 0} and {α > 0}, codim 1

Normal form theory 1/2 Each bifurcation needs to be studied? The idea is to find a polynomial CHV which improves locally a nonlinear system, in order to analyze its dynamics more easily. ẋ = Lx + R(x; α), L L(R n ), R C k (V x V α, R m ) (1) R(0; 0) = 0, dr(0; 0) = 0 Theorem 1/2 p [2, k], there are neighborhoods V 1 and V 2 of 0 in R n and R m, respectively, such that for any α V 2, there is a polynomial Φ α : R n R n of degree p with the following properties: The coefficients of the monomials of degree q in Φ α are functions of α of class C k q, and Φ 0 (0) = 0, dφ 0 (0) = 0

Normal form theory 2/2 Each bifurcation needs to be studied? Theorem 2/2 For any x V 1, the polynomial CHV x = y + Φ α (y) transforms (1) into the normal form ẏ = Ly + N α (y) + ρ(y, α) The coefficients of the monomials of degree q in N α are functions of α of class C k q, and N 0 (0) = 0, d x N 0 (0) = 0 the equality N α (e tl y) = e tl N α (y) holds for all (t, y) R R n and α V 2 the maps ρ belongs to C k (V 1 V 2, R n ) and α V 2, ρ(y; α) = o( y p )

An example Prelude to the Hopf bifurcation Consider the case L = Lemma [ 0 ω ω 0 ], ω > 0. In the basis (ζ, ζ), ζ = (1, i): L = [ iω 0 ] 0 iω Write x = y + Φ α (y), the change of variable with y = Aζ + Aζ N α (Aζ + Aζ) = AQ α ( A 2 )ζ + AQ α ( A 2 ) ζ. Proof: use N α (e tl y) = e tl N α (y). Write N α (Aζ + Aζ) = P α (A, Ā)ζ + P α (A, Ā)) ζ and note that e tl = diag ( e iωt, e iωt) which gives P α ( e iωt A, e iωt Ā ) = e iωt P α (A, Ā). Looking for the monomials satisfying this condition allows to conclude.

The machinary Consider a continuous dynamical system with a local bifurcation:

The machinary Consider a continuous dynamical system with a local bifurcation: there a center manifold x = x c + Ψ(x c ; µ). Compute Ψ with a Taylor expansion.

The machinary Consider a continuous dynamical system with a local bifurcation: there a center manifold x = x c + Ψ(x c ; µ). Compute Ψ with a Taylor expansion. project the dynamics on the center manifold ẋ c = Lx c + P c R(x c + Ψ(x c ; µ); µ)

Bifurcation conditions Consider a DS ẋ = F(x, α), x R n

Bifurcation conditions Consider a DS ẋ = F(x, α), x R n Consider an hyperbolic equilibrium x 0 for α = α 0. Remains hyperbolic for slight variations of α

Bifurcation conditions Consider a DS ẋ = F(x, α), x R n Consider an hyperbolic equilibrium x 0 for α = α 0. Remains hyperbolic for slight variations of α two ways of breaking hyperbolicity: 1 Either a simple real eigenvalue approaches zero and we have λ = 0, fold bifurcation. 2 or a pair of simple complex eigenvalues reaches the imaginary axis and we have λ 1,2 = ±iω 0, ω 0 > 0, Andronov-Hopf bifurcation.

The Fold bifurcation Also called Saddle-Node of Turning-point bifurcation Consider a DS projected on the center manifold: ẋ = F(x, α) = Lx + R(x, α), x R 1 with L = df(x 0, α 0 ). We have L = L = 0 (Fold) and e tl = Id: no constraint on the normal form N α (y), but we can still simplify it a bit. Write a Taylor expansion F(x, α) = F 0 (α) + F 1 (α)x + F 2 (α)x 2 + O(x 3 ) with F 0 (0) = 0 and F 1 (0) = df(0, 0) = 0. Change variables and parameters Some nondegeneracy and traversality conditions will appear

The Fold bifurcation Shift coordinates Use ξ = x + δ(α), δ smooth to get ξ = + ( F 1 (α) F 2 (α)δ + O(δ 2 ) ) ξ + + O(ξ 3 ) If F 2 (0) = 1 2 d 2 F(0, 0) 0, the Implicit Function Theorem applied to A(α, δ) = F 1 (α) F 2 (α)δ + δ 2 ψ(α, δ), ψ smooth, gives a smooth function δ such that δ(α) = F 1 (0) 2F 2 (0) α + O(α2 ), hence: ξ = ( F 0(0)α + O(α 2 ) ) + (F 2 (0) + O(α)) ξ 2 + O(ξ 3 )

The Fold normal form Introduce a new parameter Define µ = µ(α) = F 0 (0)α + α2 φ(α). We have µ(0) = 0, µ (0) = F 0 (0) = αf(0, 0). Hence, if α F(0, 0) 0, the Implicit Function Theorem gives a function α(µ) with α(0) = 0 such that ξ = µ + a(µ)ξ 2 + O(ξ 3 ) with a(0) = F 2 (0). Finally, use η = a(µ) ξ and β = a(µ) µ around µ = 0 to find η = µ + sη 2 + O(η 3 ), s = sign(a(0))

The Fold bifurcation Truncated system, s = 1 3.2 The normal form of the fold bifurcation 81 y y y y = f ( x, α) y = f ( x, α) y = f ( x, α) x 2 x 1 x x x x 0 α < 0 α = 0 α > 0 FIGURE 3.2. Fold bifurcation. Figure: From Kuznetsov and unstable) collide, forming at α = 0 an equilibrium with λ = 0, and disappear. This is a fold bifurcation. The term collision is appropriate,

The Fold bifurcation Topological equivalence Lemma The system ẏ = α + y 2 + O(y 3 ) is locally topologically equivalent to the system ẋ = α + x 2. Proof: use IFT to show the persistence of local equilibria y i (α). Write x i (α) = ± α. Use the homeomorphism: { x for α > 0 y = h(x, α) = a(α) + b(α)x otherwise such that y i (α) = h(x i (α), α). Theorem If ẋ = F(x, α), x R 1 is such that F(0, 0) = df(0, 0) = 0. Assume 1 d 2 F(0, 0) 0 2 α F(0, 0) 0 then the DS is topo. equivalent to ẋ = β + sx 2, s = sign ( d 2 F(0, 0) ).

The Hopf bifurcation Also called Andronov-Hopf bifurcation Consider a DS projected on the center manifold ẋ = F(x, α) = Lx + R(x, α), x R 2 Assume L df(x 0, α 0 ) has two eigenvalues ±iω and x 0 = 0, α 0 = 0: Lζ = iωζ, L ζ = iω ζ Use the CHV x = zζ + z ζ + Φ(z, z, α) (see previous slide) ż = iωz + zq( z 2, α) + ρ(z, z, α) From Q( z 2, α) = a(α) + b(α) z 2, a(α), b(α) C with a(0) = 0 we get: ż = (a(α) + iω) z + b(α) z 2 z + O((µ + z 2 ) 2 )

The Hopf bifurcation Also called Andronov-Hopf bifurcation Lemma Assume Ra (0), Rb(0) 0, then is equivalent to ż = (a(α) + iω) z + b(α) z 2 z + O((µ + z 2 ) 2 ) ż = (µ + i) z + s z 2 z + O((µ + z 2 ) 2 ), s = ±1. Lemma The system ż = (µ + i) z + s z 2 z + O((µ + z 2 ) 2 ) is locally topologically equivalent to ż = (µ + i) z + s z 2 z.

The Hopf bifurcation Also called Andronov-Hopf bifurcation Theorem If ẋ = F(x, α), x R 2 has for all sufficiently small α the equilibrium x = 0 with eigenvalues λ 1,2 (α) = µ(α) ± iω 0 (α) where µ(0) = 0, ω 0 (0) = ω > 0. Assume that 1 µ (0) 0 2 Rb 0 (Lyapunov coefficient) then the DS is topo. equivalent to ż = (µ + i) z + s z 2 z. Note: the second condition involves a lengthy expression that is not provided here.