NBA Lecture 1 Simplest bifurcations in n-dimensional ODEs Yu.A. Kuznetsov (Utrecht University, NL) March 14, 2011
Contents 1. Solutions and orbits: equilibria cycles connecting orbits other invariant sets 2. Bifurcations of n-dimensional ODEs equilibrium bifurcations local bifrcations of cycles bifurcations of homoclinic orbits to equilibria
1. Solutions and orbits u 3 Consider a smooth system u = f(u), u R n. u 2 Th. 1 If f is smooth than for any inital point u 0 there exists a unique locally defined solution t u(t) such that u(0) = u 0. u 1 Def. 1 Let I be the maximal definition interval of a solution t u(t), t I. The oriented by the advance of time image u(i) R n is called the orbit. Def. 2 Phase portrait of an ODE system is the collection of all its orbits in R n.
Def. 3 Two systems are called topologically equivalent if their phase portraits are homeomorphic, i.e. there is a continuous invertible transformation h : R n R n, u w = h(u), that maps orbits of one system onto orbits of the other, preserving their orientation. u 2 w 2 u 1 w 1
Equilibria of ODEs An equilibrium u 0 satisfies f(u 0 ) = 0 and its Jacobian matrix A = f u (u 0 ) has eigenvalues {λ 1, λ 2,..., λ n }. Linearized stability of u 0 : If R(λ j ) < 0 for j = 1,2,..., n, the equilibrium is stable; If R(λ k ) > 0 for some k {1,2,..., n}, the equilibrium is unstable. Def. 4 An equilibrium u 0 is hyperbolic if R(λ j ) 0 for j = 1,2,..., n.
Stable and unstable invariant manifolds of equilibria: If a hyperbolic equilibrium u 0 has n s eigenvalues with R(λ) < 0 and n u eigenvalues with R(λ) > 0, it has the n s -dimensional smooth invariant manifold W s composed of all orbits approaching u 0 as t, and the n u -dimensional smooth invariant manifold W u composed of all orbits approaching u 0 as t v 1 W u 1 W ss λ 3 λ 2 λ 1 u 0 Av j = λ j v j stable unstable W s v 2 v 3 W u 2
Periodic orbits (cycles) A limit cycle C 0 corresponds to a periodic solution u 0 (t+t 0 ) = u 0 (t) of u = f(u), u R n. Floquet multipliers µ 1, µ 2,..., µ n 1, µ n = 1 are the eigenvalues of M(T 0 ): Ṁ(t) f u (u 0 (t))m(t) = 0, M(0) = I n. Linearized stability of C 0 : If µ j < 1 for j = 1,2,..., n 1, the cycle is stable; If µ k > 1 for some k {1,2,..., n 1}, the cycle is unstable. Def. 5 A cycle C 0 is hyperbolic if µ j = 1 for j = 1,2,..., n 1.
Stable and unstable invariant manifolds of cycles: If a hyperbolic cycle C 0 has m s multipliers with µ < 1 and m u multipliers with µ > 1, it has the (m s +1)-dimensional smooth invariant manifold W s composed of all orbits approaching C 0 as t, and the (m u + 1)-dimensional smooth invariant manifold W u composed of all orbits approaching C 0 as t C 0 W u (C 0 ) W u (C 0 ) W s (C 0 ) W s (C 0 ) C 0
Connecting orbits Homoclinic orbits are intersections of W u and W s of an equilibrium/cycle. Heteroclinic orbits are intersections of W u and W s of two different equilibria/cycles. Γ 0 Γ 0 W u (u 0 ) W u (u ) W u (u + ) u 0 u u + W s (u 0 ) W s (u ) W s (U + )
Generically, the closure of the 2D invariant manifold near a homoclinic orbit Γ 0 to an equilibriun with real eigenvalues (saddle) in R 3 is either simple (orientable) or twisted (non-orientable): Γ 0 Γ 0 W s W s
Compact invariant manifolds 1. tori Example: 2D-torus T 2 with periodic or quasi-periodic orbits 2. spheres 3. Klein bottles
Strange (chaotic) invariant sets - have fractal structure (not a manifold); - contain infinite number of hyperbolic cycles; - demonstrate sensitive dependence of solutions on initial conditions; - can be attracting (strange attractors); - orbits can be coded by sequences of symbols (symbolic dynamics).
2. Bifurcations of n-dimensional ODEs u = f(u, α) Local (equilibrium) bifurcations Center manifold reduction: Let u 0 = 0 at α = 0 be nonhyperbolic with stable, usntable, and critical eigenvalues: I(λ) R(λ) n c n s n u Th. 2 For all sufficiently small α, there exists a local invariant center manifold Wα c of dimension n c that is locally attracting if n u = 0, repelling if n s = 0, and of saddle type if n s n u > 0. Moreover W0 c is tangent to the critical eigenspace of A = f u(0,0).
v 2 v 3 λ λ 1 3 λ 2 λ 1 λ 2 W c 0 v 1 0 0 I(v 1 ) W c 0 R(v 1 ) Remark: W0 c is not unique; however, all W 0 c expansion. have the same Taylor Th. 3 If ξ = g(ξ, α) is the restriction of u = f(u, α) to Wα, c then this system is locally topologically equivalent to ξ = g(ξ, α), ξ R n c, α R m, ẋ = x, x R n s, ẏ = +y, y R n u.
Codimension 1 bifurcations of equilibria Consider a smooth ODE system u = f(u, α), u R n, α R. Critical cases: λ 1 λ 1 λ 2 Fold (limit point, LP): λ 1 = 0; Andronov-Hopf (H): λ 1,2 = ±iω 0, ω 0 > 0.
LP normal form on W c β(α) ξ = β(α) + b(α)ξ 2 + O( ξ 3 ), b(0) 0. ξ ξ 0 β 0 β b < 0 b > 0 Approximation of equilibria: β + bξ 2 = 0 ξ 1,2 = ± β b
Generic LP bifurcation: λ 1 = 0 (b > 0) W c β W c 0 W c β β < 0 β = 0 β > 0 Collision of two equilibria.
Hopf normal form on W c β(α) ξ = (β(α) + iω(α))ξ + c(α)ξ ξ 2 + O( ξ 4 ), ω(0) = ω 0, l 1 0 First Lyapunov coeffeicient: l 1 = 1 ω 0 R(c(0)) R(ξ) R(ξ) β β I(ξ) I(ξ) l 1 < 0 l 1 > 0 Approximate cycle: { ρ = ρ(β + R(c)ρ 2 ), ϕ = ω + I(c)ρ 2, ρ 0 = β R(c)
Generic Hopf bifurcation: λ 1,2 = ±iω 0 W c β W c 0 W c β C β β < 0 β = 0 β > 0 Birth of a limit cycle.
Codimension 1 local bifurcations of cycles I(µ) dim W c α = m c Critical cases: m s m c R(µ) cyclic fold (LPC): µ 1 = 1 m u period-doubling (PD): µ 1 = 1 Neimark-Sacker (NS): µ 1,2 = e ±iθ 0, 0 < θ 0 < π, θ 0 π 2, 2π 3
Generic LPC bifurcation Periodic parameter-dependent normal form on Wβ c: dτ = 1 + ν(β) ξ + a(β)ξ 2 + O(ξ 3 ), dt dξ = β + b(β)ξ 2 + O(ξ 3 ), dt where a, b R and the O(ξ 3 )-terms are T 0 -periodic in τ. Collision and disappearance of two limit cycles (b(0) > 0): C + 1 C 2 C 0 W c β W c 0 W c β β < 0 β = 0 β > 0
Cycle manifold near LPC u 2 T LPC C 0 α 0 α α α 0 u 1
Generic PD bifurcation Periodic parameter-dependent normal form on W c β : dτ = 1 + ν(β) + a(β)ξ 2 + O(ξ 4 ), dt dξ = βξ + c(β)ξ 3 + O(ξ 4 ), dt where a, c R and the O(ξ 3 )-terms are 2T 0 -periodic in τ. Period-doubling (c(0) < 0): W c β C 1 C + 1 W c 0 C 0 W c β C 2 β < 0 β = 0 β > 0
Generic NS bifurcation Periodic parameter-dependent normal form on W c β : dτ dt dξ dt = 1 + ν(β) + a(β) ξ 2 + O( ξ 4 ), = ( β + iθ(β) T(β) ) ξ + d(β)ξ ξ 2 + O( ξ 4 ), where a R, d C and the O( ξ 4 )-terms are T 0 -periodic in τ Torus generation (R(d(0)) < 0): C 0 T 2 β < 0 β = 0 β > 0
Codim 1 bifurcations of homoclinic orbits to equilibria Homoclinic orbit to a hyperbolic equilibrium: central nonleading stable λ 1 µ 1 0 λ 2 µ 2 λ 3 nonleading unstable leading stable leading unstable µ 0 λ0 Def. 6 Saddle quantity σ = R(µ 1 ) + R(λ 1 ). Th. 4 (Homoclinic Center Manifold) Generically, there exists an invariant finitely-smooth manifold W h (α) that is tangent to the central eigenspace at the homoclinic bifurcation.
Saddle homoclinic orbit: σ = µ 1 + λ 1 Assume that Γ 0 approaches u 0 along the leading eigenvectors. W h (0) Γ 0 Γ 0 W h (0) The Poincaré map near Γ 0 : ξ ξ = β + Aξ µ 1 λ 1 +... where generically A 0, so that a unique hyperbolic cycle bifurcates from Γ 0 (stable in W h if σ < 0 and unstable in W h if σ > 0).
3D saddle homoclinic bifurcation with σ < 0: Assume that µ 2 < µ 1 < 0 < λ 1 (otherwise reverse time: t t). W u W u Γ 0 W u C β W s W s W s β < 0 β = 0 β > 0
3D saddle homoclinic bifurcation with σ > 0: Assume that µ 2 < µ 1 < 0 < λ 1 (otherwise reverse time: t t). Γ 0 W u C β W u W u A > 0 W s W s W s β < 0 β = 0 β > 0 Γ 0 W u W u W u A < 0 C β W s W s W s β < 0 β = 0 β > 0
Saddle-focus homoclinic orbit: σ = R(µ 1 ) + λ 1 3D saddle-focus homoclinic bifurcation with σ < 0: Assume that R(µ 2 ) = R(µ 1 ) < 0 < λ 1 (otherwise reverse time: t t). W u W u W u Γ 0 C β W s W s W s β < 0 β = 0 β > 0
3D saddle-focus homoclinic bifurcation with σ > 0: 10 8 6 x 3 4 2 0 2 5 0 x 2 5 10 6 4 2 0 2 x 1 4 6 8 CHAOTIC INVARIANT SET Focus-focus homoclinic orbit: σ = R(µ 1 ) + R(λ 1 ) CHAOTIC INVARIANT SET
Homoclinic orbit(s) to a non-hyperbolic equilibrium W ss W u W s Γ 0 Γ 0 One homoclinic orbit: a unique hyperbolic cycle W u W s Γ 1 Γ 0 Several homoclinic orbits: CHAOTIC INVARIANT SET