BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs

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1 BIFURCATION PHENOMENA Lecture 1: Qualitative theory of planar ODEs Yuri A. Kuznetsov August, 2010

2 Contents 1. Solutions and orbits. 2. Equilibria. 3. Periodic orbits and limit cycles. 4. Homoclinic orbits.

3 Literature 1. A.A. Andronov, E.A. Leontovich, I.I. Gordon, and A.G. Maier Qualitative Theory of Second-Order Dynamic Systems, Willey & Sons, London, L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, and L.O. Chua Methods of Qualitative Theory in Nonlinear Dynamics, Part I, World Scientific, Singapore, F. Dumortier, J. Llibre, and J.C. Artés Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, Berlin, 2006

4 1. SOLUTIONS AND ORBITS Newton s Second Law: mẍ = F(x, ẋ) { ẋ = y, ẏ = m 1 F(x, y) General planar system: { ẋ = P(x, y), ẏ = Q(x, y) where X = ( x y ) or Ẋ = f(x), X R 2,, f(x) = ( P(x, y) Q(x, y) Theorem 1 If f is smooth than for any inital point there exists y ( ) 0 x(t) a unique locally defined solution t such that x(0) = x y(t) 0 and y(0) = y 0. ). ( x0 )

5 Definition 1 Let I be the maximal definition interval of a solution t X(t), t I. The oriented by the advance of time image X(I) R 2 is called the orbit. f(x) X Vector field: X f(x) f(x) 0 is tangent to the orbit through X orbits do not cross. Definition 2 Phase portrait of a planar system is the collection of all its orbits in R 2. We draw only key orbits, which determine the topology of the phase portrait.

6 Types of orbits: 1. Equilibria: X(t) X 0 so that f(x 0 ) = Periodic orbits (cycles): X(t) X 0, X(t + T) = X(t), t R The minimal T > 0 is called the period of the cycle. 3. Connecting orbits: lim t ± X(t) = X ± with f(x ± ) = 0. If X = X + the connecting orbit is called homoclinic If X X + the connecting orbit is called heteroclinic. 4. All other orbits

7 Theorem 2 (Poincaré-Bendixson) A bounded orbit of a smooth system Ẋ = f(x), X R 2, tends to one of the following sets in the phase plane: (i) an equilibrium point; (ii) a periodic orbit; (iii) a union of equilibria and their connecting orbits.

8 2. EQUILIBRIA: Null-isoclines f(x) = 0 { P(x, y) = 0, Q(x, y) = 0. P = ( ) Px P y and Q = ( ) Qx Q y are orthogonal to P = 0 and Q = 0, resp. Q(x,y) = 0 P Q Jacobian matrix of the equilibrium X 0 : A = f X (X 0 ) = ( Px P y Q x Q y ) x=x0,y=y 0 y 0 P(x, y) = 0 If det A 0 the null-isoclines intersect transversally at X 0. If det A = 0 the null-isoclines are tangent at X 0. x 0

9 Eigenvalues of the equilibrium X 0 are the eigenvalues of A, i.e. the solutions of where λ 2 σλ + = 0, σ = λ 1 + λ 2 = SpA = P x (x 0, y 0 ) + Q y (x 0, y 0 ), = λ 1 λ 2 = det A = P x (x 0, y 0 )Q y (x 0, y 0 ) P y (x 0, y 0 )Q x (x 0, y 0 ). λ 1,2 = σ 2 ± σ 2 4 Definition 3 An equilibrium X 0 is hyperbolic if R(λ) 0. Equilibrium X 0 with λ 1 = 0 (i.e. det A = 0) is called multiple. Equilibrium X 0 with λ 1 + λ 2 = 0 (i.e. Sp A = 0) is called neutral.

10 Phase portraits of planar linear systems Ẏ = AY (n u,n s ) Eigenvalues Phase portrait Stability (0, 2) node stable focus (1, 1) saddle unstable node (2, 0) unstable focus

11 Definition 4 Two systems are called topologically equivalent if their phase portraits are homeomorphic, i.e. there is a continuous invertible transformation that maps orbits of one system onto orbits of the other, preserving their orientation. Theorem 3 (Grobman-Hartman) Consider a smooth nonlinear system Ẋ = AX + F(X), F = O( X 2 ) O(2), and its linearization Ẏ = AY. If R(λ) 0 for all eigenvalues of A, then these systems are locally topologically equivalent near the origin. Warning: A stable/unstable node is locally topologically equivalent to a stable/unstable focus.

12 Trivial topological equivalences 1. Orbital equivalence: Ẋ = f(x) Ẏ = g(y )f(y ) where g : R 2 R is smooth positive function; Y = h(x) = X preserves orbits. 2. Smooth equivalence: Ẋ = f(x) Ẏ = h X (h 1 (Y ))f(h 1 (Y )), where h : R 2 R 2 is a smooth diffeomorphism; the substitution Y = h(x) transforms solutions onto solutions: Ẏ = h X (X)Ẋ = h X (X)f(X) where X = h 1 (Y ). 3. Smooth orbital equivalence:

13 Simplest critical cases λ 1 = 0, λ 2 0 By a linear diffeomorphism, Ẋ = f(x) can be transformed into { ẋ = ax 2 + bxy + cy 2 + O(3), ẏ = λ 2 y + O(2). If a 0 then Ẋ = f(x) is locally topologically equivalent near the origin to { ẋ = ax 2, Saddle-node (a > 0): ẏ = λ 2 y. λ 2 < 0 λ 2 > 0

14 λ 1,2 = ±iω, ω > 0 By a linear diffeomorphism, Ẋ = f(x) can be transformed into { ẋ = ωy + R(x, y), R = O(2), ẏ = ωx + S(x, y), S = O(2). Introduce z = x + iy C. Then this system becomes ż = iωz + g(z, z), where ( z + z g(z, z) = R 2, z z 2i Write its Taylor expansion in z, z: ) + is ( z + z 2, z z ). 2i g(z, z) = 1 2 g 20z 2 + g 11 z z g 02 z g 21z 2 z +... Definition 5 The first Lyapunov coefficient is l 1 = 1 2ω 2R(ig 20g 11 + ωg 21 ).

15 If l 1 0 then Ẋ = f(x) is locally topologically equivalent near the origin to { ρ = l1 ρ 3, ϕ = 1, where (ρ, ϕ) are polar coordinates: z = ρe iϕ. Weak focus: stable unstable l 1 < 0 l 1 > 0

16 3. PERIODIC ORBITS AND LIMIT CYCLES Poincaré map: ξ P(ξ) = µξ + O(2), ξ P(ξ) 0 where the multiplier µ = exp ( T 0 (div f)(x0 (t)) dt ) > 0 X 0 (t) Definition 6 A cycle of the planar system is hyperbolic if µ 1. The cycle is stable if µ < 1 and is unstable if µ > 1. ξ µ > 1 ξ = ξ µ < 1 ξ

17 Theorem 4 (Bendixson-Dulac) If (div f)(x) > 0 (< 0) in a disc D R 2, then Ẋ = f(x) has no periodic orbits in D. Proof: Suppose, there is a cycle C D and let Ω be a bounded domain with Ω = C. C Pdy Qdx = f, dx 0 C but C Pdy Qdx = (div f)dx > 0 Ω (or < 0), contradiction. f = ( P Q X dx ) f = ( C Q P )

18 Implications: 1. If div(gf) > 0 (< 0) in a disc D R 2 for a smooth positive function g : R 2 R, then Ẋ = f(x) has no periodic orbits in D. 2. If div(gf) > 0 (< 0) is an annulus A R 2 for a smooth positive function g : R 2 R, then Ẋ = f(x) has at most one periodic orbit in A. 3. If f(x) 0 and div(gf) < 0 in a trapping annulus A R 2 for a smooth positive function g : R 2 R, then Ẋ = f(x) has a unique stable periodic orbit in A.

19 Example: Consider { ẋ = y P(x, y), ẏ = ax + by + αx 2 + βy 2 Q(x, y). Define in R 2. Then g(x, y) = e 2βx > 0 x (gp) + y (gp) = x (e 2βx y) + y (e 2βx (ax + by + αx 2 + βy 2 )) = 2βe 2βx y + be 2βx + 2βe 2βx y = be 2βx 0 in R 2 if b 0. no periodic orbits.

20 Reversible systems Definition 7 A smooth system Ẋ = f(x) is called reversible if f(jx) = Jf(X) for a matrix J such that J 2 = E. The transformation X JX is called an involution. If there is an orbit segment without equilibria connecting two points in the fixed subspace {Y : JY = Y } of the involution, there is a periodic orbit of Ẋ = f(x). Periodic orbits occur in continuos familes. Example: 4 { ẋ = y, ẏ = x + xy x 3, J ( x y ) = ( x y ) y x

21 Example: A prey-predator model ξη ξ = ξ (1 + αξ)(1 + βη) f 1(ξ, η), ξη η = η + (1 + αξ)(1 + βη) f 2(ξ, η), where α, β > 0 and x, y 0. There is a family of closed orbits for α = β if 0 < α = β < 1 4. since the system is reversible with involution J : (ξ, η) (η, ξ). There are no closed orbits if α β, since the choice g(ξ, η) = ξ a η b (1 + αξ)(1 + βη), with appropriate a and b implies div(gf) = (α β)ξg.

22 Planar Hamiltonian systems: H : R 2 R (smooth) { ẋ = Hy (x, y), Ḣ = H ẏ = H x (x, y). x ẋ + H y ẏ 0 H(x(t), y(t)) = h Potential system: H(x, y) = y2 2 + U(x) (reversible: y y, t t). where T = ds dh h=h0 (1) (2) (3) y h H(x,y) = h (3) (3) (2) S(h) = area inside y2 + U(x) = h (1) 2 The Lotka-Volterra prey-predator model is orbitally equivalent to a Hamiltonian system. U(x) x

23 Dissipative perturbations of 2D Hamiltonian systems { ( ẋ = Hy (x, y) + εp(x, y), P(x, y) F(x, y) = ẏ = H x (x, y) + εq(x, y), Q(x, y) ). Let X 0 (t) correspond to the T 0 -periodic orbit C 0 at ε = 0 and let Ω 0 = domain bounded by C 0. X Ω C 0 Γ 0 Theorem 5 (Pontryagin-Melnikov) If Ω 0 div F(X) dx = 0 but T0 0 div F(X0 (t)) dt 0 then there exists an annulus contaning C 0 in which the system has a unique periodic orbit C ε for all sufficiently small ε, such that C ε C 0 as ε 0.

24 Example: Van der Pol equation ẍ + x = εẋ(1 x 2 ) { ẋ = y, ẏ = x + εy(1 x 2 ), For ε = 0, H(x, y) = 1 2 (x2 + y 2 ) and X 0 (t) = with T 0 = 2π. Then C 0 div F dxdy = and T0 ( rsin t rcos t F(x, y) = ), C 0 = {(x, y) : x 2 + y 2 = r 2, r > 0} ( P(x, y) Q(x, y) C 0 Pdy Qdx = ) 2π 0 2π = ( 0 y(1 x 2 ) ). r 2 cos 2 t(1 r 2 sin 2 t)dt = π 4 r2 (4 r 2 ) div 0 F(X0 (t)) dt = (1 0 4sin2 t)dt = 2π A cycle close to r = 2 exists for small ε 0.

25 4. HOMOCLINIC ORBITS Homoclinic orbits to saddles: small Γ 0 big Γ 0 X 0 X 0 Definition 8 The real number σ = λ 1 + λ 2 = (div f)(x 0 ) is called the saddle quantity of X 0. Γ 0 Γ 0 X 0 X 0 σ < 0 σ > 0

26 Singular map: { ẋ = λ1 x ẏ = λ 2 y y Q Regular map: ξ = (η) = η λ 1 λ2 η = Q(ξ) = Aξ + O(2), A > 0. 1 ξ η η 0 1 x Poincaré map: η η = Q( (η)) = Aη λ 1 λ The homoclinic orbit is stable if σ < 0 and is unstable if σ > 0.

27 If σ = λ 1 + λ 2 = 0, then if if (div f)(x0 (t)) dt < 0 the homoclinic orbit is stable; (div f)(x0 (t)) dt > 0 the homoclinic orbit is unstable. Homoclinic orbits to saddle-nodes: X 0 X 0 Γ 0 Γ 0 codim 1 codim 2