Lecture 3 : Bifurcation Analysis D. Sumpter & S.C. Nicolis October - December 2008 D. Sumpter & S.C. Nicolis
General settings 4 basic bifurcations (as long as there is only one unstable mode!) steady state solutions (Imω c = 0) z pitchfork z transcritical z limit point periodic solutions(imω c 0) z Hopf More complex bifurcations often leading to chaos, when several unstable modes are interacting. D. Sumpter & S.C. Nicolis
3d order autocatalysis in an open well-stirred reactor 3d order autocatalysis in an open well-stirred reactor Bifurcation analysis dx dt dx dt = kax 2 k X 3 + 1 τ (X 0 X) (A + 2X k k 3X) = x 3 + λx 2 µx + µ (after scaling) Elimination of x 2 term through transformation z = x λ 3 or, dz dt = z + λ «3 + λ z + λ «2 µ z + λ «+ µ 3 3 3 dz λ 2 «2λ 3 dt = z3 + 3 µ z + 27 µλ «3 + µ (I) D. Sumpter & S.C. Nicolis
3d order autocatalysis in an open well-stirred reactor First consider case where constant term vanishes. Condition on µ and λ for this : µ = 2λ3 9 (λ 3) Eq. for z becomes Steady states : dz dt = z3 + λ3 9λ 2 9 (λ 3) z (II) (λ > 3, since µ > 0 for physical reasons) z = 0 (λ > 9) q z ± = λ λ 9 3 λ 3 (λ > 9) z = 0 (λ < 9) Notice that trivial state z = 0 becomes unstable beyond the bifurcation point λ c. The stability of bifurcating branches can be checked straightforwardly (supercritical bifurcation). This example is in fact paradigmatic : any system in the vicinity of a pitchfork bifurcation can be reduced to eq. (II) (normal form) where z is a combination of the variables (order parameter). All other variables follow z passively D. Sumpter & S.C. Nicolis
3d order autocatalysis in an open well-stirred reactor In the more general case where the constant term in (I) does not vanish, write equation as dz dt = z3 + uz + v According to the theory of cubic equations, we have the following situation for the steady states : v (1) (2) 1solution 3 solutions u D. Sumpter & S.C. Nicolis
3d order autocatalysis in an open well-stirred reactor Limit point bifurcations! z z v u u fixed, (1) v fixed, (2) D. Sumpter & S.C. Nicolis
For b > a 2 + 1 : amplified oscillations of the linearized system Nonlinearities saturate growth and lead to an attracting periodic solution represented by a closed curve in phase space (limit cycle). stable limit cycle Y unstable steady state xs = a, ys = b/a X D. Sumpter & S.C. Nicolis
Minimal nonlinear system illustrating onset of limit cycle : Poincare model dx dt dy dt = λx ω 0 y x `x 2 + y 2 = ω 0 x + λy y `x 2 + y 2 Keeping only linear terms yields eigenvalues of matrix J = λ «equal to ω0 ω = λ±iω 0 : ω 0 λ λ < 0 : damped oscillations ; λ > 0 : amplified oscillations ; λ = 0 : Hopf bifurcation point D. Sumpter & S.C. Nicolis
Nonlinear analysis : switch to polar coordinates x = rcosφ, y = rsinφ. Multiplying first eq. by x and second eq. by y and adding yields : dr = λr r3 (I) dt (formally similar to normal form of pitchfork bifurcation except that r is here positive) Multiplying first equation by y and second equation by x and subtracting yields : dφ dt = ω 0 (II) D. Sumpter & S.C. Nicolis
(I) and (II) are actually the normal form equations of a system undergoing a Hopf bifurcation. Solutions of (I) as t r = 0 for λ < 0 r = 0 for λ > 0 r = λ 1/2 for λ > 0 D. Sumpter & S.C. Nicolis
Solutions of (II) φ = φ 0 + ω 0 t In the phase plane, for λ > 0 for λ < 0 y y unstable steady state x=y=0 r stable fixed point (focus) x=y=0 x x stable limit cycle system runs on cycle at constant angular velocity D. Sumpter & S.C. Nicolis
Some global results related to limit cycles in two variable systems : 1. Bendixson s criterion dx dy = f (x, y) dt For a closed trajectory to exist, f x + g y must change sign in the (x, y) plane or vanish identically. 2. Bendixson s theorem The region bounded by a closed trajectory in (x, y) plane contains at least one fixed point (steady state solution) = g (x, y) dt 3. Poincare-Bendixson s theorem Any trajectory staying in a finite region of (x, y) phase space either approaches a fixed point or a periodic orbit. As a corollary, chaotic behavior in continuous time systems can only arise in the presence of at least three coupled variables. D. Sumpter & S.C. Nicolis