Review Sheet on Ordinary Differential Equations | MAT 119B, Study notes of Differential Equations

Download Review Sheet on Ordinary Differential Equations | MAT 119B and more Study notes Differential Equations in PDF only on Docsity! Review on ODE Hsiao-Chieh (Arcade) Tseng Grad Group in Applied Mathematics, Department of Mathematics, University of California, Davis. E-mail: hctseng@math.ucdavis.edu Website: http://www.math.ucdavis.edu/∼hctseng/ September 17, 2008 Ch2: 1D Flows • Linear stability analysis: ẋ = f(x). Step1 Solve f(x∗) = 0 and obtain equilibrium x∗. Step2 Plot phase plane (x-ẋ plot). Step3 Determine the stability of x∗. • Existence, finite time blow up. • Argument of potential function ẋ = f(x) = −∇V . • Impossibility of oscillation in 1D • Euler method Ch3: 1D Bifurcation • Bifurcation diagram (r-x plot). • Saddle node bifurcation. Ex. ẋ = r ± x2. • Transcritical bifurcation. Ex. ẋ = rx− x2. • Pitchfork: – Supercritical. Ex. ẋ = rx− x3. – Subcritical. Ex. ẋ = rx + x3 − x5. • Nondimensionalization. • Imperfect bifurcation: two or more parameters. 1 Ch5: Linear 2D Systems • Linear stability analysis: ẋ = Ax, x = [ x y ] . Step1 Solve the eigenvalues λ1, λ2, and corresponding eigenvectors of A. The character- istic equation yields λ2 − τλ + ∆ = 0. Step2 Classification of fixed points, (∆-τ diagram, ∆ = λ1λ2, τ = λ1 + λ2. ). Step3 Plot the phase portrait (x-y plot). • Stable and unstable manifold: The stable manifold of a saddle point x∗, defined as the set of initial condition vecx0 such tat x(t) → x∗ as t → ∞. Likewise, the unstable manifold of x∗ is the set of initial conditions such that x(t) → x∗ as t → −∞. • Attracting: x∗ is attracting if there is a δ > 0 such that limt→∞ x(t) = x∗ whenever ‖x(0)− x∗‖ < δ. • Globally attracting: x∗ is called globally attracting if it attracts all trajectories in the phase plane. • Liapunov stable: x∗ is Liapunov stable for each  > 0 there is a δ > 0 such that ‖x(t)− x∗‖ <  whenever t ≥ 0 and ‖x(0)− x∗‖ < δ. • Asymptotic stable: x∗ is asymptotic stable if it is both attracting and Liapunov stable. Ch6: Linearization Near Fixed Points • Sketch phase portrait. – Nullclines: the curve where either ẋ = 0 or ẏ = 0. – Different trajectories never intersect. • Linearization ẋ = f(x): – Find the Jacobian matrix A of f at each equilibrium x∗. – Analize the linearized system ẋ = Ax and observe the stability at x∗. – If the linearized system predicts a saddle, node, or a spiral, then the fixed point really is a saddle, node, or spiral for the original nonlinear system ( i.e., except the centers ). • Polar coordinates: x = r cos θ, y = r sin θ, xẋ + yẏ = rṙ , θ̇ = xẏ − yẋ r2 . • Stability of fixed points: 2 Then the system has a unique stable limit cycle surrounding the origin in the phase plane. (exisence of closed orbits) • Simple harmonic oscillation: – Equation: ẍ + x = 0, x(0) = a, ẋ(0) = b. – Rewrite the equation, we have ẋ = y, x(0) = a ẏ = −x, y(0) = b, and thus ṙ = 0, r(0) = √ a2 + b2 θ̇ = −1, θ(0) = tan−1 ( b a ) . Anyway, the general solution is x(t) = a cos(t) + b sin(t), or x = r cos(θ) on the phase plane. • Relaxation oscillation: – Equation: ẍ + µf(x)ẋ + x = 0, µ  1. – Solution steps: Step1 Let ẇ = ẍ + µf(x)ẋ = −x. Step2 Solve for w, and we get w = ẋ + µF (x) where d dt F (x) = f(x)ẋ. Step3 Write the system ẋ = w − µF (x) = µ ( w µ − F (x) ) ẇ = −x, and let y = w µ . Step4 Now we have ẋ = µ(y − F (x)) ẏ = −1 µ x, Step5 Plot the nullcline y = F (x) on the Lienard plane (x-y plot). Assume y − F (x) ≈ O(1), then |ẋ| ≈ O(µ)  1 faster, whereas |ẏ| ≈ O( 1 µ )  1 slower. 5 • Weakly nonlinear oscillators: – Equation: ẍ + x + h(x, ẋ) = 0, 0 ≤   1. – Solution steps (regular perturbation): Step1 Write x(t; ) = x0(t) + x1(t) +  2x2(t) + . . .. Step2 The method poorly works if it yields a secular term, i.e., a term that grows without bounded as t →∞. – Solution steps (two timing): Step1 Let τ = t denote the fast O(1) time, and T = t the slow time. Treat these two times as if they were independent variables. Step2 Write x(t; ) = x0(τ, T )+ x1(τ, T )+  2x2(τ, T )+ . . .. In the following we just deal with O(1) and O() terms. Step3 By chain rule (ẋ = dx dt ) we have ẋ = ∂τx + ∂T x = ∂τx0 + (∂T x0 + ∂τx1) +O(2) ẍ = ∂ττx0 + (∂ττx1 + 2∂Tτx0) +O(2) Step4 Substitute the above to the original equation, collect O(1) and O() terms. We have O(1) : ∂ττx0 + x0 = 0 O() : ∂ττx1 + x1 = −2∂Tτx0 − h(x0, ∂τx0). Observe that the O(1) equation is the simple harmonic oscillation and thus x0 = r cos(τ + φ), ∂τx0 = −r sin(τ + φ) where r = r(T ), φ = φ(T ) are the slowly-varying amplitude and phase of x0. Step5 Derive differential equation for r and φ. Substitute the above x0 to the O() equation, it’s RHS yields (the prime denotes d dT ) 2r′ sin(τ + φ) + 2rφ′ cos(τ + φ)− h(r cos(τ + φ),−r sin(τ + φ)). Step6 Let θ = τ + φ. Observe that h is a 2π-periodic function and thus h(θ) = a0 + ∞∑ k=1 ak cos kθ + bk sin kθ where k ≥ 1, a0 = 1 2π ∫ 2π 0 h(θ) dθ, ak = 1 π ∫ 2π 0 h(θ) cos kθ dθ, bk = 1 π ∫ 2π 0 h(θ) sin kθ dθ. Step7 To avoid secular term happen in the RHS we need r′ = b1/2, rφ ′ = a1/2. Thus, we have the averaged equation (the prime denotes d dT ) r′ = 1 2π ∫ 2π 0 h(θ) sin θ dθ, r(0) ≈ √ x(0)2 + ẋ(0)2 rφ′ = 1 2π ∫ 2π 0 h(θ) cos θ dθ, φ(0) ≈ tan−1( ẋ(0) x(0) ). 6 Ch8: 2D Bifurcation • Saddle node bifurcation. Ex: ẋ = µ− x2, ẏ = −y. • Transcritical bifurcation. Ex: ẋ = µx− x2, ẏ = −y. • Supercritical pitchfork. Ex: ẋ = µx− x3, ẏ = −y. • Subcritical pitchfork. Ex: ẋ = µx + x3, ẏ = −y. • Hopf bifurcation. – Supercritical Hopf bifurcation. Ex: ṙ = µr − r3 θ̇ = ω + br2 for constant frequency ω and dependence of frequency on amplitude b. Here µc = 0. – Subcritical Hopf bifurcation. Ex: ṙ = µr + r3 − r5 θ̇ = ω + br2 for constant ω and b. Here µc = 0. – Degenerate Hopf bifurcation. Ex: ẍ + µẋ + sin x = 0 (no limit cycles, recall that a limit cycle is an isolated closed orbit.) • Global bifurcation of cycles – Saddle-node bifurcation of cycles. (ref: subcritical Hopf bifurcation). – Infinite-period bifurcation. Ex: ṙ = r(1− r2) θ̇ = µ− sin θ – Homoclinic bifurcation. • Poincarè maps (existence of closed orbits): – Definition: Consider an n-dimensional system ẋ = f(x). Let S be an n − 1 dimensional surface of section. S is required to be transverse to the flow, i.e., all trajectories starting on S flow through it, not parallel to it. The Poincarè map P is a mapping from S to itself, obtained by following trajectories from one intersection with S to the next. Let xk ∈ S denotes the kth intersection, then the Poincarè map is defined by xk+1 = P (xk). 7