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TWO DIMENSIONAL FLOWS Lecture 4: Linear and Nonlinear Systems 4. Linear and Nonlinear Systems in 2D In higher dimensions, trajectories have more room to manoeuvre, and hence a wider range of behaviour is possible. 4.1 Linear systems: deﬁnitions and examples A 2-dimensional linear system has the form ˙ x = ax + by ˙ y = cx + dy where a, b, c, d are parameters. Equivalently, in vector notation ˙ x = Ax (1) where a b x A= and x= (2) c d y The Linear property means that if x1 and x2 are solutions, then so is c1x1 + c2x2 for any c1 and c2. The solutions of x = Ax can be visualized ˙ as trajectories moving on the (x, y) plane, or phase plane. 1 Example 4.1.1 m¨ + kx = 0 i.e. the simple x harmonic oscillator Fig. 4.1.1 The state of the system is characterized by ˙ x and v = x ˙ x = v k ˙ v = − x m ˙ ˙ i.e. for each (x, v) we obtain a vector (x, v) ⇒ vector ﬁeld on the phase plane. 2 As for a 1-dimensional system, we imagine a ﬂuid ﬂowing steadily on the phase plane with a local velocity given by (x, v) = (v, −ω 2x). ˙ ˙ Fig. 4.1.2 • Trajectory is found by placing an imag- inary particle or phase point at (x0, v0) and watching how it moves. • (x, v) = (0, 0) is a ﬁxed point: static equilibrium! • Trajectories form closed orbits around (0, 0): oscillations! 3 The phase portrait looks like... Fig. 4.1.3 • NB ω 2x2 + v 2 is constant on each ellipse. This is simply the energy Example 4.1.2 ˙ x a 0 x = ˙ y 0 −1 y 4 The phase portraits for these uncoupled equa- tions are... Fig. 4.1.4 Solution is x x0eat = y y0e−t 5 Some terminology... • x∗ = 0 is an attracting ﬁxed point in Figs (a) - (c) since x(t) → x∗ as t → ∞. • x∗ = 0 is called Lyapunov Stable in Figs (a) - (d) since all trajectories that start suﬃciently close to x∗ remain close to it for all time. • Fig. (d) shows that a ﬁxed point can be Lyapunov stable but not attracting ⇒ it is neutrally stable. It is also possible for a ﬁxed point to be attracting but not Lyapunov stable! • If a ﬁxed point is both Lyapunov sta- ble and attracting, we’ll call it stable, or sometimes asymptotically stable • x∗ is unstable in Fig. (e) because it is neither attracting nor Lyapunov stable 6 4.2 Classiﬁcation of Linear Systems Consider a general 2 ×2 matrix A such that ˙ x = Ax To solve: try x(t) = eλtv (v is a constant vector) ⇒ λeλt v = eλt Av ⇒ Av = λv Hence if we obtain the eigenvectors v and eigenvalues λ, we will have two independent a b solutions x(t). Recall that A = has c d eigenvalues λ1 and λ2, where τ+ τ 2 − 4∆ τ− τ 2 − 4∆ λ1 = λ2 = 2 2 with τ = trace(A) = a + d ∆= det(A) = ad − bc 7 • Useful check when calculating eigenval- ues: λ1 + λ2 = τ and λ1λ2 = ∆ ˙ x 1 1 x Example 4.2.1 = ˙ y 4 −2 y 1 • ⇒ λ1 = 2 with v1 = λ1 > 0 1 hence solution grows 1 • ⇒ λ2 = −3 with v2 = λ2 < 0 −4 hence solution decays Fig. 4.2.1 8 • straight line trajectories in Fig. 4.2.1 are the eigenvectors v1 and v2 Example 4.2.2 Consider λ2 < λ1 < 0 Fig. 4.2.2 • Both solutions decay exponentially! 9 Example 4.2.3 What happens if λ1, λ2 are complex? Fixed point is either... Fig. 4.2.3 • If λ1, λ2 are purely imaginary, all solutions are periodic • If λ1 = λ2 we get a star node or a degen- erate node 10 Classiﬁcation of Fixed Points λ1,2 = 1 (τ ± τ 2 − 4∆), 2 where ∆ = λ1λ2 and τ = λ1 + λ2 Fig. 4.2.4 11 4.3 Phase Portraits ˙ Recall x = f (x), i.e. x˙1 = f1(x1, x2) x˙2 = f2(x1, x2) where x = (x1, x2) and f (x) = (f1(x), f2(x)) (not necessarily linear now). The trajectories x(t) wind their way through the phase plane. The entire phase plane is ﬁlled with trajec- tories! 4.4 Example of a phase portrait - Shows a sample of the qualitatively diﬀerent trajectories 12 Fig. 4.4.1 • Fixed points A, B and C satisfy f (x∗ ) = 0 and correspond to steady states or equi- libria • Closed orbit D corresponds to periodic solutions, i.e. x(t + T ) = x(t) for all t for some T > 0 • The existence and uniqueness theorem given for 1-dimensional systems can be generalized to 2-dimensional systems ... fortunately ⇒ diﬀerent trajectories never intersect! 13 4.5 Fixed points and Linearization This is the same idea as for 1-dimensional systems ˙ x = f (x, y) ˙ y = g(x, y) Suppose (x∗, y ∗) is a ﬁxed point. Expand around (x∗, y ∗) using u = x−x∗ and v = y−y ∗. u = x = f (x∗ + u, y ∗ + v) ˙ ˙ ∂f ∂f ∗ ∗ = f (x , y ) + u +v + O(u2, v 2, uv) ∂x ∂y ∂f ∂f ≃ u +v ∂x ∂y Similarly ∂g ∂g ˙ v≃u +v ∂x ∂y Hence a small disturbance around (x∗, y ∗) evolves as ∂f ∂f ˙ u u = ∂x ∂g ∂y ∂g ˙ v v ∂x ∂y where the matrix is known as the Jacobian matrix A at (x∗, y ∗), and is the multivariable equivalent of f ′ (x∗) for 1-D systems. 14 Example 4.5.1 x = −x + x3 ˙ ˙ y = −2y ˙ Fixed points occur where x = 0 and y = ˙ 0 simultaneously. Hence x = 0 or x = ±1 and y = 0 ⇒ 3 ﬁxed points (0, 0), (1, 0) and (−1, 0) Jacobian matrix A ˙ ∂x ˙ ∂x −1 + 3x2 0 A = ∂x ˙ ∂y ∂y ˙ ∂y = 0 −2 ∂x ∂y −1 0 At (0, 0) A= ⇒ stable node 0 −2 2 0 At (±1, 0) A= ⇒ both are sad- 0 −2 dle points. 15 Fig. 4.5.1 In general, we must obtain ﬁxed points by ˙ ˙ solving x = 0 and y = 0 simultaneously. e.g. ˙ x = x(3 − x − 2y) ˙ y = y(2 − x − y) yields ﬁxed points (0, 0), (0, 2), (3, 0) and (1, 1) In general, A will not be diagonal at (x∗, y ∗). Hence we must diagonalize A, i.e. ﬁnd eigen- values λ1 and λ2 and eigenvectors v1 and v2 of A 16 Basically, we are doing the same here as be- fore for 2D linear systems, since we are treat- ing the nonlinear system as linear near (x∗, y ∗). Knowledge of λ1 and λ2, and v1 and v2, en- ables us to sketch the phase portrait near (x∗, y ∗). The ﬁxed points can be classiﬁed according to their stability as follows: • If Re(λ1) > 0 and Re(λ2) > 0 ⇒ repeller (unstable node) • If Re(λ1) < 0 and Re(λ2) < 0 ⇒ attractor (stable node) • If Re(λ1) > 0 but Re(λ2) < 0 (or vice versa) ⇒ saddle • If λ1 and λ2 are both imaginary ⇒ centre 17 4.6 Example: Rabbits vs Sheep An example of the Lotka-Volterra model of competition between two species (e.g. rab- bits and sheep) grazing the same food supply (grass). • Each species grows to its carrying capac- ity in the absence of the other - logistic growth (rabbits faster...!) • When species encounter each other, the larger (sheep) has an advantage. • Conﬂicts occur at a rate proportional to the size of each population. Conﬂicts re- duce the growth rate of each species (but more for rabbits). A model encapsulating these properties could be (see above!) ˙ x = x(3 − x − 2y) ˙ y = y(2 − x − y) 18 Fixed points at 3 0 (0, 0) where A = ⇒ λ = 3, 2 0 2 −1 0 (0, 2) where A = ⇒ λ = −1, −2 −2 −2 −1 −2 √ (1, 1) where A = ⇒ λ = −1 ± 2 −1 −1 −3 −6 (3, 0) where A = ⇒ λ = −3, −1 0 −1 (0, 0): λ = 3, 2 ⇒ unstable node (repeller) λ = 2 ⇒ v = (0, 1) “slow eigendirection′′ λ = 3 ⇒ v = (1, 0) “fast eigendirection′′ General rule... Trajectories are tangential to the slow eigendirection (i.e. smallest |λ|) at a node 19 (0, 2): λ = −1, −2 ⇒ stable node (attrac- tor) • Once again... Trajectories are tangential to the slow eigendirection at a node • Here λ = −1 ⇒ v = (1, −2) is the slow eigendirection. √ (1, 1): λ = −1 ± 2 ⇒ saddle point 20 (3, 0): λ = −3, −1 ⇒ stable node (attrac- tor) Putting these together, the phase portrait becomes.... Fig. 4.6.1 NB: You don’t really need to calculate the eigenvectors to get the right shape! 21 Biological interpretation... • In general, one species eventually drives the other to extinction; which species even- tually dominates depends on initial pop- ulations x0 = (x0, y0) • Basin of attraction of an attracting ﬁxed point x∗ deﬁned as the set of initial con- ditions x0 such that x → x∗ as t → ∞. • In this case, basin boundary is the stable manifold of the saddle node at (1, 1) Fig. 4.6.2 22 4.7 Conservative Systems ˙ Consider x = f (x). A conserved quantity of this system is a real-valued continuous func- tion E(x) that is constant on trajectories i.e. dE/dt = 0. Example 4.7.1 m¨ = −dV (z)/dz = F (z) z ˙ Take x = z and y = z ⇒ ˙ x = y 1 ˙ y = F (x) m 1 ˙ E(z) = 2 mz 2 +V (z) is the total energy, which is constant 1 ⇒ E(x) ≡ my 2 + V (x) 2 dE(x) = 0 dt since total energy is constant. 23 ¨ Example 6.5.2 θ + sin θ = 0 e.g. undamped simple pendulum ˙ θ = ν ˙ ν = − sin θ Fixed points at (θ ∗, ν ∗) = (kπ, 0) 0 1 (0, 0) : A= ⇒ λ = ±i ⇒ centre −1 0 (oscillations) 1 Energy E(θ, ν) = 2 ν 2 − cos θ is conserved, since dE ˙ ˙ ¨ = ν ν + sin θ θ = ν[θ + sin θ] = 0 dt 0 1 (π, 0) : A= ⇒ λ = ±1 ⇒ saddle 1 0 24 Phase portrait becomes... Fig. 4.7.1 25

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two-dimensional flows, two dimensional, DIMENSIONAL FLOWS, stream function, circular cylinder, fixed points, Reynolds number, the wake, the wall, phase portrait

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