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 ofx ˙ = Ax can be visualized as trajectories moving on the (x,y) plane, or phase plane. 1 Example 4.1.1 mx¨ + kx = 0 i.e. the simple 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, ω2x). − 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 ω2x2 + v2 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 at x x0e = t y ! y0e− ! 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) λeλtv = eλtAv ⇒ 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∆ λ = − λ = − − 1 q 2 2 q 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 λ , λ are purely imaginary, all solutions • 1 2 are periodic If λ = λ we get a star node or a degen- • 1 2 erate node 10 Classification of Fixed Points λ = 1(τ τ 2 4∆), where 1,2 2 ± − ∆= λ1λ2 andq τ = λ1 + λ2 Fig. 4.2.4 11 4.3 Phase Portraits Recallx ˙ = 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 x Fixed points A, B and C satisfy f( ∗) = 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 2 2 = f(x∗,y∗)+ u + v + O(u ,v ,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 ∂y v˙ ∂g ∂g v ! ∂x ∂y ! where the matrix is known as the Jacobian A matrix 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 wherex ˙ = 0 andy ˙ = 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˙ 2 ∂x ∂y 1 + 3x 0 A = = − ∂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 solvingx ˙ = 0 andy ˙ = 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) A In general, 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(λ ) > 0 and Re(λ ) > 0 • 1 2 repeller (unstable node) ⇒ If Re(λ ) < 0 and Re(λ ) < 0 • 1 2 attractor (stable node) ⇒ If Re(λ ) > 0 but Re(λ ) < 0 (or vice • 1 2 versa) saddle ⇒ If λ and λ are both imaginary centre • 1 2 ⇒ 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 Considerx ˙ = 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 mz¨ = dV (z)/dz = F (z) − Take x = z and y =z ˙ ⇒ x˙ = y 1 y˙ = F (x) m 1 2 E(z)= 2mz˙ +V (z) is the total energy, which is constant 1 E(x) my2 + V (x) ⇒ ≡ 2 dE(x) = 0 dt since total energy is constant. 23 Example 6.5.2 θ¨+ sin θ = 0 e.g.