NONLINEAR DYNAMICAL SYSTEMS Math 21b, O. KnillSUMMARY. For linear systems ˙x = Ax, the eigenvalues of A determine the behavior completely. For nonlinearsystems explicit formulas for solutions are no more available in general. It even also happen that orbits go gooff to infinity in finite time like in ˙x = x2with solution x(t) = −1/(t −x(0)). With x(0) = 1 it reaches infinityat time t = 1. Linearity is often too crude. The exponential growth ˙x = ax of a bacteria colony for example isslowed down due to lack of food and the logistic model ˙x = ax(1 − x/M) would be more accurate, where Mis the population size for which bacteria starve so much that the growth has stopped: x(t) = M, then ˙x(t) = 0.Nonlinear systems can be investigated with qualitative methods. In 2 dimensions ˙x = f (x, y), ˙y = g(x, y),where chaos does not happen, the analysis of equilibrium points and linear approximation at thosepoints in general allows to understand the system. Also in higher dimensions, where ODE’s can have chaoticsolutions, the analysis of equilibrium points and linear approximation at those points is a place, where linearalgebra becomes useful.EQUILIBRIUM POINTS. A point x0is called an equilibrium point of ˙x = f(x) if f(x0) = 0. If x(0) = x0then x(t) = x0for all times. The system ˙x = x(6−2x−y), ˙y = y(4−x−y) for example has the four equilibriumpoints (0, 0), (3, 0), (0, 4), (2, 2).JACOBIAN MATRIX. If x0is an equilibrium point for ˙x = f (x) then [A]ij=∂∂xjfi(x) is called the Jacobianat x0. For two dimensional systems˙x = f(x, y)˙y = g(x, y)this is the 2 ×2 matrixA="∂f∂x(x, y)∂f∂y(x, y)∂g∂x(x, y)∂g∂y(x, y)#.The linear ODE ˙y = Ay with y = x − x0approximates the nonlinear system well near the equilibrium point.The Jacobian is the linear approximation of F = (f, g) near x0.VECTOR FIELD. In two dimensions, we can draw the vector field by hand: attaching a vector (f(x, y), g(x, y))at each point (x, y). To find the equilibrium points, it helps to draw the nullclines {f(x, y) = 0}, {g(x, y) = 0}.The equilibrium points are located on intersections of nullclines. The eigenvalues of the Jacobeans at equilibriumpoints allow to draw the vector field near equilibrium points. This information is sometimes enough to drawthe vector field by hand.MURRAY SYSTEM (see handout) ˙x = x(6 −2x −y), ˙y = y(4 −x −y) has the nullclines x = 0, y = 0, 2x + y =6, x + y = 5. There are 4 equilibrium points (0, 0), (3, 0), (0, 4), (2, 2). The Jacobian matrix of the system atthe point (x0, y0) is6 − 4x0− y0−x0−y04 − x0− 2y0. Note that without interaction, the two systems would belogistic systems ˙x = x(6 − 2x), ˙y = y(4 − y). The additional −xy is the competition.Equilibrium Jacobean Eigenvalues Nature of equilibrium(0,0)6 00 4λ1= 6, λ2= 4 Unstable source(3,0)−6 −30 1λ1= −6, λ2= 1 Hyperbolic saddle(0,4)2 0−4 −4λ1= 2, λ2= −4 Hyperbolic saddle(2,2)−4 −2−2 −2λi= −3 ±√5 Stable sinkUSING TECHNOLOGY (Example: Mathematica). Plot the vector field:Needs["Graphics‘PlotField‘"]f[x_,y_]:={x(6-2x-y),y(5-x-y)};PlotVectorField[f[x,y],{x,0,4},{y,0,4}]Find the equilibrium solutions:Solve[{x(6-2x-y)==0,y(5-x-y)==0},{x,y}]Find the Jacobian and its eigenvalues at (2, 2):A[{x_,y_}]:={{6-4x,-x},{-y,5-x-2y}};Eigenvalues[A[{2,2}]]Plotting an orbit:S[u_,v_]:=NDSolve[{x’[t]==x[t](6-2x[t]-y[t]),y’[t]==y[t](5-x[t]-y[t]),x[0]==u,y[0]==v},{x,y},{t,0,1}]ParametricPlot[Evaluate[{x[t],y[t]}/.S[0.3,0.5]],{t,0,1},AspectRatio->1,AxesLabel->{"x[t]","y[t]"}]VOLTERRA-LODKA SYSTEMSare systems of the form˙x = 0.4x − 0.4xy˙y = −0.1y + 0.2xyThis example has equilibriumpoints (0, 0) and (1/2, 1).It describes for example a tunashark population. The tuna pop-ulation x(t) becomes smaller withmore sharks. The shark populationgrows with more tuna. Volterra ex-plained so first the oscillation of fishpopulations in the Mediterrian sea.EXAMPLE: HAMILTONIAN SYS-TEMS are systems of the form˙x = ∂yH(x, y)˙y = −∂xH(x, y)where H is called the energy. Usu-ally, x is the position and y the mo-mentum.THE PENDULUM: H(x, y) = y2/2 −cos(x).˙x = y˙y = −sin(x)x is the angle between the pendulumand y-axes, y is the angular velocity,sin(x) is the potential.(See homework). Hamiltonian systems preserve energy H(x, y) becauseddtH(x(t), y(t)) = ∂xH(x, y) ˙x +∂yH(x, y) ˙y = ∂xH(x, y)∂yH(x, y) −∂yH(x, y)∂xH(x, y) = 0. Orbits stay on level curves of H.EXAMPLE: LIENHARD SYSTEMSare differential equations of the form¨x + ˙xF0(x) + G0(x) = 0. With y =˙x + F (x), G0(x) = g(x), this gives˙x = y − F (x)˙y = −g(x)VAN DER POL EQUATION ¨x+(x2−1) ˙x + x = 0 appears in electricalengineering, biology or biochemistry.Since F (x) = x3/3 − x, g(x) = x.˙x = y − (x3/3 − x)˙y = −xLienhard systems have limit cycles. A trajectory always ends up on that limit cycle. This is useful forengineers, who need oscillators which are stable under changes of parameters. One knows: if g(x) > 0 for x > 0and F has exactly three zeros 0, a, −a, F0(0) < 0 and F0(x) ≥ 0 for x > a and F (x) → ∞ for x → ∞, then thecorresponding Lienhard system has exactly one stable limit cycle.CHAOS can occur for systems ˙x = f(x) in three dimensions. For example, ¨x = f(x, t) can be written with(x, y, z) = (x, ˙x, t) as ( ˙x, ˙y, ˙z) = (y, f (x, z), 1). The system ¨x = f(x, ˙x) becomes in the coordinates (x, ˙x) theODE ˙x = f(x) in four dimensions. The term chaos has no uniform definition, but usually means that one canfind a copy of a random number generator embedded inside the system. Chaos theory is more than 100 yearsold. Basic insight had been obtained by Poincar´e. During the last 30 years, the subject exploded to its ownbranch of physics, partly due to the availability of computers.ROESSLER SYSTEM˙x = −(y + z)˙y = x + y/5˙z = 1/5 + xz − 5.7zLORENTZ SYSTEM˙x = 10(y − x)˙y = −xz + 28x − y˙z = xy −8z3These two systems are examples, where one can observe strange attractors.THE DUFFING SYSTEM¨x +˙x10− x + x3− 12 cos(t) = 0˙x = y˙y = −y/10 − x + x3− 12 cos(z)˙z = 1The Duffing system models a metal-lic plate between magnets. Otherchaotic examples can be obtainedfrom mechanics like the drivenpendulum ¨x + sin(x) −cos(t) =
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