F and G have continuous second-order derivatives. Assume Equation (1.1) possesses an equilibrium point (x*,y*) so that

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1 Characterizing solutions (continued) C4. Stability analysis for nonlinear dynamical systems In many economic problems, the equations describing the evolution of a system are non-linear. The behaviour of the nonlinear system around an equilibrium point (if it eists) can be approimated by that of a linear system. Consider the nonlinear system: = F(, y) y = Gy (, ) F and G have continuous second-order derivatives. Assume Equation (1.1) possesses an equilibrium point (*,y*) so that F( *, y*) = G ( *, y*) = 0 For and y close to the equilibrium, we can linearize the nonlinear system around the steady state. (1.1) F(, y) = F( *, y*) + F ( *, y*)( *) + F ( y y*) + H. OT. Gy (, ) = G ( *, y*) + G( *, y*)( *) + G( y y*) + HOT.. But since F( *, y*) = G ( *, y*) = 0, and if we denote F, Fy, G, G y evaluated at (*,y*) by a 11, a 1, a 1, and a respectively, then the linearized system can be approimated by: y y (1.) a11 a1 * y = a1 a y y* (1.3) Devine z 1 as the deviation -* and z as the deviation y-y*, then we can rewrite Equation as z a a z = z a1 a z (1.4) Ec 655, 3.1.6, Characterizing the solution to optimal control problems 19

2 which is eactly the format of the linear dynamical system analyzed in the previous section. Theorem: Assume a11a a1a1 0. The qualitative behaviour of the trajectories of the nonlinear system (Equation (1.1)) in the neighbourhood of the equilibrium point (*,y*) is that same as that of the trajectories of the corresponding linear system (Equation (1.4)) around the origin, with the eception that if the origin is a center, then (*y*) may be either a center or a focus. (See Léonard and Van Long, page 101) C.5. A nonlinear fishery problem Consider a firm that owns a fishery and can catch fish at zero cost. The firm has a monopoly on fish sales and sells into a market at price p. The growth of the fish biomass follows a continuous time logistic growth function. Market demand is linear. There is no constraint on the fish catch in each period. The monopolist s problem is: rt ma h Rhe ( ) dt 0 (1.5) subject to = g(, t) h( t) (0) = 0 h is the harvest rate at time t, is the stock of biomass, R( ) is the revenue function, and g( ) is a logistic growth function of the form: g(, h, t) = a + b, a > 0; b < 0. (1.6) The revenue function can be written as: Rh ( ) = f( hh ), where h is the inverse demand function. (1.7) Ec 655, 3.1.6, Characterizing the solution to optimal control problems 0

3 The Hamiltonian is: First order conditions are: Ec 655, 3.1.6, Characterizing the solution to optimal control problems 1

4 We end up with two autonomous differential equations: R ( h) h= ( r g) R ( h ) = g(, t) h= a+ b h (1.8) (1.9) Equations (1.8) and (1.9) can be used to find the optimal harvest rate given ( 0,h 0 ), determine a steady state solution, and assess the stability of the steady state. Find the isoclines 1. h = 0 when r = g. g = a+ b. r a a r Hence h= 0 when = = + b b b.. Sketch the isoclines: = a+ b h= 0 0 h= a+ b dh d h = a+ b, = b< 0 d d a At the maimum: = b (1.10) Ec 655, 3.1.6, Characterizing the solution to optimal control problems

5 h(t) (t) Determine stability of the dynamical system Write the Jacobian matri of partial derivatives: where h = F( h, ) = G( h, ). A a a F F 11 1 h = a1 a Gh G (1.11) Stability is indicated by the eigenvalues, λ, of the matri A at the steady state solution. F G h h λ F = 0 G λ (1.1) Ec 655, 3.1.6, Characterizing the solution to optimal control problems 3

6 Numerical Eample Let r=0.1. Solve for * and h*: = G h = h (1.13) (, ) f( h) = p= 1 h (inverse demand function) Rh ( ) = (1 hh ) (1.14) Find the values of the Jacobian matri of A and the eigenvalues. What do you conclude about the steady state solution? Sketch the phase diagram and determine the direction of motion in each isosector. Ec 655, 3.1.6, Characterizing the solution to optimal control problems 4

7 Sectors I and III are convergent isosectors. (*,h*) is a saddle point. Isosectors I and III each contain a trajectory which convergest to (*, h*) and is referred to as a separtri. The two separatrices define the optimal solution trajectories for our infinite horizon problem. If we could calculate the separatri cuves we would have an eplicit optimal harvest policy, h*=h*(). This is a closed loop control policy h* depends only on the current value of the state variable. Separatrices can be calculated numerically. Ec 655, 3.1.6, Characterizing the solution to optimal control problems 5