Project 6.2 Phase Plane Portraits of Almost Linear Systems

Transcription

1 Project 6.2 Phase Plane Portraits of Almost Linear Sstems Interesting and complicated phase portraits often result from simple nonlinear perturbations of linear sstems. For instance, the figure below shows a phase plane portrait for the almost linear sstem d d = cos( + 1) = cos( + 1). (1) Among the seven critical points marked with dots, we see Apparent spiral points in the first and third quadrants of the -plane; Apparent saddle points in the second and fourth quadrants, plus another one on the positive -ais; A critical point of undetermined character on the negative -ais; and An apparentl "ver weak" spiral point at the origin -- meaning one that is approached ver slowl as t increases or decreases (according as it is a sink or a source). ' = - cos( + - 1) ' = cos( - + 1) Project

2 Some ODE software sstems can automaticall locate and classif critical points. For instance, Fig in the tet shows a screen produced b John Polking's MATLAB pplane program (cited in Project 6.1). It indicates that the fourth-quadrant critical point in the figure above has approimate coordinates (1.5708, ), and that the coefficient matri of the associated linear sstem has the positive eigenvalue λ and the negative eigenvalue λ It therefore follows from Theorem 2 in Section 6.2 that this critical point is, indeed, a saddle point of the almost linear sstem in (1). With a general computer algebra sstem, ou ma have to do a bit of work ourself or tell the computer precisel what to do in order to find and classif a critical point. In the sections below, we illustrate this procedure using Maple, Mathematica, and MATLAB. Once the critical-point coordinates a = , b = indicated above have been found, the substitution = u+ a, = v+ b ields the translated sstem du dv = ( v)cos( u v) = f( u, v) = ( u)cos( u v) = g( u, v). (2) If we substitute u = v = 0 in the Jacobian matri J = we get the coefficient matri A =! f u g u! f v g v " $ # of the linear sstem corresponding to the almost linear sstem in (2). " $# (3) (4) Alternativel, one can circumvent the translated sstem in (2) b looking at the Talor epansions f(, ) = D f( a, b)( a) + D f( a, b)( b) + g (, ) = Dgab (, )( a) + Dgab (, )( b) + (5) of the right-hand side functions in the original sstem (1), and retaining onl the linear terms in this epansion. We see from (5) that 172 Chapter 6

3 A =! Dfab (, ) Dfab (, ) Dgab (, ) Dgab (, ) " $# (6) is the coefficient matri of the linearization of the sstem (1) that results when we substitute u = a, v = b and retain onl the terms that are linear in u and v. In an event, we can then use our computer algebra sstem to find the eigenvalues λ and λ of the matri A, thereb verifing that the critical point (1.5708, ) of (1) is, indeed, a saddle point. Use a computer algebra sstem to find and classif similarl the other critical points of (1) indicated in the figure above. Then investigate similarl an almost linear sstem of our own construction. One convenient wa to construct such a sstem is to start with a linear or almost linear sstem and insert sine or cosine factors resembling the ones in (1). For instance: 1. = cos, = sin = + cos, = sin 3. = cos( 2 + ), = sin( 3) 4. = 2 cos( + ), = + 2 cos( ) Using Maple After we enter the right-hand side functions in (1), f := -*cos(+-1): g := *cos(--1): we can proceed to solve numericall for a solution near (1.5, 2): soln := fsolve({f=0,g=0}, {,}, =1..2, =-3..-1); { = , = } Thus our critical point (a, b) is given approimatel b a := rhs(soln[1]); b := rhs(soln[2]); a := b := To classif this critical point, we proceed to calculate first the partial derivatives Project

4 f := evalf(subs(=a,=b,diff(f,))): f := evalf(subs(=a,=b,diff(f,))): g := evalf(subs(=a,=b,diff(g,))): g := evalf(subs(=a,=b,diff(g,))): evaluated at (a, b), and then the Jacobian matri in (6): with(linalg): A := matri(2,2, [f,f,g,g]); Finall, its eigenvalues are given b eigenvals(a); , Thus the eigenvalues λ and λ are real with opposite signs, so the critical point (1.5708, ) is, indeed, a saddle point of the sstem in (1). Using Mathematica After we enter the right-hand side functions in (1), f = -*Cos[+-1]; g = *Cos[-+1]; we can proceed to solve numericall for a solution near (1.5, 2): soln = FindRoot[{f == 0, g == 0}, {,1.5}, {,-2}] { -> , -> } Thus our critical point (a, b) is given approimatel b a = /. soln b = /. soln To classif this critical point, we proceed to set up the Jacobian matri in (6), A = { {D[f,], D[f,]}, {D[g,], D[g,]}} /. {->a, ->b}; evaluated at (a, b), and calculate its eigenvalues 174 Chapter 6

5 Eigenvalues[A] { , } Thus the eigenvalues λ and λ are real with opposite signs, so the critical point (1.5708, ) is, indeed, a saddle point of the sstem in (1). Using MATLAB We want to solve the equations f(, ) = 0, g(, ) = 0 where f and g are the right-hand side functions in our sstem (1). However, the student edition of MATLAB does not include a function for the solution of sstems of equations. Our strateg is therefore to use the MATLAB function fminsearch to minimize the function h (, ) = f(, ) + g (, ) = cos( + 1) + cos( + 1) This function is defined as a function of the vector v = [;] b the m-file h.m consisting of the lines function z = h(v) = v(1); = v(2); z = (-*cos(+-1))^2 + (*cos(-+1))^2; Evidentl a minimal point where h(, )= 0 will be a critical point of the sstem in (1). Hence the commands soln = fminsearch('h',[1.5;-2]) soln = a = soln(1); b = soln(2); ield the approimate critical point (1.5708, ). To classif this critical point, we proceed to set up the Jacobian matri in (6). First we define the inline functions f = '-*cos(+-1)'; g = '*cos(-+1)'; calculate their partial derivatives f = diff(f,''); g = diff(g,''); f = diff(f,''); g = diff(g,''); and evaluate these partial derivatives at the point (a, b): Project

6 sms f = subs(f, {,}, {a,b}); f = subs(f, {,}, {a,b}); g = subs(g, {,}, {a,b}); g = subs(g, {,}, {a,b}); Then the eigenvalues of the Jacobian matri are given b A = [f f g g]; eig(a) ans = Thus the eigenvalues λ and λ are real with opposite signs, so the critical point (1.5708, ) is, indeed, a saddle point of the sstem in (1). 176 Chapter 6