Local Phase Portrait of Nonlinear Systems Near Equilibria

Transcription

1 Local Phase Portrait of Nonlinear Sstems Near Equilibria [1] Consider 1 = , = 3 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating sstem near the equilibrium; sketch the phase portrait of the linear approimating sstem; sketch the local phase portrait of the original nonlinear sstem ( ) near the equilibrium; determine whether the equilibrium is stable or unstable with respect to the nonlinear sstem ( ). [] Consider 1 = , = + 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating sstem near the equilibrium; sketch the phase portrait of the linear approimating sstem; sketch the local phase portrait of the original nonlinear sstem ( ) near the equilibrium; determine whether the equilibrium is stable or unstable with respect to the nonlinear sstem ( ). [3] Consider 1 = 1, = 1 1. ( ) (a) Find all equilibrium solutions of the sstem ( ). (b) For each equilibrium point, give the linear approimating sstem near the equilibrium; sketch the phase portrait of the linear approimating sstem; sketch the local phase portrait of the original nonlinear sstem ( ) near the equilibrium; determine whether the equilibrium is stable or unstable with respect to the nonlinear sstem ( ). 1 Turn over for the answers

2 Answers: [1] (a) (, ), (15, ), (, 1), (6, 1). (b) Near (, ): the linear approimating sstem is 1 = 6 1, =. (, ) is a repulsive node. Linear Approimation Near (,) Nonlinear Flow Near (,) Near (15, ): the linear approimating sstem is 1 = 6( 1 15) 5, = 3. (15, ) is an attractive node. Linear Approimation Near (15,) Nonlinear Flow Near (15,) Near (, 1): the linear approimating sstem is 1 = 3 1, = 63 1 ( 1). (, 1) is an attractive node. Linear Approimation Near (,1) Nonlinear Flow Near (,1)

3 Near (6, 1): the linear approimating sstem is 1 = ( 1 6) 18( 1), = 36( 1 6) ( 1). (6, 1) is a saddle. Linear Approimation Near (6,1) Nonlinear Flow Near (6,1) For our reference, the global phase portrait of the nonlinear sstem is included below: Global Phase Portrait 3

4 [] (a) (, ), (5, ), (, 3). (b) Near (, ): the linear approimating sstem is 1 = 5 1, =. (, ) is a saddle. Linear Approimation Near (,) Nonlinear Flow Near (,) Near (5, ): the linear approimating sstem is 1 = 5( 1 5) 5, = 3. (5, ) is a saddle. Linear Approimation Near (5,) Nonlinear Flow Near (5,) Near (, 3): the linear approimating sstem is 1 = ( 1 ) ( 3), = 3( 1 ). (, 3) is an attractive spiral focus. Linear Approimation Near (,3) Nonlinear Flow Near (,3)

5 For our reference, the global phase portrait of the nonlinear sstem is included below: Global Phase Portrait

6 [3] (a) (, ). (b) Near (, ): the linear approimating sstem is 1 = 1, = 1. [ ] The Jacobian matri J(, ) = 1 1 has eigenvalues λ 1 = and λ = 3. This allows us to sketch the phase portrait for the linear approimating sstem, which is shown on the right. In particular, (, ) is a stable (but not asmptoticall stable) equilibrium, with repsect to the linear approimating sstem. This, however, does not automaticall give the picture for the original nonlinear sstem. Because one of the eigenvalues of J(, ) is λ 1 =, the local dnamics of the nonliear sstem near (, ) ma be significantl different from that of the linear sstem. The linear approimation alone is not enough to determine the nonlinear dnamics. In order to determine the local dnamics of the nonlinear sstem in such cases, we need to use more advanced techniques (the theor of center manifolds and so on). Indeed, it can be proved that (, ) is an unstable equilibrium with respect to the original nonlinear sstem; hence, for this particular sstem, the local nonlinear dnamics is actuall different from the linear dnamics. But we onl find this after a much more complicated nonlinear analsis, which we did not (and will not) cover in this course. In summar, in the present eercise, linear analsis was helpful in giving a partial information, but unfortunatel was not enough to answer all the questions asked. Well, I still hope that ou learned a thing or two b attempting to solve this eercise. For our reference and viewing pleasure, the phase portrait of the nonlinear sstem of the present eercise is shown on the right. 6