Notes-PhasePlane

Portrait approximates the local behavior of the

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portrait approximates the local behavior of the original nonlinear system near the critical point. To wit, start with the lineariztions of F and G (recall that such a linearization is just the 3 lowest order terms in the Taylor series expansion of each function) about the critical point ( x , y ) = ( α , β ).
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© 2008 Zachary S Tseng D -2 - 23 x ′ = F ( x , y ) ≈ F ( α , β ) + F x ( α , β )( x α ) + F y ( α , β )( y β ) y ′ = G ( x , y ) ≈ G ( α , β ) + G x ( α , β )( x α ) + G y ( α , β )( y β ) But since ( α , β ) is a critical point, so F ( α , β ) = 0 = G ( α , β ), the above linearizations become x ′ ≈ F x ( α , β )( x α ) + F y ( α , β )( y β ) y ′ ≈ G x ( α , β )( x α ) + G y ( α , β )( y β ) As before, the critical point could be translated to (0, 0) and still retains its type and stability, using the substitutions χ = x α and γ = y β . After the translation, the approximated system becomes x ′ = F x ( α , β ) x + F y ( α , β ) y y ′ = G x ( α , β ) x + G y ( α , β ) y It is now a homogeneous linear system with a coefficient matrix A = ) , ( ) , ( ) , ( ) , ( β α β α β α β α y x y x G G F F . That is, it is a matrix calculate by plugging in x = α and y = β into the matrix of first partial derivatives J = y x y x G G F F . This matrix of first partial derivatives, J , is often called the Jacobian matrix . It just needs to be calculated once for each nonlinear system. For each critical point of the system, all we need to do is to compute the coefficient matrix of the linearized system about the given critical point ( x , y ) = ( α , β ), and then use its eigenvalues to determine the (approximated) type and stability.
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© 2008 Zachary S Tseng D -2 - 24 Example : x ′ = x y y ′ = x 2 + y 2 − 2 The critical points are at (1, 1) and (−1, −1). The Jacobian matrix is J = y x 2 2 1 1 . At (1, 1), the linearized system has coefficient matrix: A = 2 2 1 1 . The eigenvalues are 2 7 3 i r ± = . The critical point is an unstable spiral point. At (−1, −1), the linearized system has coefficient matrix: A = 2 2 1 1 . The eigenvalues are 2 17 1 ± = r . The critical point is an unstable saddle point. The phase portrait is shown on the next page.
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© 2008 Zachary S Tseng D -2 - 25 (1, 1) is an unstable spiral point. (−1, −1) is an unstable saddle point.
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© 2008 Zachary S Tseng D -2 - 26 Example : x ′ = x xy y ′ = y + 2 xy The critical points are at (0, 0) and (−1/2, 1). The Jacobian matrix is J = + x y x y 2 1 2 1 At (0, 0), the linearized system has coefficient matrix: A = 1 0 0 1 . There is a repeated eigenvalue r = 1. A linear system would normally have an unstable proper node (star point) here. But as a nonlinear system it actually has an unstable node. (Didn’t I say that this approximation using linearization is not always 100% accurate?) At (−1/2, 1), the linearized system has coefficient matrix: A = 0 2 2 / 1 0 .
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