ia-dyn-chapter11

# Then it is a stable focus if α 0 ie if the eigen

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then it is a stable focus . If α = 0 (i.e., if the eigen- values are purely imaginary) then the solution just goes round and round a closed loop (an ellipse with semi-axes a and b ): this is a centre . The sense of the rotation (i.e., either clockwise or anticlockwise) is best discovered on a case-by-case basis (usually by considering the sign of either f or g close to the equilibrium point). Note that e 1 and e 2 are not necessarily orthogonal. Similarly, for complex eigenvalues α ± i β , a and b are not necessarily orthogonal; so in this case the diagrams above might need to be skewed. In the case of a centre ( α = 0) this would result in a sheared ellipse: but a sheared ellipse is still an ellipse (albeit one with different axes of symmetry). The diagrams are qualitatively correct. Using the Trace and Determinant In practice, it is not necessary to find the exact values of the eigenvalues and eigenvectors; the signs of the eigenvalues (or of their real parts) are all that is required to perform the categorisation. The characteristic equation for J is ( f x - λ )( g y - λ ) - f y g x = 0 or, equivalently, λ 2 - + Δ = 0 where T = f x + g y is the trace and Δ = f x g y - f y g x the determinant of J . We note that the eigenvalues are real iff T 2 - 0; and that the product of the eigenvalues is Δ while their sum is T . This enables us to deduce the required signs. 77

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Summary of Results Δ T 2 - Eigenvalues of J Classification - ve [+ve] Real, opposite signs Saddle +ve +ve Real, same signs Node: T < 0 stable T > 0 unstable +ve - ve Complex conjugate pair Focus: T < 0 stable T = 0 centre T > 0 unstable In the above analysis we have ignored the following degenerate cases: One of the eigenvalues is zero, which occurs iff Δ = 0. In this situation we would have to consider second-order terms in the Taylor series, which we could neglect in the analysis above. These terms could either stabilise or destabilise the equilibrium point. The two eigenvalues are equal (and real), which occurs iff T 2 - 4Δ = 0. In this situation there may be two non-parallel eigenvectors, just as normal; or there may be only one eigenvector, in which case the general solution is not of the form given above. In either case, the classification still holds: the equilibrium point is an unstable node if λ 1 = λ 2 > 0 and a stable node if λ 1 = λ 2 < 0. The phase diagram looks somewhat different, however: a star node (in the case of two non-parallel eigenvectors) or an inflected node (in the case of only one). 11.3 The Phase Plane for a Conservative System A second-order differential equation for a variable x ( t ) can always be converted to two first-order differential equations by defining y = ˙ x . For example, a general force equation in one dimension, m ¨ x = F ( x, ˙ x ) , can be converted to ˙ x = y, ˙ y = 1 m F ( x, y ) which is of the form given in § 11.1 for a plane autonomous system.
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