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X x x ? μ μ ? = ′ = ′ 1 2 2 2 2 1 2 tan x x x x

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Unformatted text preview: x x x λ μ μ λ +- = ′ + = ′ 1 2 2 2 2 1 2 tan , x x x x r = + = θ ( 29 ( 29 2 1 1 2 2 1 2 2 2 1 1 / / sec , 2 2 2 x dt dx x dt dx x dt d dt dx x dt dx x dt dr r- = + = θ θ ( 29 ( 29 2 1 1 2 2 1 2 2 2 1 1 sec , x x x x x x x x x r r ′- ′ = ′ ′ + ′ = ′ θ θ μ θ λ- = ′ = ′ , r r Case 4: Spiral Point (3 of 5) Solving the differential equations we have These equations are parametric equations in polar coordinates of the solution trajectories to our system x ' = Ax . Since μ > 0, it follows that θ decreases as t increases, so the direction of motion on a trajectory is clockwise. If λ < 0, then r → 0 as t → ∞ , while r → ∞ if λ > 0. Thus the trajectories are spirals, which approach or recede from the origin depending on the sign of λ , and the critical point is called a spiral point in this case. μ θ λ- = ′ = ′ , r r ) ( , , θ θ θ μ θ λ = +- = = t ce r t Case 4: Phase Portraits (4 of 5) The phase portrait along with several graphs of x 1 versus t are given below. Frequently the terms spiral sink and spiral source are used to refer to spiral points whose trajectories approach, or depart from, the critical point. 1 2 2 2 2 1 2 2 1 2 1 , tan , θ μ θ θ λ μ μ λ λ +- = = = + = - = ′ ′ = ′ t ce r x x x x r x x x x t Ax x Case 4: General System (5 of 5) It can be shown that for any system with complex eigenvalues λ ± i μ , where λ ≠ 0, the trajectories are always spirals. They are directed inward or outward, respectively, depending on whether λ is negative or positive. The spirals may be elongated and skewed with respect to the coordinate axes, and the direction may be either clockwise or counterclockwise. See text for more details. Case 5: Pure Imaginary Eigenvalues (1 of 2) Suppose the eigenvalues are λ ± i μ , where λ = 0 and μ real. Systems having eigenvalues ± i μ are typified by As in Case 4, using polar coordinates r , θ leads to The trajectories are circles with center at the origin, which are traversed clockwise if μ > 0 and counterclockwise if μ < 0. A complete circuit about the origin occurs in a time interval of length 2 π / μ , so all solutions are periodic with period 2 π / μ . The critical point is called a center . 1 2 2 1 x x x x μ μ μ μ- = ′ = ′ ⇔ - = ′ x x , θ μ θ +- = = t c r Case 5: Phase Portraits (2 of 2) In general, when the eigenvalues are pure imaginary, it is possible to show that the trajectories are ellipses centered at the origin....
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x x x μ μ = ′ = ′ 1 2 2 2 2 1 2 tan x x x x r = =...

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