Notes-PhasePlane

Neutrally stable it is not asymptotically stable and

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neutrally stable ). It is NOT asymptotically stable and one should not confuse them. 6. When the real part λ is nonzero. The trajectories still retain the elliptical traces as in the previous case. However, with each revolution, their distances from the critical point grow/decay exponentially according to the term e λt . Therefore, the phase portrait shows trajectories that spiral away from the critical point to infinite-distant away (when λ > 0). Or trajectories that spiral toward, and converge to the critical point (when λ < 0). This type of critical point is called a spiral point . It is asymptotically stable if λ < 0, it is unstable if λ > 0.
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© 2008 Zachary S Tseng D -2 - 13 Complex eigenvalues, with real part zero (purely imaginary numbers) Type: Center Stability: Stable (but not asymptotically stable); sometimes it is referred to as neutrally stable.
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© 2008 Zachary S Tseng D -2 - 14 Complex eigenvalues, with nonzero real part Type: Spiral Point Stability: It is unstable if the eigenvalues have positive real part; asymptotically stable if the eigenvalues have negative real part.
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© 2008 Zachary S Tseng D -2 - 15 Summary of Stability Classification Asymptotically stable – All trajectories of its solutions converge to the critical point as t . A critical point is asymptotically stable if all of A ’s eigenvalues are negative, or have negative real part for complex eigenvalues. Unstable – All trajectories (or all but a few, in the case of a saddle point) start out at the critical point at t → − , then move away to infinitely distant out as t . A critical point is unstable if at least one of A ’s eigenvalues is positive, or has positive real part for complex eigenvalues. Stable (or neutrally stable ) – Each trajectory move about the critical point within a finite range of distance. It never moves out to infinitely distant, nor (unlike in the case of asymptotically stable) does it ever go to the critical point. A critical point is stable if A ’s eigenvalues are purely imaginary. In short, as t increases, if all (or almost all) trajectories 1. converge to the critical point asymptotically stable , 2. move away from the critical point to infinitely far away unstable , 3. stay in a fixed orbit within a finite (i.e., bounded) range of distance away from the critical point stable (or neutrally stable ).
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© 2008 Zachary S Tseng D -2 - 16 Nonhomogeneous Linear Systems with Constant Coefficients Now let us consider the nonhomogeneous system x ′ = Ax + b . Where b is a constant vector. The system above is explicitly: x 1 ′ = a x 1 + b x 2 + g 1 x 2 ′ = c x 1 + d x 2 + g 2 As before, we can find the critical point by setting x 1 ′ = x 2 ′ = 0 and solve the resulting nonhomgeneous system of algebraic equations. The origin will no longer be a critical point, since the zero vector is never a solution of a nonhomogeneous linear system. Instead, the unique critical point (as long as A has nonzero determinant, there remains exactly one critical point) will be located at the solution of the system of algebraic equations: 0 = a x 1 + b x 2 + g 1 0 = c
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