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Phase Portrait Behavior [1] Eigenvalue, Trace, Determinant Phase Portrait Shape λ 1 & λ 2 are real and of opposite sign; Determinant < 0 Saddle (unstable) λ 1 & λ 2 are real and of the same sign, and λ 1 ≠ λ 2; 0 < determinant < (trace 2 / 4) Node (stable if trace < 0, unstable if trace > 0) λ 1 & λ 2 have both a real and imaginary ...
The signs of the eigenvalues indicate the phase plane's behaviour: If the signs are opposite, the intersection of the eigenvectors is a saddle point . If the signs are both positive, the eigenvectors represent stable situations that the system diverges away from, and the intersection is an unstable node .
Phase portrait showing saddle-node bifurcation. Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.
But the topological conjugacy in this context does provide the full geometric picture. In effect, the nonlinear phase portrait near the equilibrium is a thumbnail of the phase portrait of the linearized system. This is the meaning of the following regularity results, and it is illustrated by the saddle equilibrium in the example below.
As can be seen by the animation obtained by plotting phase portraits by varying the parameter , When α {\displaystyle \alpha } is negative, there are no equilibrium points. When α = 0 {\displaystyle \alpha =0} , there is a saddle-node point.
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In control theory, a continuous linear time-invariant system (LTI) is exponentially stable if and only if the system has eigenvalues (i.e., the poles of input-to-output systems) with strictly negative real parts (i.e., in the left half of the complex plane). [1]
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