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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 ...
where λ 1 and λ 2 are the eigenvalues, and (k 1, k 2), (k 3, k 4) are the basic eigenvectors. The constants c 1 and c 2 account for the nonuniqueness of eigenvectors and are not solvable unless an initial condition is given for the system. The above determinant leads to the characteristic polynomial:
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.
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.
Plot of the Duffing map showing chaotic behavior, where a = 2.75 and b = 0.15. Phase portrait of a two-well Duffing oscillator (a differential equation, rather than a map) showing chaotic behavior. The Duffing map (also called as 'Holmes map') is a discrete-time dynamical system. It is an example of a dynamical system that exhibits chaotic behavior
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.
In quantum mechanics, the intrinsic parity is a phase factor that arises as an eigenvalue of the parity operation ′ = (a reflection about the origin). [1] To see that the parity's eigenvalues are phase factors, we assume an eigenstate of the parity operation (this is realized because the intrinsic parity is a property of a particle species) and use the fact that two parity transformations ...
In quantum computing, the quantum phase estimation algorithm is a quantum algorithm to estimate the phase corresponding to an eigenvalue of a given unitary operator.Because the eigenvalues of a unitary operator always have unit modulus, they are characterized by their phase, and therefore the algorithm can be equivalently described as retrieving either the phase or the eigenvalue itself.