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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 ...
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.
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:
If the eigenvalue is equal to one, the bifurcation is either a saddle-node (often called fold bifurcation in maps), transcritical or pitchfork bifurcation. If the eigenvalue is equal to −1, it is a period-doubling (or flip) bifurcation, and otherwise, it is a Hopf bifurcation. Examples of local bifurcations include: Saddle-node (fold) bifurcation
The roots of the corresponding scalar polynomial equation, λ 2 = λ, are 0 and 1. Thus any projection has 0 and 1 for its eigenvalues. The multiplicity of 0 as an eigenvalue is the nullity of P, while the multiplicity of 1 is the rank of P. Another example is a matrix A that satisfies A 2 = α 2 I for some scalar α. The eigenvalues must be ± ...
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 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 ...