Linear stability
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form , where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part. If all the eigenvalues have negative real part, then the solution is called linearly stable. Other names for linear stability include exponential stability or stability in terms of first approximation.[1][2] If there exist an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".[3]
Examples[edit]
Ordinary differential equation[edit]
The differential equation
To derive the linearization at x = 1, one writes , where r = x − 1. The linearized equation is then ; the linearized operator is A1 = −1, the only eigenvalue is , hence this stationary point is linearly stable.
Nonlinear Schrödinger Equation[edit]
The nonlinear Schrödinger equation
It should be mentioned that linear stability does not automatically imply stability; in particular, when k = 2, the solitary waves are unstable. On the other hand, for 0 < k < 2, the solitary waves are not only linearly stable but also orbitally stable.[6]
See also[edit]
- Asymptotic stability
- Linearization (stability analysis)
- Lyapunov stability
- Orbital stability
- Stability theory
- Vakhitov–Kolokolov stability criterion
References[edit]
- ^ V.I. Arnold, Ordinary Differential Equations. MIT Press, Cambridge, MA (1973)
- ^ P. Glendinning, Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge university press, 1994.
- ^ V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations", Princeton Univ. Press (1960)
- ^ H. Berestycki and P.-L. Lions (1983). "Nonlinear scalar field equations. I. Existence of a ground state". Arch. Rational Mech. Anal. 82 (4): 313–345. Bibcode:1983ArRMA..82..313B. doi:10.1007/BF00250555. S2CID 123081616.
- ^ N.G. Vakhitov and A.A. Kolokolov (1973). "Stationary solutions of the wave equation in the medium with nonlinearity saturation". Radiophys. Quantum Electron. 16 (7): 783–789. Bibcode:1973R&QE...16..783V. doi:10.1007/BF01031343. S2CID 123386885.
- ^ Manoussos Grillakis, Jalal Shatah, and Walter Strauss (1987). "Stability theory of solitary waves in the presence of symmetry. I". J. Funct. Anal. 74: 160–197. doi:10.1016/0022-1236(87)90044-9.
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