Linear stability

Last updated

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 linearlystable. Other names for linear stability include exponential stability or stability in terms of first approximation. [1] [2] If there exists 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]

Contents

Examples

Ordinary differential equation

The differential equation

has two stationary (time-independent) solutions: x = 0 and x = 1. The linearization at x = 0 has the form . The linearized operator is A0 = 1. The only eigenvalue is . The solutions to this equation grow exponentially; the stationary point x = 0 is linearly unstable.

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

The nonlinear Schrödinger equation

where u(x,t) C and k > 0, has solitary wave solutions of the form . [4]

To derive the linearization at a solitary wave, one considers the solution in the form . The linearized equation on is given by

where

with

and

the differential operators. According to Vakhitov–Kolokolov stability criterion, [5] when k > 2, the spectrum of A has positive point eigenvalues, so that the linearized equation is linearly (exponentially) unstable; for 0 < k ≤ 2, the spectrum of A is purely imaginary, so that the corresponding solitary waves are linearly stable.

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

Related Research Articles

<span class="mw-page-title-main">Wave equation</span> Differential equation important in physics

The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves or electromagnetic waves. It arises in fields like acoustics, electromagnetism, and fluid dynamics.

<span class="mw-page-title-main">Partial differential equation</span> Type of differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

<span class="mw-page-title-main">Eigenfunction</span> Mathematical function of a linear operator

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as

In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form:

In mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the linear partial differential equation:

In mathematics, a hyperbolic partial differential equation of order is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along any non-characteristic hypersurface. Many of the equations of mechanics are hyperbolic, and so the study of hyperbolic equations is of substantial contemporary interest. The model hyperbolic equation is the wave equation. In one spatial dimension, this is

In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc. (any pair of variables). It is a two-dimensional case of the general n-dimensional phase space.

In linear algebra, it is often important to know which vectors have their directions unchanged by a given linear transformation. An eigenvector or characteristic vector is such a vector. Thus an eigenvector of a linear transformation is scaled by a constant factor when the linear transformation is applied to it: . The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor .

In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time-delay systems, systems with aftereffect or dead-time, hereditary systems, equations with deviating argument, or differential-difference equations. They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector. Four points may give a possible explanation of the popularity of DDEs:

  1. Aftereffect is an applied problem: it is well known that, together with the increasing expectations of dynamic performances, engineers need their models to behave more like the real process. Many processes include aftereffect phenomena in their inner dynamics. In addition, actuators, sensors, and communication networks that are now involved in feedback control loops introduce such delays. Finally, besides actual delays, time lags are frequently used to simplify very high order models. Then, the interest for DDEs keeps on growing in all scientific areas and, especially, in control engineering.
  2. Delay systems are still resistant to many classical controllers: one could think that the simplest approach would consist in replacing them by some finite-dimensional approximations. Unfortunately, ignoring effects which are adequately represented by DDEs is not a general alternative: in the best situation, it leads to the same degree of complexity in the control design. In worst cases, it is potentially disastrous in terms of stability and oscillations.
  3. Voluntary introduction of delays can benefit the control system.
  4. In spite of their complexity, DDEs often appear as simple infinite-dimensional models in the very complex area of partial differential equations (PDEs).
<span class="mw-page-title-main">Inverse scattering transform</span> Method for solving certain nonlinear partial differential equations

In mathematics, the inverse scattering transform is a method that solves the initial value problem for a nonlinear partial differential equation using mathematical methods related to wave scattering. The direct scattering transform describes how a function scatters waves or generates bound-states. The inverse scattering transform uses wave scattering data to construct the function responsible for wave scattering. The direct and inverse scattering transforms are analogous to the direct and inverse Fourier transforms which are used to solve linear partial differential equations.

<span class="mw-page-title-main">Reaction–diffusion system</span> Type of mathematical model

Reaction–diffusion systems are mathematical models that correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the substances are transformed into each other, and diffusion which causes the substances to spread out over a surface in space.

<span class="mw-page-title-main">Benjamin–Bona–Mahony equation</span>

The Benjamin–Bona–Mahony equation is the partial differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to their derivatives.

In mathematics, the slow manifold of an equilibrium point of a dynamical system occurs as the most common example of a center manifold. One of the main methods of simplifying dynamical systems, is to reduce the dimension of the system to that of the slow manifold—center manifold theory rigorously justifies the modelling. For example, some global and regional models of the atmosphere or oceans resolve the so-called quasi-geostrophic flow dynamics on the slow manifold of the atmosphere/oceanic dynamics, and is thus crucial to forecasting with a climate model.

The Vakhitov–Kolokolov stability criterion is a condition for linear stability of solitary wave solutions to a wide class of U(1)-invariant Hamiltonian systems, named after Soviet scientists Aleksandr Kolokolov and Nazib Vakhitov . The condition for linear stability of a solitary wave with frequency has the form

In mathematical physics and the theory of partial differential equations, the solitary wave solution of the form is said to be orbitally stable if any solution with the initial data sufficiently close to forever remains in a given small neighborhood of the trajectory of

Derrick's theorem is an argument by physicist G. H. Derrick which shows that stationary localized solutions to a nonlinear wave equation or nonlinear Klein–Gordon equation in spatial dimensions three and higher are unstable.

Riemann invariants are mathematical transformations made on a system of conservation equations to make them more easily solvable. Riemann invariants are constant along the characteristic curves of the partial differential equations where they obtain the name invariant. They were first obtained by Bernhard Riemann in his work on plane waves in gas dynamics.

In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial differential equations.

References

  1. V.I. Arnold, Ordinary Differential Equations. MIT Press, Cambridge, MA (1973)
  2. P. Glendinning, Stability, instability and chaos: an introduction to the theory of nonlinear differential equations. Cambridge university press, 1994.
  3. V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations", Princeton Univ. Press (1960)
  4. 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.
  5. 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.
  6. 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 .{{cite journal}}: CS1 maint: multiple names: authors list (link)