Linear dynamical systems are
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
whose
evaluation functions are
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
. While dynamical systems, in general, do not have
closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties. Linear systems can also be used to understand the qualitative behavior of general dynamical systems, by calculating the
equilibrium points
In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.
Formal definition
The point \tilde\in \mathbb^n is an equilibrium point for the differential equation
:\frac = \ma ...
of the system and approximating it as a linear system around each such point.
Introduction
In a linear dynamical system, the variation of a state vector
(an
-dimensional
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
denoted
) equals a constant matrix
(denoted
) multiplied by
. This variation can take two forms: either
as a
flow, in which
varies
continuously with time
:
or as a mapping, in which
varies in
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
steps
:
These equations are linear in the following sense: if
and
are two valid solutions, then so is any
linear combination
of the two solutions, e.g.,
where
and
are any two
scalars
Scalar may refer to:
*Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
*Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
. The matrix
need not be
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
.
Linear dynamical systems can be solved exactly, in contrast to most nonlinear ones. Occasionally, a nonlinear system can be solved exactly by a change of variables to a linear system. Moreover, the solutions of (almost) any nonlinear system can be well-approximated by an equivalent linear system near its
fixed points. Hence, understanding linear systems and their solutions is a crucial first step to understanding the more complex nonlinear systems.
Solution of linear dynamical systems
If the initial vector
is aligned with a
right eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of
the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, the dynamics are simple
:
where
is the corresponding
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
;
the solution of this equation is
:
as may be confirmed by substitution.
If
is
diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
, then any vector in an
-dimensional space can be represented by a linear combination of the right and
left eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s (denoted
) of the matrix
.
:
Therefore, the general solution for
is
a linear combination of the individual solutions for the right
eigenvectors
:
Similar considerations apply to the discrete mappings.
Classification in two dimensions
The roots of the
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The chara ...
det(A - λI) are the eigenvalues of A. The sign and relation of these roots,
, to each other may be used to determine the stability of the dynamical system
:
For a 2-dimensional system, the characteristic polynomial is of the form
where
is the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ...
and
is the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of A. Thus the two roots are in the form:
:
:
,
and
and
. Thus if
then the eigenvalues are of opposite sign, and the fixed point is a saddle. If
then the eigenvalues are of the same sign. Therefore, if
both are positive and the point is unstable, and if
then both are negative and the point is stable. The
discriminant
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the origi ...
will tell you if the point is nodal or spiral (i.e. if the eigenvalues are real or complex).
See also
*
Linear system
In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator.
Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.
As a mathematical abstraction o ...
*
Dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
*
List of dynamical system topics
This is a list of dynamical system and differential equation topics, by Wikipedia page. See also list of partial differential equation topics, list of equations.
Dynamical systems, in general
*Deterministic system (mathematics)
*Linear system
* P ...
*
Matrix 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 funct ...
Dynamical systems