In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
theory of
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 ...
, an exponential dichotomy is a property of an
equilibrium point
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 ...
that extends the idea of
hyperbolicity In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations (depending quantitatively on a nonnegative real number δ) between points. The definition, introduced by Mikhael Gromov, generalizes the metric properti ...
to non-
autonomous systems.
Definition
If
:
is a
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 ...
non-autonomous dynamical system in R
''n'' with
fundamental solution matrix
Fundamental may refer to:
* Foundation of reality
* Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental"
* Fundamentalism, the belief in, and usually the strict adherence to, the simple or "fundamental" idea ...
Φ(''t''), Φ(0) = ''I'', then the equilibrium point 0 is said to have an ''exponential dichotomy'' if there exists a (constant)
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 ...
''P'' such that ''P''
2 = ''P'' and positive constants ''K'', ''L'', α, and β such that
:
and
:
If furthermore, ''L'' = 1/''K'' and β = α, then 0 is said to have a ''uniform exponential dichotomy''.
The constants α and β allow us to define the ''spectral window'' of the equilibrium point, (−α, β).
Explanation
The matrix ''P'' is a projection onto the stable subspace and ''I'' − ''P'' is a projection onto the unstable subspace. What the exponential dichotomy says is that the norm of the projection onto the stable subspace of any orbit in the system
decays exponentially as ''t'' → ∞ and the norm of the projection onto the unstable subspace of any orbit decays exponentially as ''t'' → −∞, and furthermore that the stable and unstable subspaces are conjugate (because
).
An equilibrium point with an exponential dichotomy has many of the properties of a hyperbolic equilibrium point in
autonomous systems. In fact, it can be shown that a hyperbolic point has an exponential dichotomy.
References
* Coppel, W. A. ''Dichotomies in stability theory'', Springer-Verlag (1978),
Dynamical systems
Dichotomies
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