Exponential dichotomy
Encyclopedia
In the mathematical
theory of dynamical systems, an exponential dichotomy is a property of an equilibrium point that extends the idea of hyperbolicity
to non-autonomous system
s.
is a linear
non-autonomous dynamical system in Rn with fundamental solution matrix Φ(t), Φ(0) = I, then the equilibrium point 0 is said to have an exponential dichotomy if there exists a (constant) matrix
P such that P2 = 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).
).
An equilibrium point with an exponential dichotomy has many of the properties of a hyperbolic equilibrium point in autonomous system
s. In fact, it can be shown that a hyperbolic point has an exponential dichotomy.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
theory of dynamical systems, an exponential dichotomy is a property of an equilibrium point that extends the idea of hyperbolicity
Hyperbolic equilibrium point
In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds. Near a hyperbolic point the orbits of a two-dimensional, non-dissipative system resemble hyperbolas. This fails to hold in general...
to non-autonomous system
Autonomous system
Autonomous system may refer to:* Autonomous system , a collection of IP networks and routers under the control of one entity* Autonomous system , a system of ordinary differential equations which does not depend on the independent variable* Autonomous robot, robots which can perform desired tasks...
s.
Definition
Ifis a linear
Linear system
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 general, nonlinear case....
non-autonomous dynamical system in Rn with fundamental solution matrix Φ(t), Φ(0) = I, then the equilibrium point 0 is said to have an exponential dichotomy if there exists a (constant) matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
P such that P2 = 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 system
Autonomous system (mathematics)
In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable...
s. In fact, it can be shown that a hyperbolic point has an exponential dichotomy.