# Lyapunov characteristic exponent

of a solution of a linear system

$$\lambda _ {x ( t) } = \ {\lim\limits _ {t \rightarrow + \infty } } bar \ \frac{1}{t} \ \mathop{\rm ln} | x ( t) | ,$$

where $x ( t) \neq 0$ is a solution of the linear system of ordinary differential equations

$$\tag{1 } \dot{x} = A ( t) x ;$$

here $x \in \mathbf R ^ {n}$ and $A ( \cdot )$ is a mapping $\mathbf R \rightarrow \mathop{\rm Hom} ( \mathbf R ^ {n} , \mathbf R ^ {n} )$, summable on every interval. In coordinates,

$$x ( t) = ( x ^ {1} ( t) \dots x ^ {n} ( t) ) ,$$

$$\dot{x} ^ {i} = \sum _ {j=1} ^ { n } a _ {j} ^ {i} ( t) x ^ {j} ,\ i = 1 \dots n ,$$

where $a _ {j} ^ {i} ( t)$ are functions summable on every interval and

$$| x | = \sqrt {\sum _ {i=1} ^ { n } | x ^ {i} | ^ {2} }$$

(or any other equivalent norm; $\lambda _ {x ( t) }$ does not depend on the choice of the norm in $\mathbf R ^ {n}$ or in $\mathbf C ^ {n}$).

Lyapunov's theorem. Suppose that

$${\lim\limits _ {t \rightarrow + \infty } } bar \ \frac{1}{t} \int\limits _ { 0 } ^ { t } \| A ( \tau ) \| d \tau < + \infty ;$$

equivalently:

$${\lim\limits _ {t \rightarrow + \infty } } bar \ \frac{1}{t} \int\limits _ { 0 } ^ { t } | a _ {j} ^ {i} ( \tau ) | d \tau < + \infty ,\ \ i , j = 1 \dots n .$$

Then for any solution $x ( t) \neq 0$ of the system (1) the Lyapunov characteristic exponent $\lambda _ {x ( t) }$ is a real number (that is, $\neq \pm \infty$). The following assertions hold for the Lyapunov characteristic exponents of non-zero solutions of (1):

1) $\lambda _ {\alpha x ( t) } = \lambda _ {x ( t) }$, $\alpha \neq 0$;

2) $\lambda _ {x _ {1} ( t) + x _ {2} ( t) } \leq \max ( \lambda _ {x _ {1} ( t) } , \lambda _ {x _ {2} ( t) } )$;

3) there exists a set of linearly independent solutions of (1), denoted by $\{ x _ {i} ( t) \} _ {i=1} ^ {n}$, such that for any other $n$ linearly independent solutions $\widehat{x} _ {i} ( t)$, $i = 1 \dots n$, of (1), numbered in decreasing order of the Lyapunov characteristic exponents, that is, $\lambda _ {\widehat{x} _ {i} ( t) } \geq \lambda _ {\widehat{x} _ {j} ( t) }$ for $i \leq j$, the following inequalities hold:

$$\lambda _ {\widehat{x} _ {i} ( t) } \geq \ \lambda _ {x _ {i} ( t) } ,\ \ i = 1 \dots n .$$

A fundamental system of solutions $\{ x _ {i} ( t) \} _ {i=1} ^ {n}$ with this property is called normal. Such a normal system has the properties:

a) the family of numbers $\lambda _ {i} ( A) = \lambda _ {x _ {i} ( t) }$, $i = 1 \dots n$, does not depend on the choice of the normal fundamental system;

b) for any solution $x ( t) \neq 0$ of (1) its Lyapunov characteristic exponent $\lambda _ {x ( t) }$ is equal to some $\lambda _ {i} ( A)$;

c) $\lambda _ {i} ( A) \geq \lambda _ {j} ( A)$, $i \leq j$.

The numbers $\lambda _ {1} ( A) \geq \dots \geq \lambda _ {n} ( A)$ are called the Lyapunov characteristic exponents of the system (1); the number $\lambda _ {1} ( A)$ is often called the leading Lyapunov characteristic exponent of (1).

The set of all Lyapunov characteristic exponents of non-zero solutions of (1) is called the spectrum.

## Contents

### Special case.

1) A system with constant coefficients (that is, $A ( t) \equiv A ( 0)$). In this case the $\lambda _ {i} ( A)$ are equal to the real parts of the eigen values of the operator $A ( 0)$( the matrix $\| a _ {j} ^ {i} \|$).

2) A system with periodic coefficients (that is, $A ( t + T ) \equiv A ( t)$, $T > 0$). In this case

$$\lambda _ {i} ( A) = \frac{1}{T} \mathop{\rm ln} | \mu _ {i} | ,$$

where $\mu _ {i}$ are the multipliers of the system (1), numbered in non-increasing order of their moduli (each is taken as many times as its multiplicity).

The role of the Lyapunov characteristic exponent in the theory of Lyapunov stability is based on the following assertion: If $\lambda _ {1} ( A) < 0$( $> 0$), then the solutions of (1) are asymptotically stable (respectively, unstable, cf. Asymptotically-stable solution). From $\lambda _ {1} ( A) < 0$ it does not follow that the null solution of the system

$$\dot{x} = A ( t) x + O ( | x | ^ {2} )$$

is Lyapunov stable; however, if it is also known that the system (1) is a regular linear system, then this conclusion is valid (Lyapunov's theorem).

Suppose that the system $\dot{x} = B ( t) x$ is obtained by a small perturbation of a system (1) satisfying the condition

$$\sup _ {t \in \mathbf R } \| A ( t) \| < + \infty ;$$

that is, the distance between them, defined by the formula

$$\tag{2 } d ( A , B ) = \ \sup _ {t \in \mathbf R } \| A ( t) - B ( t) \| ,$$

is small. For $n > 1$ this does not imply that the quantity

$$| \lambda _ {1} ( A) - \lambda _ {1} ( B) |$$

is small (it is implied if the system (1) has constant or periodic coefficients, and also for certain other systems); in other words, the functionals $\lambda _ {i} ( A)$ are not everywhere continuous on the space of systems (1) ( $\sup _ {t \in \mathbf R } \| A ( t) \| < + \infty$), endowed with the given metric (2).

Lyapunov characteristic exponents were introduced by A.M. Lyapunov, not only for solutions of the system (1), but also for arbitrary functions on $\mathbf R ^ {+}$( see ).

How to Cite This Entry:
Lyapunov characteristic exponent. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lyapunov_characteristic_exponent&oldid=53915
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article