Lyapunov characteristic exponent

of a solution of a linear system

The limes superior

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

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.

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 [1]).

References

[1]	A.M. Lyapunov, "Stability of motion" , Acad. Press (1966) (Translated from Russian)
[2]	B.F. Bylov, R.E. Vinograd, D.M. Grobman, V.V. Nemytskii, "The theory of Lyapunov exponents and its applications to problems of stability" , Moscow (1966) (In Russian)
[3]	N.A. Izobov, "Linear systems of ordinary differential equations" J. Soviet Math. , 5 (1976) pp. 46–96 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 71–146

Comments

Presently, Lyapunov (characteristic) exponents are used at a much wider scale. For a survey see [a6]. First, the matrix A may be a stochastic time-dependent function. Lyapunov exponents are also used in relation with a system of non-linear differential equations

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

having a strange attractor (or repellor) $ s ( t) $ as limit solution, see [a7]. The system linearized in $ s ( t) $ is of the form (1) with

$$ A ( t) = \left \{ \frac{\partial f _ {i} }{\partial x _ {j} } ( s( t) ) \right \} _ {( n \times n ) } . $$

One of the exponents is zero. The occurrence of one or more positive exponents indicates that $ s ( t) $ is a strange attractor. For a conservative system the sum of the Lyapunov exponents is zero, while for a dissipative system the sum is negative. The capacity $ D $ of a strange attractor is a fractal number related to the Hausdorff dimension. J.L. Kaplan and J.A. Yorke made the conjecture

$$ D = j - \frac{1}{\lambda _ {j+} 1 } \sum _ { i= } 1 ^ { j } \lambda _ {i} $$

with $ 0 < \sum _ {i=} 1 ^ {j} \lambda _ {i} < - \lambda _ {j+} 1 $( $ \lambda _ {1} \geq \dots \geq \lambda _ {n} $).

The concept of Lyapunov exponents extends to non-linear stochastic systems, as well as to iteration mappings

$$ x ( t + 1 ) = A ( t) x ( t) , $$

see [a6] and [a7].

In a yet more general deterministic setting, let $ X $ be a compact subset of a Hilbert space $ H $, and let $ f : X \rightarrow H $ be a mapping such that $ f ( X) = X $. The mapping $ f $ is supposed to satisfy the following uniform differentiability condition: For each $ x \in X $ there is a linear compact operator $ L ( x) : H \rightarrow H $ such that

$$ \sup _ {x,y } \frac{\| f ( y) - f ( x) - L ( x) ( y - x ) \| }{\| y - x \| } \rightarrow 0 $$

as $ \epsilon \rightarrow 0 $, where $ \| \cdot \| $ denotes the norm in $ H $, and where the supremum is taken over all $ x , y $ with $ 0 < \| x - y \| \leq \epsilon $.

For a compact linear operator $ L $, let $ \alpha _ {1} ( L) \geq \alpha _ {2} ( L) \geq \dots $ be the eigen values of $ ( L ^ {*} L ) ^ {1/2} $. For each positive integer $ d $, let $ \omega _ {d} ( L) = \alpha _ {1} ( L) \dots \alpha _ {d} ( L) $, while for a non-integer positive real number $ d = n + s $, $ 0 < s < 1 $, define $ \omega _ {d} ( L) = \omega _ {n} ( L) ( \alpha _ {n+} 1 ( L) ) ^ {s} $.

Suppose that $ f $ and $ L $ are such that $ \sup _ {x \in X } \| L ( x) \| < \infty $.

For each $ x \in X $, let $ L _ {p} ( x) = L ( f ^ { p- 1 } ( x) ) \circ \dots L ( f ( x) ) \circ L ( x) $, where $ f ^ { r } $ denotes the $ r $- th iterate of $ f $. Define the (local) Lyapunov numbers and Lyapunov exponents by

$$ \Lambda _ {j} ( x) = \lim\limits \sup _ {p \rightarrow \infty } \{ \alpha _ {j} ( L _ {p} ( x) ) \} ^ {1/p} , $$

$$ \mu _ {j} ( x) = \mathop{\rm ln} \Lambda _ {j} ( x) . $$

The uniform Lyapunov exponents $ \mu _ {i} $ and uniform Lyapunov numbers $ \Lambda _ {i} $ in this setting are defined as follows:

$$ \overline \omega \; _ {j} ( p) = \sup _ {x \in X } \omega _ {j} ( L _ {p} ( x) ) , $$

$$ \Pi _ {j} = \lim\limits _ {p \rightarrow \infty } ( \overline \omega \; _ {j} ( p) ) ^ {1/p} , $$

$$ \Lambda _ {1} \dots \Lambda _ {j} = \Pi _ {j} ,\ j = 1 , 2 \dots $$

$$ \mu _ {j} = \mathop{\rm ln} \Lambda _ {j} ,\ j = 1 , 2 ,\dots . $$

Let $ n $ be the smallest integer such that $ \mu _ {1} + \dots + \mu _ {n} \geq 0 $ and $ \mu _ {1} + \dots + \mu _ {n} + \mu _ {n+} 1 < 0 $. The number $ d _ {L} ( X) = n + | \mu _ {n+} 1 | ^ {-} 1 ( \mu _ {1} + \dots + \mu _ {n} ) $ is then called the Lyapunov dimension of $ X $. One has (see [a2]) $ d _ {H} ( X) \leq d _ {L} ( X) $, where $ d _ {H} ( X) $ is the Hausdorff dimension of $ X $.

References

[a1]	W. Kliemann, "Analysis of nonlinear stochastic systems" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 43–102
[a2]	P. Constantin, C. Foias, R. Temam, "Attractors representing turbulent flows" , Amer. Math. Soc. (1985)
[a3]	L.S. Young, "Capacity of attractors" Ergod. Th. Dynam. Systems , 1 (1981) pp. 381–383
[a4]	L.S. Young, "Dimension, entropy, and Lyapunov exponents" Ergod. Th. Dynam. Systems , 2 (1982) pp. 109–124
[a5]	P. Fredrickson, J.L. Kaplan, E.D. Yorke, J.A. Yorke, "The Lyapunov dimension of strange attractors" J. Diff. Eq. , 49 (1983) pp. 185–207
[a6]	L. Arnold (ed.) V. Wihstutz (ed.) , Lyapunov exponents , Lect. notes in math. , 1186 , Springer (1986)
[a7]	H.G. Schuster, "Deterministic chaos" , Physik-Verlag (1988)
[a8]	E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) pp. Chapts. 13–17
[a9]	J. Guckenheimer, P. Holmes, "Non-linear oscillations, dynamical systems, and bifurcations of vector fields" , Springer (1983)