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Lyapunov Exponents. III Steven Finch November 21, 2007 Our ongoing study encompasses both discrete iteration [1] and continuous ?ow [2]; the system dynamics can be either deterministic or stochastic. Let A denote a real m × m matrix and B, X denote real m-vectors. Consider the di?erence equation Xn = A Xn?1 + B εn , X0 arbitrary

where εn is scalar N(0, 1) white noise. Order the complex eigenvalues λ1 , λ2 , . . ., λm of A so that λ1 has maximum modulus. When |λ1 | > 1, it follows that

1 n

ln |Xn | → ln |λ1 | > 0

almost surely as n → ∞

which indicates that no convergence to stationarity can occur. The quantity ln |λ1 | is the Lyapunov exponent of the system, since the derivative of the linear transformation x 7→ A X is itself. Consider instead the di?erential equation dXt = A Xt dt + B dWt , X0 arbitrary

where Wt is scalar Brownian motion with unit variance. The corresponding ?ow is Xt = eA t ?X0 +

? Zt

0

e?A s B dWs ?

?

and the complex eigenvalues of eA are eλ1 , eλ2 , . . ., eλm . Here, however, we order λ1 , λ2 , . . ., λm so that λ1 has maximum real part (which implies that eλ1 has maximum modulus). When Re(λ1 ) > 0, the interpretation of

1 t

ln |Xt | → Re(λ1 )

almost surely as t → ∞

is exactly as before. An informal proof is to choose X0 to be the dominant eigenvector of A or of eA , respectively, and to choose B = 0; then |Xn | = |An X0 | = |λ1 |n |X0 |

0

or

|Xt | = |eA t X0 | = |eλ1 |t |X0 |,

c Copyright ° 2007 by Steven R. Finch. All rights reserved.

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Lyapunov Exponents. III

2

respectively. See [3] for special treatment of the case m = 1. The probability density of ln |Xt | ? t Re(λ1 ) ln |Xn | ? n ln |λ1 |, is also of interest, and turns out to be doubly-exponential [3, 4]. Additive noise does not enter the formula for Lyapunov exponents; multiplicative noise contrasts in this regard. Let A, B denote real m × m matrices. The equations Xn = A Xn?1 + B Xn?1 εn , dXt = A Xt dt + B Xt dWt , X0 arbitrary; X0 arbitrary

require more intricate analysis. Let us focus only on the continuous-time case for now, leaving the discrete-time case for later. In the event A and B commute, that is, A B = B A, it can be proved that [5] Xt = exp

??

A ? 1 B 2 t + B Wt X0. 2

There is, however, no consequential formula for the Lyapunov exponent that is valid for all m ≥ 1 and all A, B. Set m = 1 or m = 2. Let us adhere to the convention of replacing A by A + 1 B 2 : 2 dXt = A + 1 B 2 Xt dt + B Xt dWt , 2

? ?

?

?

X0 arbitrary.

If m = 1, A = a and B = σ > 0, then the random variable ln |Xt /X0 | is normally distributed with mean a t and variance σ 2 t (the process Xt is often called geometric Brownian motion). Clearly

1 t

ln |Xt | → a

?

almost surely as t → ∞.

! ?

Stability is unchanged by noise in this example. The same can be said if m = 2, A= a 0 0 b and B= σ 0 0 σ

!

where a > b and σ > 0. If instead B= then it can be proved that [6, 7]

1 t

?

0 ?σ σ 0

?

!

ln |Xt | → 1 (a + b) + 1 (a ? 2 2

a?b 2 σ2 b) ? a?b ? I0 2 σ2

I1

?

almost surely

Lyapunov Exponents. III

3

where I0 , I1 are modi?ed Bessel functions [8]. For example, when a = 1, b = ?2 and σ = 10, the Lyapunov exponent has value ?0.4887503163... [9]. For the same a and b, the Lyapunov exponent has value 0.3941998582... when σ = 1, and is zero precisely when σ = 1.4560286969... [6]. More noise implies enhanced stability in this example. If instead [10, 11] A= then

1 t

?

0 1 0 0

!

and

B=

?

0 0 σ 0

!

√ π 31/3 π κ = 1/6 = 2/3 . 12 Γ(1/3)2 2 Γ(1/6) What happens when the bottom row of A is nonzero? If A=

?

and

ln |Xt | → κ σ 2/3 = (0.2893082598...)σ 2/3

almost surely

0 1 ?α 2β

!

and

B=

?

0 0 σ 0

!

where β 2 > α, then more complicated formulation emerges. We avoid the hypergeometric functions in [12, 13], preferring modi?ed Bessel functions (of both integer and fractional types). Let 4 (β 2 ? α) σ2 δ= γ= , 2 9γ 2 for convenience. De?ne

∞ ? ? ? 3 Z √ 1 2 3 ? 2 f (α, β, γ) = z exp ? γ z + β ? α z dz 3 12 2π 2 0 ?√ ? ?√ ? ? ?1 1 ?√ ?2 ?√ ? ?√ ? 1 1 δ I? 1 δ δ δ I2 δ δ 2 I? 2 2 2 3 δ 3 I1 δ 2 I1 3 3 3 3 3 3 = + + + 1 γ γ (β 2 ? α) γ 3 ? ?2 1 2 ?√ ?2 ? ?2 ?√ ? ?√ ? 1 6 2 3 γ 3 δ 3 I2 δ δ I4 δ 2 2 3 (β 2 ? α) δ 6 I 1 3 3 3 3 3 + + 5 2 (β 2 ? α) γ3 ? ?1 ?√ ? ?√ ? 1 1 2 2 3 (β 2 ? α) 2 δ 3 I 2 δ I5 δ 3 3 3 , 4 γ3 ∞ ? ? ? ? 1 3 Z 1 √ exp ? γ 2 z 3 + β 2 ? α z dz g(α, β, γ) = 3 z 12 2π 2 0

Lyapunov Exponents. III

? ?1

2 3

3 1 3

4

?√ ?

δ

=

?

3 2

I? 1

3 3

?√ ?2

δ

+ I? 1

3

3 Ai = then

1 t

?

?? ? 2

δ

1 3

?2

+ Bi

1

2γ 3

?? ? 2

3 2

3

γ3

?√ ?

1

δ I1

3

δ + I1

3

?√ ?2 ?

δ

δ

1 3

?2 !

ln |Xt | → β +

γ f (α, β, γ) 2 g(α, β, γ)

almost surely.

For example, when α = 1 and |β| > 1 is ?xed, the Lyapunov exponent is decreasing as a function of γ ∈ (0, (β 2 ? 1)3/2 γ0 ) and increasing for γ ∈ ((β 2 ? 1)3/2 γ0 , ∞), where γ0 = 1.6946141069.... At criticality, we have 1 f (1, β, (β 2 ? 1)3/2 γ0 ) = (1.4567743021...) 2 ? 1)3/2 γ ) 2 ? 1)γ g(1, β, (β (β 0 0 2 = (1 + 0.8848441574...)?1/2 ; (β 2 ? 1)γ0 further stabilization by noise beyond this point is impossible. As another example, when α = 1 and γ > 0 is ?xed, we have f (1, β, γ) lim+ = |β|→1 g(1, β, γ)

? !2/3

4 γ

κ=

2 2/3 κσ , γ

consistent with preceding zero-row results. The constant κ = 0.2893082598... also appears in [14], but reasons for this connection are unclear. Explicit expressions like the above are quite rare in this area. We hope to report later on [15], which seems to present a promising approach but unfortunately gives no examples. References [1] S. R. Finch, Lyapunov exponents, unpublished note (2007). [2] S. R. Finch, Lyapunov exponents. II, unpublished note (2007). [3] S. R. Finch, Another look at AR(1), arXiv:0710.5419. [4] S. R. Finch, Extreme value constants, Mathematical Constants, Cambridge Univ. Press, 2003, pp. 363—367. [5] L. Arnold, Stochastic Di?erential Equations: Theory and Applications, Wiley, 1974, pp. 125—144, 176—201; MR0443083 (56 #1456)

Lyapunov Exponents. III

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[6] P. H. Baxendale, Moment stability and large deviations for linear stochastic di?erential equations, Probabilistic Methods in Mathematical Physics, Proc. 1985 Katata/Kyoto conf., ed. K. It? and N. Ikeda, Academic Press, 1987, pp. 31—54; o MR0933817 (89c:60068). [7] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Di?erential Equations, Springer-Verlag, 1992, pp. 103—126, 148—152, 232—241, 396—399, 540—548; MR1214374 (94b:60069). [8] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1972, pp. 374—377; MR1225604 (94b:00012). [9] D. Talay, Approximation of upper Lyapunov exponents of bilinear stochastic di?erential systems, SIAM J. Numer. Anal. 28 (1991) 1141—1164; MR1111458 (92i:60119). [10] S. T. Ariaratnam and W. C. Xie, Lyapunov exponent and rotation number of a two-dimensional nilpotent stochastic system, Dynam. Stability Systems 5 (1990) 1—9; MR1057870 (91h:60086). [11] W. C. Xie, Lyapunov exponents and moment Lyapunov exponents of a twodimensional near-nilpotent system, Trans. ASME J. Appl. Mech. 68 (2001) 453— 461; MR1836484. [12] P. Imkeller and C. Lederer, An explicit description of the Lyapunov exponents of the noisy damped harmonic oscillator, Dynam. Stability Systems 14 (1999) 385—405; MR1746113 (2000m:34122). [13] P. Imkeller and C. Lederer, Some formulas for Lyapunov exponents and rotation numbers in two dimensions and the stability of the harmonic oscillator and the inverted pendulum, Dyn. Syst. 16 (2001) 29—61; MR1835906 (2002j:60100). [14] B. Derrida and E. Gardner, Lyapounov exponent of the one-dimensional Anderson model: weak disorder expansions, J. Physique 45 (1984) 1283—1295; MR0763431 (85m:82098). [15] A. Leizarowitz, Exact results for the Lyapunov exponents of certain linear Ito systems, SIAM J. Appl. Math. 50 (1990) 1156—1165; MR1053924 (91j:93120).

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