Power-law sensitivity to initial conditions—New entropic representation

• Published in: Chaos, solitons and fractals Vol. 8; no. 6; pp. 885 - 891
• Main Authors: Tsallis, C., Plastino, A.R., Zheng, W.-M.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01-06-1997
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/S0960-0779(96)00167-1

The
exponential sensitivity to the initial conditions of chaotic systems (e.g.
D = 1) is characterized by the Liapounov exponent λ, which is, for a large class of systems, known to
equal the Kolmogorov-Sinai entropy
K. We unify this type of sensitivity with a weaker, herein exhibited,
power-law one through (for a dynamical variable
x)
lim
Δx(0)→0
[
Δ
x(t)]
[
Δ
x(0)]
= [1 + (1 − q)λ
qt]
1
(1 − q)
(equal to e
λ
1
t
for
q = 1, and proportional, for large
t, to
t
1
(1 − q)
for
q ≠ 1;. We show that gl (∀q), where K
q is the generalization of
K within the non-extensive thermostatistics based upon the generalized entropic form
S
q ≡
(1 − ∑
ip
i
q)
(q − 1)
(
hence, S
1 = −Σ
ip
i
lnp
i)
. The well-known theorem λ
1 =
K
1 (Pesin equality) is thus extended to arbitrary
q. We discuss the logistic map at its threshold to chaos, at period doubling bifurcations and at tangent bifurcations, and find
q ≈ 0.2445, q =
5
3
and q =
3
2
, respectively. 05.45. +
b; 05.20. −
y; 05.90. +
m.