Power-law Sensitivity Research Articles

Nonextensivity and Multifractality in Low-Dimensional Dissipative Systems

Power-law sensitivity to initial conditions at the edge of chaos provides a natural relation between the scaling properties of the dynamics attractor and its degree of nonextensivity as prescribed in the generalized statistics recently introduced by one of us (C.T.) and characterized by the entropic index $q$. We show that general scaling arguments imply that $1/(1-q) = 1/\alpha_{min}-1/\alpha_{max}$, where $\alpha_{min}$ and $\alpha_{max}$ are the extremes of the multifractal singularity spectrum $f(\alpha)$ of the attractor. This relation is numerically checked to hold in standard one-dimensional dissipative maps. The above result sheds light on a long-standing puzzle concerning the relation between the entropic index $q$ and the underlying microscopic dynamics.

Power-law sensitivity to initial conditions within a logisticlike family of maps: Fractality and nonextensivity

Power-law sensitivity to initial conditions, characterizing the behaviour of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear 1D logistic-like maps $x_{t+1} = 1 - a | x_t |^z, (z > 1; 0 < a \le 2; t=0,1,2,...)$ The main ingredient of our approach is the generalized deviation law $\lim_{\Delta x(0) -> 0} \Delta x(t) / \Delta x(0)} = [1+(1-q)\lambda_q t]^{1/(1-q)}$ (equal to $e^{\lambda_1 t}$ for q=1, and proportional, for large t, to $t^{1/(1-q)}$ for $q \ne 1; q \in R$ is the entropic index appearing in the recently introduced nonextensive generalized statistics). The relation between the parameter q and the fractal dimension d_f of the onset-to-chaos attractor is revealed: q appears to monotonically decrease from 1 (Boltzmann-Gibbs, extensive, limit) to -infinity when d_f varies from 1 (nonfractal, ergodic-like, limit) to zero.

Power-law sensitivity to initial conditions—New entropic representation

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.

Criticality at the metastability limit in random-ferromagnet Ising models.

We have examined zero-temperature metastable states of random-bond Ising models (random ferromagnets and spin glasses, in one, two, and three dimensions) for criticality (power-law sensitivity to single-spin-flip perturbations) and for limit cycles, with the following results. We found no evidence of criticality in metastable states obtained by quenches from high temperature. Near the metastability limit in a magnetic field, random ferromagnets in two and three dimensions are critical; the metastable states were generated by starting from a ground state and ramping a contrary field. Under a cycled magnetic field, both spin glasses and random ferromagnets quickly enter simple limit cycles.

Resonance-sensitivity in dynamic Hopf bifurcations under fluid loading

Under steady fluid loading, elastic structures are liable to exhibit dynamic bifurcations to limit cycles: such unimodal instabilities are referred to as galloping while such multimodal instabilities are referred to as flutter. The trace of limit cycles energing from the critical equilibrium state can be either super-critical and stable, in analogy with a stable symmetric static bifurcation, or sub-critical and unstable, in analogy with an unstable symmetric static bifurcation. Galloping of a bluff body in a steady flow can be of the unstable type, and we might expect some form of imperfection sensitivity, although in contrast to static bifurcations, a Hopf bifurcation is actually topologically stable under the operation of a single control parameter: the form of the Hopf bifurcation cannot be rounded off or destroyed by imperfections as in the static case. However, since the dynamic instabilities are associated with a well defined and non-zero circular frequency we might expect the failure ‘load’ to be sensitive to resonant periodic forcing, and this is here shown to be the case, with a two-thirds power law sensitivity analogous to the static cusp. The conclusion is drawn that the concept of structural stability, vital as it is to good mathematical modelling, must be examined with care, particular attention being given to any restrictions on the class of allowable perturbations.