We study the nature of electronic states in a tight-binding one-dimensional model with the on-site energies exhibiting long-range correlated disorder and nonrandom hopping amplitudes. The site energies describe the trace of a fractional Brownian motion with a specified spectral density $S\left(k\right)\propto 1/k^{\alpha }$. Using a renormalization group technique, we show that for long-range correlated energy sequences with persistent increments ( $\alpha >2\mathrm{}$) the Lyapunov coefficient (inverse localization length) vanishes within a finite range of energy values revealing the presence of an Anderson-like metal-insulator transition.

Power-law sensitivity to the initial conditions at the edge of chaos provides a natural relation between the scaling properties of the dynamics attractor and its degree of nonextensivity within the generalized statistics recently introduced by one of the authors (C.T.) and characterized by the entropic index $q$. We show that general scaling arguments imply that $1/(1-q)=1\mathrm{}/{\alpha }_{\min }-1\mathrm{}/{\alpha }_{\max }$, where ${\alpha }_{\min }$ and ${\alpha }_{\max }$ are the extremes of the multifractal singularity spectrum $f\left(\alpha \right)$ of the attractor. This relation is numerically verified in standard $D=1\mathrm{}$ dissipative maps.

An important application involving two-species reaction-diffusion systems relates to the problem of finding the best statistical strategy for optimizing the encounter rate between organisms. We investigate the general problem of how the encounter rate depends on whether organisms move in Lévy or Brownian random walks. By simulating a limiting generalized searcher-target model (e.g., predator-prey, mating partner, pollinator-flower), we find that Lévy walks confer a significant advantage for increasing encounter rates when the searcher is larger or moves rapidly relative to the target, and when the target density is low.

Power-law sensitivity to initial conditions, characterizing the behavior of dynamical systems at their critical points (where the standard Liapunov exponent vanishes), is studied in connection with the family of nonlinear one-dimensional logisticlike maps $x_{t+1\mathrm{}}=1\mathrm{}-a|x_t{|}^z$ ($z>\mathrm{}1;$ $0<a<~2;$ $t=0,1,2,\dots$). The main ingredient of our approach is the generalized deviation law $limop{}_{\Delta x\left(0\right)\mathrm{}\to 0}\left[\Delta x\right(t)/\Delta x(0\left)\mathrm{}\right]=[1+(1-q){\lambda }_{q}t{]}^{1/\left(1-q\right)}$ (equal to $e^{{\lambda }_{1}t}$ for $q=1,$ and proportional, for large $t,$ to $t^{1/\left(1-q\right)}$ 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 $-\infty$ when $d_f$ varies from 1 (nonfractal, ergodiclike, limit) to zero.