<title>Fractal dimension and nonlinear dynamical processes</title>

The concept of industry dynamism claims a central role in models of organizational adaptation. However, the development of new ways to study it has waned. Building on the literatures on nonlinear dynamical systems and information complexity, we introduce a fractal approach as a useful lens to industry dynamism and a fresh alternative and complement to prevailing approaches. This differs conceptually from existing methods in highlighting nonlinearity and recognizing endogenous and stable sources of apparent unpredictability. Further, it uses the fractal dimension, a measure of the jaggedness in a time series, which offers several advantages over existing dispersion-based measures of unpredictability. We apply the fractal approach in an exploratory longitudinal study of the turbulent US network television industry and demonstrate its ability to uncover distinct aspects of industry dynamism.

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Chaos in driven Alfvén systems: unstable periodic orbits and chaotic saddles

Nonlinear Processes in Geophysics ◽

10.5194/npg-14-17-2007 ◽

2007 ◽

Vol 14 (1) ◽

pp. 17-29 ◽

Cited By ~ 15

Author(s):

A. C.-L. Chian ◽

W. M. Santana ◽

E. L. Rempel ◽

F. A. Borotto ◽

T. Hada ◽

...

Keyword(s):

Periodic Orbits ◽

Chaotic Attractor ◽

Unstable Periodic Orbits ◽

Gap Filling ◽

Intermittent Turbulence ◽

Dynamical Processes ◽

Nonlinear Dynamical ◽

Derivative Nonlinear Schrödinger Equation ◽

Low Dimensional ◽

Chaotic Saddle

Abstract. The chaotic dynamics of Alfvén waves in space plasmas governed by the derivative nonlinear Schrödinger equation, in the low-dimensional limit described by stationary spatial solutions, is studied. A bifurcation diagram is constructed, by varying the driver amplitude, to identify a number of nonlinear dynamical processes including saddle-node bifurcation, boundary crisis, and interior crisis. The roles played by unstable periodic orbits and chaotic saddles in these transitions are analyzed, and the conversion from a chaotic saddle to a chaotic attractor in these dynamical processes is demonstrated. In particular, the phenomenon of gap-filling in the chaotic transition from weak chaos to strong chaos via an interior crisis is investigated. A coupling unstable periodic orbit created by an explosion, within the gaps of the chaotic saddles embedded in a chaotic attractor following an interior crisis, is found numerically. The gap-filling unstable periodic orbits are responsible for coupling the banded chaotic saddle (BCS) to the surrounding chaotic saddle (SCS), leading to crisis-induced intermittency. The physical relevance of chaos for Alfvén intermittent turbulence observed in the solar wind is discussed.

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