Dimensions and Entropies in Chaotic Systems pp 158-170 | Cite as

# Characterization of Chaotic Instabilities in an Electron-Hole Plasma in Germanium

## Abstract

Helical instabilities in an electron-hole plasma in Ge in parallel dc electric and magnetic fields are known to exhibit chaotic behavior. By fabricating probe contacts along the length of a Ge crystal we study the spatial structure of these instabilities, finding two types: (i) spatially coherent and temporally chaotic helical density waves characterized by strange attractors of measured fractal dimension d ~ 3, and (ii) beyond the onset of spatial incoherence, instabilities of indeterminately large fractal dimension d ≥ 8. In the first instance, calculations of the fractal dimension provide an effective means of characterizing the observed chaotic instabilities. However, in the second instance, these calculations do not provide a means of determining whether the observed plasma turbulence is of stochastic or of deterministic (i.e., chaotic) origin.

## Keywords

Phase Space Fractal Dimension Chaotic System Chaotic Dynamic Strange Attractor## Preview

Unable to display preview. Download preview PDF.

## References

- 1.for example, H. L. Swinney: Physica (Utrecht)
**7D**, 3 (1983);ADSGoogle Scholar - see also
*The Physics of Chaos and Related Problems*, edited by S. Lundqvist, Phys. Scr. T9 (1985).Google Scholar - 2.D. Ruelle and F. Takens: Comm. Math. Phys.
**20**, 167 (1971);MathSciNetADSMATHCrossRefGoogle Scholar - E. Ott: Rev. Mod. Phys.
**53**, 655 (1981).MathSciNetADSMATHCrossRefGoogle Scholar - 3.for example, J. D. Farmer, E. Ott, J. A. Yorke: Physica (Utrecht)
**7D**, 153 (1983).MathSciNetADSGoogle Scholar - 4.A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano: Physica
**16D**, 285 (1985).MathSciNetADSGoogle Scholar - 5.J. P. Crutchfield and N. H. Packard: Int. J. Theor. Phys.
**21**, 433 (1982);MathSciNetMATHCrossRefGoogle Scholar - Physica 7D, 201 (1983);MathSciNetADSGoogle Scholar
- P. Grassberger and I. Procaccia: Phys. Rev. A
**28**, 2591 (1983).ADSCrossRefGoogle Scholar - 6.I. L. Ivanov and S. M. Ryvkin: Zh. Tekh. Fiz. 28, 774 (1958)Google Scholar
- [Sov. Phys. Tech. Phys.
**3**, 722 (1958)].Google Scholar - 7.C. E. Hurwitz and A. L. McWhorter: Phys. Rev.
**134**, A1033 (1964).ADSCrossRefGoogle Scholar - 8.G. A. Held, C. Jeffries and E. E. Haller: Phys. Rev. Lett. 52, 1037 (1984);ADSCrossRefGoogle Scholar
*Proceedings of the Seventeenth International Conference on the Physics of Semiconductors, San Francisco, 1984*, edited by D. J. Chadi and W. A. Harrison (Springer-Verlag, New York, 1985), p. 1289.Google Scholar- 9.G. A. Held and C. Jeffries: Phys. Rev. Lett. 55, 887 (1985).ADSCrossRefGoogle Scholar
- 10.for example, J. P. Gollub and S. V. Benson in
*Pattern Formation and Pattern Recognition*edited by H. Haken (Springer-Verlag, Berlin, 1979), p. 74.CrossRefGoogle Scholar - 11.N. H. Packard, J. P. Crutchfield, J. D. Farmer and R. S. Shaw: Phys. Rev. Lett. 45, 712(1980).Google Scholar
- 12.Unconventional notation is used in Eq. (1) to avoid confusion with the notation of Eq. (2).Google Scholar
- 13.P. Grassberger and I. Procaccia: Phys. Rev. Lett. 50, 346 (1983).MathSciNetADSCrossRefGoogle Scholar
- 14.H. G. E. Hentschel and I. Procaccia: Physica
**8D**, 435 (1983).MathSciNetADSGoogle Scholar - 15.A. Brandstäter
*et al*: Phys. Rev. Lett. 51, 1442 (1983).MathSciNetADSCrossRefGoogle Scholar - 16.H. L. Swinney and J. P. Gollub: to appear in Physica D.Google Scholar
- 17.This has been observed in calculations of fractal, information and correlation dimensions for a system consisting of a driven p-n junction in series with an inductor and a resistor. G. A. Held and C. Jeffries, unpublished.Google Scholar
- 18.F. Takens: “Detecting Strange Attractors in Turbulence”, in
*Lecture Notes in Mathematics 898*, edited by D. A. Rand and L. S. Young (Springer-Verlag, Berlin, 1981), p. 366.Google Scholar - 19.H. S. Greenside, A. Wolf, J. Swift and T. Pignaturo: Phys. Rev. A 25, 3453 (1982).ADSGoogle Scholar
- 20.The pointwise dimension is defined in reference 3. Similar definitions of dimension have been given in references 11, 20, 21, and 31 therein.Google Scholar
- 21.This is the method used by A. Brandstater
*et al*, in reference 15. See also reference 13.Google Scholar - 22.A. Ben-Mizrachi, I. Procaccia and P. Grassberger: Phys. Rev. A
**29**, 975 (1984).ADSCrossRefGoogle Scholar - 23.A fractal dimension of this magnitude (~10
^{10}) is experimentally unattain able for two reasons. First, an inordinate number of data points would be required, as discussed in Section 4. Second, one would need to measure signals with a very large bandwidth Δf ~ 1/τ, where τ is the shortest fluctuation time; typically Δf ~ 10^{9}— 10^{14}Hz for stochastic noise in conducting media. Thus, fractal dimensions of this magnitude are operationally meaningless.Google Scholar - 24.B. B. Mandelbrot:
*The Fractal Geometry of Nature*(W. H. Freeman and Company, New York, 1983), p.310.Google Scholar - 25.S. Ciliberto and J. P. Gollub: J. Fluid Mech. 158, 381 (1985).MathSciNetADSCrossRefGoogle Scholar
- 26.H. Froehling, J. P. Crutchfield, J. D. Farmer, N. H. Packard, and R. Shaw: Physica
**3D**, 605 (1981).MathSciNetADSGoogle Scholar