How Common is Chaos?

J. C. Sprott
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
Received 1 July 1992; accepted for publication 18 November 1992

ABSTRACT

The solutions of about 4 x 10^7 low-dimensional, low-order polynomial maps and ordinary differential equations were classified as either fixed point, limit cycle, chaotic, or unstable. Of those cases for which the solutions are stable, representing candidate models for the real world, typically a few percent were found to be chaotic.

Ref: J. C. Sprott, Physics Letters A 173, 21-24 (1993)

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Table 1. The percentage of stable solutions of various types for maps and ordinary differential equations of different dimensions and orders.

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   Type   Dim   Order Fixed point    Limit cycle    Chaotic 
-------------------------------------------------------------------     
   map    1     2     38.14±0.88%    34.97±0.84%    26.90±0.74% 
   map    1     3     40.03±0.87%    36.21±0.83%    23.76±0.67% 
   map    1     4     42.44±0.90%    35.62±0.83%    21.95±0.65% 
   map    1     5     44.18±0.85%    33.17±0.73%    22.65±0.61% 
   map    2     2     50.09±0.76%    38.82±0.66%    11.10±0.36% 
   map    2     3     53.93±0.85%    36.28±0.69%     9.79±0.36% 
   map    3     2     57.24±0.59%    38.19±0.48%     4.57±0.17% 
   map    3     3     59.74±0.53%    36.35±0.41%     3.91±0.14% 
   map    4     2     60.48±0.44%    37.22±0.35%     2.29±0.09%   

   ODE    3     2     94.08±0.33%     5.54±0.08%     0.38±0.02%
   ODE    3     3     92.45±0.44%     7.09±0.12%     0.46±0.03%
   ODE    4     2     90.87±0.53%     8.46±0.16%     0.67±0.05%
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Fig. 1. Lyapunov exponent versus A for the logistic map showing that the system is chaotic (L > 0) over 13% of the stable region from -2 < L < 4.
[Figure 1]

Fig. 2. The fraction of stable solutions within a hypercube of size 2a centered on the origin (solid circles) and the fraction of the stable solutions that are chaotic (open circles) for two-dimensional quadratic maps.
[Figure 2]

Fig. 3. The fraction of stable solutions within a hypercube of size 2a centered on the origin (solid circles) and the fraction of the stable solutions that are chaotic (open circles) for three-dimensional quadratic maps.
[Figure 3]