# J. C. Sprott

Department of Physics, University of Wisconsin, Madison, WI 53706, USA
April 18, 1998
(Revised November 2, 2004)

Below are a number of common chaotic systems and their parameters (some representing new previously unpublished calculations), collected here for convenience.  The least significant digit is only a best estimate.  A good project would be to improve the precision of the values and to add other cases.

### Logistic map

• Xn+1 = AXn(1 - Xn)
• Usual parameter: A = 4
• Lyapunov exponent (base-e): l = ln(2) = 0.693147181...
• Kaplan-Yorke dimension: DKY = 1.0 (exact value)
• Correlation dimension: D2 = 1.0 (exact value, converges slowly)
• Ref: R. May, Nature 261, 45-67 (1976)

### Hénon map

• Xn+1 = 1 + Yn - aXn2
• Yn+1 = bXn
• Usual parameters: a = 1.4, b = 0.3
• Lyapunov exponents (base-e): l = 0.41922, -1.62319
• Kaplan-Yorke dimension: DKY = 1.25827
• Correlation dimension: D2 = 1.220 + 0.036
• Ref: M. Hénon, Commun. Math. Phys. Phys. 50, 69-77 (1976)

### Chirikov (standard) map

• Xn+1 = Xn + Yn+1 mod 2pi
• Yn+1 = Yn + k sin Xn mod 2pi
• Usual parameter: k = 1
• Lyapunov exponents (base-e): l = 0.10497, -0.10497
• Kaplan-Yorke dimension: DKY = 2.0 (exact value)
• Correlation dimension: D2 = 1.954 + 0.077
• Ref: B. V. Chirikov, Physics Reports 52, 263-379 (1979)

### Lorenz attractor

• dx/dt = s(y - x)
• dy/dt = -xz + rx - y
• dz/dt = xy - bz
• Usual parameters: s = 10, r = 28, b = 8/3
• Lyapunov exponents (base-e): l = 0.9056, 0, -14.5723
• Kaplan-Yorke dimension: DKY = 2.06215
• Correlation dimension: D2 = 2.068 + 0.086
• Ref: E. N. Lorenz, J. Atmos. Sci. 20, 130-141 (1963)

### Rössler attractor

• dx/dt = -y - z
• dy/dt = x + ay
• dz/dt = b + z(x - c)
• Usual parameters: a = b = 0.2, c = 5.7
• Lyapunov exponents (base-e): l = 0.0714, 0, -5.3943
• Kaplan-Yorke dimension: DKY = 2.0132
• Correlation dimension: D2 = 1.991 + 0.065 (converges slowly)
• Ref: O. E. Rössler, Phys. Lett. 57A, 397-398 (1976)

### Ueda attractor

• dx/dt = y
• dy/dt = -x3 - ky + B sin z
• dz/dt = 1
• Usual parameters: B = 7.5, k = 0.05
• Lyapunov exponents (base-e): l = 0.1034, 0, -0.1534
• Kaplan-Yorke dimension: DKY = 2.6741
• Correlation dimension: D2 = 2.675 + 0.132
• Ref: Y. Ueda, J. Stat. Phys. 20, 181-196 (1979)
• ### Simplest quadratic dissipative chaotic flow

• dx/dt = y
• dy/dt = z
• dz/dt = -Az + y2 - x
• Usual parameter: A = 2.017
• Lyapunov exponents (base-e): l = 0.0551, 0, -2.0721
• Kaplan-Yorke dimension: DKY = 2.0266
• Correlation dimension: D2 = 2.187 + 0.075 (converges slowly)
• Ref: J. C. Sprott, Phys. Lett. A 228, 271-274 (1977)
• ### Simplest piecewise linear dissipative chaotic flow

• dx/dt = y
• dy/dt = z
• dz/dt = -Az - y - |x| + 1
• Usual parameter: A = 0.6
• Lyapunov exponents (base-e): l = 0.0362, 0, -0.6362
• Kaplan-Yorke dimension: DKY = 2.0569
• Correlation dimension: D2 = 2.131 + 0.072 (converges slowly)
• Ref: S. J. Linz and J. C. Sprott, Phys. Lett. A 259, 240-245 (1999)

• A more extensive list of such systems is included in the paper Improved Correlation Dimension Calculation, and an even more extensive list (62 cases) is in Appendix A of the book Chaos and Time-Series Analysis by J. C. Sprott (Oxford University Press, 2003).