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Nonlinear stiffness, Lyapunov exponents, and attractor dimension. (English) Zbl 0949.37014
Summary: The author proposes that stiffness may be defined and quantified for nonlinear systems using Lyapunov exponents, and demonstrates the relationship that exists between stiffness and the fractal dimension of a strange attractor: That stiff chaos is thin chaos.

##### MSC:
 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 34D08 Characteristic and Lyapunov exponents of ordinary differential equations 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
##### Keywords:
Lyapunov exponents; fractal dimension; strange attractor; chaos
RODAS
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##### References:
 [1] J.H.E. Cartwright, O. Piro, The dynamics of Runge-Kutta methods, Int. J. Bifurcation and Chaos 2 (1992) 427-49. · Zbl 0876.65061 [2] R.C. Aiken, editor, Stiff Computation, Oxford University Press, 1985. [3] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer, 2nd ed., 1996. · Zbl 0859.65067 [4] R.M. Corless, Defect-controlled numerical methods and shadowing for chaotic differential equations, Physica D 60 (1992) 323-334. · Zbl 0779.34035 [5] J.D. Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, 1991. · Zbl 0745.65049 [6] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.A. Vetterling, Numerical Recipes in C, Cambridge University Press, 2nd ed., 1992. · Zbl 0778.65003 [7] B. van der Pol, On relaxation-oscillations, Phil. Mag. (7) 2 (1927) 978-992. · JFM 52.0450.05 [8] M.L. Cartwright, J.E. Littlewood, On nonlinear differential equations of the second order. I. The equation $$ÿ+k(1−y\^{}\{2\})ẏ+y=bλk cos(λt+a)$$, k large, J. Lond. Math. Soc. 20 (1945) 180-189. · Zbl 0061.18903 [9] E.A. Jackson, Perspectives of Nonlinear Dynamics, Vol. 1, Cambridge University Press, 1989. · Zbl 0701.70001 [10] K. Tomita, Periodically forced nonlinear oscillators, in: A.V. Holden (Ed.) Chaos, Manchester University Press, 1986. [11] J.M.T. Thompson, H.B. Stewart, Nonlinear Dynamics and Chaos, Wiley, 1986. · Zbl 0601.58001 [12] J.L. Kaplan, J.A. Yorke, Chaotic Behavior of Multidimensional Difference Equations, Vol. 730 of Lecture Notes in Mathematics, pp. 204-227, Springer, 1979. · Zbl 0448.58020 [13] P. Grassberger, I. Procaccia, Measuring the strangeness of strange attractors, Physica D 9 (1983) 189-208. · Zbl 0593.58024 [14] R.H. Abraham, C.D. Shaw (Eds.) Dynamics: The Geometry of Behavior, Addison-Wesley, 1992. [15] E.A. Jackson, Perspectives of Nonlinear Dynamics, Vol. 2, Cambridge University Press, 1990. · Zbl 0769.58018 [16] R. Shaw, Strange attractors, chaotic behavior, and information flow, Z. Naturforsch. 36a (1981) 80-112. · Zbl 0599.58033
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