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Many papers used positive Lyapunov exponent as an indicator that a map has chaotic behavior and having sensitive dependence on initial conditions.

See for example:

  1. Hua, Z., Zhou, Y., Pun, C. M., & Chen, C. P. (2015). 2D Sine Logistic modulation map for image encryption. Information Sciences, 297, 80-94.

  2. Zhu, H., Zhao, Y., & Song, Y. (2019). 2D logistic-modulated-sine-coupling-logistic chaotic map for image encryption. IEEE Access, 7, 14081-14098.

However, there are papers shows that there are maps has positive Lyapunov exponent and do not have a sensitive dependence on initial conditions property, and vice versa.

See for example:

  1. Balibrea, F., & Caballero, M. V. (2013). Stability of orbits via Lyapunov exponents in autonomous and nonautonomous systems. International Journal of Bifurcation and chaos, 23(07), 1350127.

  2. Balibrea, F., & Caballero, M. V. (2014). Examples of Lyapunov Exponents in Two-Dimensional Systems. In Nonlinear Maps and their Applications (pp. 9-15). Springer, New York, NY.

The question: Does the Lyapunov exponent is a sufficient indicator of chaotic behavior and the sensitive dependence on initial consultations or not?

Crypt01
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  • See this question, as well as others. Short answer is no. The linear system $\dot{x} = x$ has a positive Lyapunov exponent but is clearly not chaotic. – Steve Heim Feb 23 '20 at 06:37
  • @SteveHeim Is this the case for discrete dynamical systems, which are the systems in the first two papers.? – Crypt01 Feb 23 '20 at 09:28
  • @SteveHeim and your example is unbounded which is not the case in the first two papers, sorry, this is not indicated in the question. – Crypt01 Feb 23 '20 at 17:57

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