|Paper title:||Dynamic Lyapunov Indicator (DLI): A Perfect Indicator for Evolutionary System|
|Published in:||Issue 2, (Vol. 10) / 2016Download|
|Author(s):||SAHNI Niteesh, SARMA Til Prasad, SAHA Lal Mohan|
|Abstract.||Recently proposed indicators along with their abilities for identification of chaotic motion have been described. These are Fast Lyapunov Exponents (FLI), Smaller Alignment Indices (SALI) and Dynamic Lyapunov Indicator (DLI). Working performances of these indicators are verified for some two and three dimensional systems. For clear identification of regular and chaotic motion, Lyapunov exponents (LCEs) are obtained and represented through plots. Numerical data for indicators FLI, SALI and DLI have been calculated and represented through plots and nature of these plots are analyzed and discussed. This leads to establish the efficiency of these indicators. Investigations made here reveal that the indicator DLI works perfectly and FLI and SALI are not consistent as per their definitions.|
|Keywords:||Attractors, Lyapunov Exponents, Bifurcation, Period Adding, Chaos Indicators|
1. H. Poincaré, New Methods of Celestial Mechanics (1892), ISBN 1563961172 (3 vols.; Eng. trans., (1967))
2. H. Poincaré, The Foundations of Science, (1902–1908), New York: Science Press; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908).
3. E. N. Lorenz, Deterministic nonperiodic flows, J. Atmos. Sci., 20, (1963) 130-141.
4. Robert M. May, Simple mathematical models with very complicated dynamics, Nature, 261, (1976) 459 – 467.
5. Hao-Bai-Lin, Chaos II. (1990), World Scientific.
6. C. Froeschle, R. Gonczi and E. Lega, The Fast Lyapunov Indicator: A simple tool to detect weak chaos, Application to the structure of the main asteroidal belt, Space Sci. 45, (1997)881 – 886.
7. E. Lega, C. Froeschle, On the Relationship between Fast Lyapunov Indicator and Periodic Orbits for Symplectic Mappings, Celestial Mechanics and Dynamical Astronomy,81, (2001) 129-147.
8. Ch. Skokos, Alignment indices: a new simple method for determining the ordered or chaotic nature of orbits? J. Phys. A: Math Gen., 34, (2001) 10029 – 10043.
9. Ch. Skokos, Ch. Antonopoulos, T. C. Bountis and M. N. Vrahatis, Detecting order and chaos in Hamiltonian systems by SALI method, J. Phys. A: Math. Gen. 37, (2004) 6269 – 6284.
10. L. M. Saha, and M. Budhraja, The Largest Eigenvalue: An Indicator of Chaos? Int.J. of Appl.Math and Mech, 3(1), (2007) 61-71.
11. Dumitru Deleanu, Dynamic Lyapunov Indicator: a practical tool for distinguishing between ordered and chaotic orbits in discrete dynamical systems, (2012), Recent Researches in Computational Techniques, Non-Linear Systems and Control, 117-122.
12. L. M. Saha and R. Tehri, Application of recent indicators of regularity and chaos to discrete maps, Int. J. of Appl. Math and Mech. 6 (1) (2010) 86-93.
13]Mridula Budhraja, Narender Kumar and L. M. Saha, The 0-1 Test Applied to Peter-de-Jong Map (2012). IJEIT, 2, (2012) No. 6 253 – 257.
14. A. Arneodo, P. Coullet and C. Tresser, A renormalization group with periodic behavior, (1979), Phys. Lett. 70 A, 74 –76.
15. A. Arneodo, P. Coullet and C. Tresser, Occurrence of strange attractors in three – dimensional Volterra equations, (1980), Phys. Lett. 79 A, 259 – 263.
16. A. Arneodo, P. Coullet and C. Tresser, A possible new mechanism for onset of Turbulence, (1981), Phys. Lett. 81 A, 197 – 201.
17. A. Arneodo, P. Coullet and C. Tresser, Oscillations with chaotic behavior: an illustration of the theorem by Shilnikov, (1982), J. Stat. Phys. 27, 171 – 182.
18. RobertVan Buskirk and Carson Jeffries, Observation of chaotic dynamics of coupled nonlinear oscillators, (1985) Phys. Rev. A, 31, no. 5, (1985) 3332-3357.
19. S. Ushiki, Sur les liaisons-cols des systèmes dynamiques analytiques, (1980), C. R. Acad. Sci. Paris, 291(7):447–449.
20. R. Q. Grafton and J. Silva-Echenique, How to Manage Nature? Strategies, Predator-Prey Models, and Chaos, Marine Resource Economics, 12, (1997) 127–143.
21. Abd-Elalim A. Elsadany, Dynamical complexities in a discrete-time food chain. Computational Ecology and Software, 2(2) (2012) 124-139.
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