ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Astafyev G. B., Koronovskii A. A., Hramov A. E., Hramova A. E. Transient processes in Henon map: Part 1. Periodical dynamics. Izvestiya VUZ. Applied Nonlinear Dynamics, 2003, vol. 11, iss. 4, pp. 124-147. DOI: 10.18500/0869-6632-2003-11-4-124-147

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Language: 
Russian
Heading: 
Methodical Papers on Nonlinear Dynamics
Article type: 
Article
UDC: 
517.9
DOI: 
10.18500/0869-6632-2003-11-4-124-147

Transient processes in Henon map: Part 1. Periodical dynamics

Autors: 
Astafyev Gennadiy Borisovich, Saratov State University
Koronovskii Aleksei Aleksandrovich, Saratov State University
Hramov Aleksandr Evgenevich, Immanuel Kant Baltic Federal University
Hramova Anastasia Evgenevna, Saratov State University
Abstract: 

In this рарег we analyze the transient processes in two-dimensional dynamical system with discrete time (Henon map) demonstrating periodical oscillations. The method оf calculation оf the transient process duration and its dependence оn the accuracy of determination of the process and on the initial conditions is considered. Mechanisms leading to complication of this dependence are revealed. In the second part of the work we plan to consider the dependence of the mean duration of the transient process when controlling parameters change.

Key words: 
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Acknowledgments: 
This work was supported by the Russian Foundation Basic Research (projects 01-02-17392 and 02-02-16351).
Reference: 
  1. Venkatesan А, Lakshmanan M. Different routes to chaos via strange nonchaotic attractors in а quasiperiodically forced system. Phys. Rev. E. 1998;58(3):3008-3016. DOI: 10.1103/PhysRevE.58.3008.
  2. Depassier MC, Murа J. Variational approach to а class of nonlinear oscillators with several limit cycles. Phys. Rev. Е. 2001;64(5):056217. DOI: 10.1103/PhysRevE.64.056217.
  3. Kuznetsov AP, Kuznetsov SP. Critical dynamics for one-dimensional maps part 1: feigenbaum's scenario. Izvestiya VUZ. Applied Nonlinear Dynamics. 1993;1(1):15-33 (in Russian).
  4. Kuznetsov AP, Kuznetsov SP, Sataev IR. Critical dynamics of one-dimensional mappings. Part 2. Two-parameter transition to chaos. Izvestiya VUZ. Applied Nonlinear Dynamics. 1993;1(3):17-35 (in Russian).
  5. Kuznetsov AP, Savin AV. On the problem of the boundary of chaos and typical structures on the parameter plane of non-autonomous discrete mappings with period doublings. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(4):25-36 (in Russian).
  6. Koronovsky AA, Trubetskov DI, Khramov AE. Influence of an external signal on self-oscillations in the distributed system “helical electron beam—backward electromagnetic wave”. Radiophysics and Quantum Electronics. 2002;45(9):706–724. DOI: 10.1023/A:1021797418844.
  7. Kal’yanov EV. Transients in an autostochastic oscillator with delayed feedback. Tech. Phys. Lett. 2000;26(8):656–658. DOI: 10.1134/1.1307804.
  8. Bezruchko BP, Dikanev TV, Smirnov DA. Role оf transient processes for reconstruction оf model equations from time series. Phys. Rev. Е. 2001;64(3):036210. DOI: 10.1103/PhysRevE.64.036210.
  9. Bezruchko BP, Dikanev TV, Smirnov DA. Global reconstruction of the equations of a dynamic system based on the temporal implementation of the transition process. Izvestiya VUZ. Applied Nonlinear Dynamics. 2001;9(3):3 (in Russian).
  10. Koronovsky AA, Trubetskov DI, Khramov AE, Khramova AE. Universal scaling laws of transient processes. Proc. Acad. Sci. USSR. 2002;383(3):322 (in Russian).
  11. Pisarchik AN. Controlling the multistability of nonlinear systems with coexisting attractors. Phys. Rev. Е. 2001;64(4):046203. DOI: 10.1103/physreve.64.046203.
  12. Morua APS, Grebogi С. Output functions and fractal dimensions in dynamical systems. Phys. Rev. Lett. 2001;86(13):2778-2781. DOI: 10.1103/PhysRevLett.86.2778.
  13. Janosi IM, Tél Т. Time-series analysis оf transient chaos. Phys. Rev. Е. 1994;49(4):2756-2763. DOI: 10.1103/PhysRevE.49.2756.
  14. Dhamala M, Lаi Y-C, Kostelich EJ. Detecting unstable periodic orbits from transient chaotic time series. Phys. Rev. Е. 2000;61(6):6485-6489. DOI: 10.1103/physreve.61.6485.
  15. Dhalama M, Lai Y-С, Kostelich EJ. Analyses of transient chaotic time series. Phys. Rev. Е. 2001;64(5):056207. DOI: 10.1103/PhysRevE.64.056207.
  16. Zhu L, Raghu A, Lai Y-C. Experimental observation of superpersistent chaotic transients. Phys. Rev. Lett. 2001;86(18):4017-4020. DOI: 10.1103/PhysRevLett.86.4017.
  17. Grebogi C, Оtt E, Yorke JA. Chaotic attractors in crisis. Phys. Rev. Lett. 1982;48(22):1507-1510. DOI: 10.1103/PhysRevLett.48.1507.
  18. Grebogi C, Ott E, Romeiras F, Yorke JA. Critical exponents for crisis-induced intermittency. Phys. Rev. А. 1987;36(11):5365-5380. DOI: 10.1103/physreva.36.5365.
  19. Szabé KG, Lai Y-C, Tel T, Grebogi С. Topological scaling and gap filling at crisis. Phys. Rev. Е. 2000;61(5):5019-5032. DOI: 10.1103/PhysRevE.61.5019.
  20. Stewart HB, Ueda Y. Double crisis in two-parameter dynamical system. Phys. Rev. Lett. 1995;75(13):2478-2481. DOI: 10.1103/PhysRevLett.75.2478.
  21. Gallas JAC, Grebogi C, Yorke JA. Vertices in parameter space: double crisis which destroy chaotic attractors. Phys. Rev. Lett. 1993;71(9):1359-1362. DOI: 10.1103/PhysRevLett.71.1359.
  22. Meucci R, Gadomski W, Ciofini M, Arecchi FT. Transient statistics in stabilizing periodic orbit. Phys. Rev. Е. 1995;52(5):4676-4680. DOI: 10.1103/PhysRevE.52.4676.
  23. Aston PJ, Marriot PK. Waiting time paradox applied to transient times. Phys. Rev. Е. 1998;57(1):1181-1182. DOI: 10.1103/PhysRevE.57.1181.
  24. Paar V, Buljan H. Bursts in the chaotic trajectory lifetimes preceding controlled periodic motion. Phys. Rev. Е. 2000;62(4):4869-4872. DOI: 10.1103/PhysRevE.62.4869.
  25. Koronovsky AA, Trubetskov DI, Khramov AE, Khramova AE. Universal rules of transient processes. Radiophysics and Quantum Electronics. 2002;45(10):806–812. DOI: 10.1023/A:1022436501970.
  26. Koronovsky AA, Khramov AE, Khromova IA. Average duration of transient processes in dynamic systems with discrete time. Izvestiya VUZ. Applied Nonlinear Dynamics. 2003;11(1):36-46 (in Russian).
  27. Henon M. On the numerical computation of Poincaré maps. Physica D. 1982;5(2-3):412-414. DOI: 10.1016/0167-2789(82)90034-3.
  28. Kaufmann Z, Lustfeld H. Comparison of averages of flows and maps. Phys. Rev. E. 2001;64(5):055206. DOI: 10.1103/physreve.64.055206.
  29. Hénon M. A two-dimensional mapping with a strange attractor. Commun. Math. Phys. 1976;50(1):69-77. DOI: 10.1007/BF01608556.
  30. Hénon M. Two-dimensional mapping with a strange attractor. In: Shilnikov LP, Sinai YG, editors. Strange Attractors. Moscow: Mir; 1981. P. 152 (in Russian).
  31. Koronovsky AA, Starodubov AB, Khramov AE. Methodology for determining the duration of the transition process for a dynamic system with discrete time, which is in the mode of chaotic oscillations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2002;10(5):25 (in Russian).
  32. Koronovskii AA, Starodubov AV, Khramov AE. A method for determining the transient process duration in dynamic systems in the regime of chaotic oscillations. Tech. Phys. Lett. 2003;29(4):323–326. DOI: 10.1134/1.1573304.
  33. Anishchenko VS. Attractors of dynamic systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(1):109 (in Russian).
  34. Efimov NV. Quadratic Forms and Matrices. Moscow: Fizmatgiz; 1963. 168 p. (in Russian).
  35. Ango A. Mathematics for Electrical and Radio Engineers. Moscow: Nauka; 1967. 780 p. (in Russian).
  36. Grebogi C, Ott E, Yorke JA. Fractal basin boundaries, long lived chaotic trancients, and unstable-unstable pair bifurcation. Phys. Rev. Lett. 1983;50(13):935-938. DOI: 10.1103/PhysRevLett.50.935.
  37. Grebogi C, Ott E, Jorke JA. Metamorphoses оf basin boundaries in nonlinear dynamical systems. Phys. Rev. Lett. 1986;56(10):1011-1014. DOI: 10.1103/physrevlett.56.1011.
  38. Parker TS, Chua LO. Practical Numerical Algorithms for Chaotic Systems. Berlin: Springer-Verlag; 1989. 348 p. DOI: 10.1007/978-1-4612-3486-9.
  39. You ZP, Kostelich EJ, Yorke JA. Calculating stable and unstable manifolds. Int. J. Bifurc. Chaos. 1991;1(1):605-623.
  40. Kostelich EJ, Yorke JA, You Z. Plotting stable manifolds: error estimates and noninvertible maps. Physica D. 1996;93(3-4):210-222. DOI: 10.1016/0167-2789(95)00309-6.
  41. Kuznetsov SP. Dynamic Chaos. Moscow: Fizmatlit; 2001. 296 p. (in Russian).
  42. Koronovskii AA, Khramov AE. Variation of the dependence of the transient process duration on the initial conditions in systems with discrete time. Tech. Phys. Lett. 2002;28(8):648–651. DOI: 10.1134/1.1505539.
Received: 
28.02.2003
Accepted: 
05.03.2003
Available online: 
30.11.2023
Published: 
31.12.2003
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