December  2006, 16(4): 757-781. doi: 10.3934/dcds.2006.16.757

Dissipative and weakly--dissipative regimes in nearly--integrable mappings

1. 

Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy

2. 

Observatoire de Nice, B.P. 229, 06304 Nice Cedex 4, France, France

Received  May 2005 Revised  April 2006 Published  September 2006

We consider a dissipative standard map--like system, which is governed by two parameters measuring the strength of the dissipation and of the perturbation. In order to investigate the dynamics, we follow a numerical and an analytical approach. The numerical study relies on the frequency analysis and on the computation of the differential fast Lyapunov indicators. The analytical approach is based on the computation of a suitable normal form for dissipative systems, which allows us to derive an analytic expression of the frequency.
    We explore different kinds of attractors (invariant curves, periodic orbits, strange attractors) and their relation with the choice of the perturbing function and of the main frequency of motion (i.e., the frequency of the invariant trajectory of the unperturbed system). In this context we also investigate the occurrence of periodic attractors by looking at the relationship between their periods and the parameters ruling the mapping. Particular attention is devoted to the investigation of the weakly chaotic regime and its transition to the conservative case.
Citation: Alessandra Celletti, Claude Froeschlé, Elena Lega. Dissipative and weakly--dissipative regimes in nearly--integrable mappings. Discrete and Continuous Dynamical Systems, 2006, 16 (4) : 757-781. doi: 10.3934/dcds.2006.16.757
[1]

Dong Chen. Positive metric entropy in nondegenerate nearly integrable systems. Journal of Modern Dynamics, 2017, 11: 43-56. doi: 10.3934/jmd.2017003

[2]

Fuzhong Cong, Jialin Hong, Hongtian Li. Quasi-effective stability for nearly integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 67-80. doi: 10.3934/dcdsb.2016.21.67

[3]

Marcel Guardia. Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2829-2859. doi: 10.3934/dcds.2013.33.2829

[4]

Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. Persistence of lower dimensional elliptic invariant tori for a class of nearly integrable reversible systems. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 1237-1249. doi: 10.3934/dcdsb.2010.14.1237

[5]

Fuzhong Cong, Hongtian Li. Quasi-effective stability for a nearly integrable volume-preserving mapping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1959-1970. doi: 10.3934/dcdsb.2015.20.1959

[6]

Jianlu Zhang. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6419-6440. doi: 10.3934/dcds.2019278

[7]

Aristophanes Dimakis, Folkert Müller-Hoissen. Bidifferential graded algebras and integrable systems. Conference Publications, 2009, 2009 (Special) : 208-219. doi: 10.3934/proc.2009.2009.208

[8]

Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873

[9]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407

[10]

Carles Simó, Dmitry Treschev. Stability islands in the vicinity of separatrices of near-integrable symplectic maps. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 681-698. doi: 10.3934/dcdsb.2008.10.681

[11]

Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683

[12]

Shanshan Guo, Zhong Tan. Energy dissipation for weak solutions of incompressible liquid crystal flows. Kinetic and Related Models, 2015, 8 (4) : 691-706. doi: 10.3934/krm.2015.8.691

[13]

Danping Ding, Lixin Tian, Gang Xu. The study on solutions to Camassa-Holm equation with weak dissipation. Communications on Pure and Applied Analysis, 2006, 5 (3) : 483-492. doi: 10.3934/cpaa.2006.5.483

[14]

Michele Coti Zelati, Piotr Kalita. Smooth attractors for weak solutions of the SQG equation with critical dissipation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1857-1873. doi: 10.3934/dcdsb.2017110

[15]

Sonomi Kakizaki, Akiko Fukuda, Yusaku Yamamoto, Masashi Iwasaki, Emiko Ishiwata, Yoshimasa Nakamura. Conserved quantities of the integrable discrete hungry systems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 889-899. doi: 10.3934/dcdss.2015.8.889

[16]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109

[17]

Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 61-84. doi: 10.3934/dcds.2001.7.61

[18]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001

[19]

Álvaro Pelayo, San Vű Ngọc. First steps in symplectic and spectral theory of integrable systems. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3325-3377. doi: 10.3934/dcds.2012.32.3325

[20]

Boris S. Kruglikov and Vladimir S. Matveev. Vanishing of the entropy pseudonorm for certain integrable systems. Electronic Research Announcements, 2006, 12: 19-28.

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (78)
  • HTML views (0)
  • Cited by (7)

[Back to Top]