American Institute of Mathematical Sciences

Advanced
October  2008, 20(4): 989-1011. doi: 10.3934/dcds.2008.20.989

Asymptotics of the Arnold tongues in problems at infinity

Victor Kozyakin 1, , Alexander M. Krasnosel’skii 2, and  Dmitrii Rachinskii 3,

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane 19, Moscow 127994 GSP-4, Russian Federation
2. 

Institute for Information Transmission Problems, Russian Academy of Sciences, 19 Bol.Karetny Lane, Moscow GSP-4, 127994, Russia; National Research University Higher School of Economics, 20 Myasnitskaya Street, Moscow 101000, Russian Federation
3. 

Department of Applied Mathematics, University College, Cork, Ireland

Received  February 2007 Revised  October 2007 Published  January 2008

We consider discrete time systems $x_{k+1}=U(x_{k};\lambda)$, $x\in\R^{N}$, with a complex parameter $\lambda$, and study their trajectories of large amplitudes. The expansion of the map $U(\cdot;\lambda)$ at infinity contains a principal linear term, a bounded positively homogeneous nonlinearity, and a smaller vanishing part. We study Arnold tongues: the sets of parameter values for which the large-amplitude periodic trajectories exist. The Arnold tongues in problems at infinity generically are thick triangles [4]; here we obtain asymptotic estimates for the length of the Arnold tongues and for the length of their triangular part. These estimates allow us to study subfurcation at infinity. In the related problems on small-amplitude periodic orbits near an equilibrium, similarly defined Arnold tongues have the form of narrow beaks. For standard pictures associated with the Neimark-Sacker bifurcation of smooth discrete time systems at an equilibrium, the Arnold tongues have asymptotically zero width except for the strong resonance points. The different shape of the tongues in the problem at infinity is due to the non-polynomial form of the principal homogeneous nonlinear term of the map $U(\cdot;\lambda)$: this form implies non-degeneracy of the nonlinear terms in the expansion of the map iterations and non-degeneracy of the corresponding resonance functions.
Keywords: periodic trajectory, discrete time system, Bifurcation at infinity, Poincare map, Arnold tongue, subfurcation, positively homogeneous nonlinearity, saturation, invariant set, rotation of vector fields., subharmonics.
Mathematics Subject Classification: 37G15, 34C25, 34B1.
Citation: Victor Kozyakin, Alexander M. Krasnosel’skii, Dmitrii Rachinskii. Asymptotics of the Arnold tongues in problems at infinity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 989-1011. doi: 10.3934/dcds.2008.20.989
[1]

Alexander Krasnosel'skii, Alexei Pokrovskii. On subharmonics bifurcation in equations with homogeneous nonlinearities. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 747-762. doi: 10.3934/dcds.2001.7.747
[2]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247
[3]

Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 107-116. doi: 10.3934/dcdss.2008.1.107
[4]

Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure & Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433
[5]

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809
[6]

Alessandro Fonda, Rafael Ortega. Positively homogeneous equations in the plane. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 475-482. doi: 10.3934/dcds.2000.6.475
[7]

Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165
[8]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623
[9]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020145
[10]

A.M. Krasnosel'skii, Jean Mawhin. The index at infinity of some twice degenerate compact vector fields. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 207-216. doi: 10.3934/dcds.1995.1.207
[11]

José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078
[12]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[13]

Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755
[14]

C. R. Chen, S. J. Li. Semicontinuity of the solution set map to a set-valued weak vector variational inequality. Journal of Industrial & Management Optimization, 2007, 3 (3) : 519-528. doi: 10.3934/jimo.2007.3.519
[15]

Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669
[16]

Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 451-470. doi: 10.3934/dcds.2003.9.451
[17]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[18]

Ying Zhang. Wave breaking and global existence for the periodic rotation-Camassa-Holm system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2243-2257. doi: 10.3934/dcds.2017097
[19]

Zhirong He, Weinian Zhang. Critical periods of a periodic annulus linking to equilibria at infinity in a cubic system. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 841-854. doi: 10.3934/dcds.2009.24.841
[20]

Pierre-Étienne Druet. Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences