American Institute of Mathematical Sciences

Advanced
April  1998, 4(2): 379-391. doi: 10.3934/dcds.1998.4.379

Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space

Paolo Perfetti 1,

1. 

Dipartimento di matematica II Università di Roma, via della Ricerca Scientifica 00133 Roma

Received  August 1997 Published  February 1998

We consider the hamiltonian $H=1/2(I_1^2+I_2^2)+\varepsilon(\cos\varphi_1-1) (1+\mu(\sin\varphi_2+\cos t))$ $I\in\mathbb{R}^2$ ("Arnol'd model about diffusion"); by means of fixed point theorems, the existence of the stable and unstable manifolds (whiskers) of invariant, "a priori unstable tori", for any vector-frequency $(\omega,1)\in\mathbb{R}^2$ is proven. Our aim is to provide detailed proofs which are missing in Arnol'd's paper, namely prove the content of the Assertion B pag.583 of [A]. Our proofs are based on technical tools suggested by Arnol'd i.e. the contraction mapping method together with the "conical metric" (see the footnote ** of pag. 583 of [A]).
Keywords: Fixed point theorems, Arnol'd model..
Mathematics Subject Classification: 37C2.
Citation: Paolo Perfetti. Fixed point theorems in the Arnol'd model about instability of the action-variables in phase-space. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 379-391. doi: 10.3934/dcds.1998.4.379
[1]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248
[2]

Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1947-1955. doi: 10.3934/dcdss.2020152
[3]

Nicholas Long. Fixed point shifts of inert involutions. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297
[4]

Zhihong Xia, Peizheng Yu. A fixed point theorem for twist maps. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022045
[5]

Sebastián Ferrer, Francisco Crespo. Parametric quartic Hamiltonian model. A unified treatment of classic integrable systems. Journal of Geometric Mechanics, 2014, 6 (4) : 479-502. doi: 10.3934/jgm.2014.6.479
[6]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic and Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707
[7]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255
[8]

Enrique Fernández-Cara, Arnaud Münch. Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods. Mathematical Control and Related Fields, 2012, 2 (3) : 217-246. doi: 10.3934/mcrf.2012.2.217
[9]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017
[10]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709
[11]

Yong Ji, Ercai Chen, Yunping Wang, Cao Zhao. Bowen entropy for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6231-6239. doi: 10.3934/dcds.2019271
[12]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692
[13]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979
[14]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381
[15]

Anna Maria Candela, Giuliana Palmieri. Some abstract critical point theorems and applications. Conference Publications, 2009, 2009 (Special) : 133-142. doi: 10.3934/proc.2009.2009.133
[16]

Faker Ben Belgacem. Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters. Inverse Problems and Imaging, 2012, 6 (2) : 163-181. doi: 10.3934/ipi.2012.6.163
[17]

Thomas Hudson. Gamma-expansion for a 1D confined Lennard-Jones model with point defect. Networks and Heterogeneous Media, 2013, 8 (2) : 501-527. doi: 10.3934/nhm.2013.8.501
[18]

Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467
[19]

Teck-Cheong Lim. On the largest common fixed point of a commuting family of isotone maps. Conference Publications, 2005, 2005 (Special) : 621-623. doi: 10.3934/proc.2005.2005.621
[20]

Mircea Sofonea, Cezar Avramescu, Andaluzia Matei. A fixed point result with applications in the study of viscoplastic frictionless contact problems. Communications on Pure and Applied Analysis, 2008, 7 (3) : 645-658. doi: 10.3934/cpaa.2008.7.645

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy