American Institute of Mathematical Sciences

Advanced
January  2007, 18(1): 135-157. doi: 10.3934/dcds.2007.18.135

Area contraction in the presence of first integrals and almost global convergence

Dirk Aeyels 1, , Filip De Smet 1, and  Bavo Langerock 1,

1. 

SYSTeMS Research Group, Dept. of Electrical Energy, Systems and Automation, Ghent University, Technologiepark-Zwijnaarde 914, 9052 Zwijnaarde, Belgium, Belgium

Received  April 2006 Revised  November 2006 Published  February 2007

We investigate the evolution of the area of multi-dimensional surfaces along the flow of a dynamical system with known first integrals, and we formulate sufficient conditions for area contraction.
   These results, together with known results about the Hausdorff dimension and the box-counting dimension of invariant sets, are applied to systems exhibiting almost global convergence/asymptotic stability. This leads to a generalization of a well-known result on almost global convergence of a system, based on the use of density functions. We conclude with an example.
Keywords: Hausdorff dimension, $k$-contracting vector fields, first integrals, almost global convergence..
Mathematics Subject Classification: 34D23, 37C1.
Citation: Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete & Continuous Dynamical Systems, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135
[1]

Jaume Llibre, Marco Antonio Teixeira. Regularization of discontinuous vector fields in dimension three. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 235-241. doi: 10.3934/dcds.1997.3.235
[2]

C. Alonso-González, M. I. Camacho, F. Cano. Topological invariants for singularities of real vector fields in dimension three. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 823-847. doi: 10.3934/dcds.2008.20.823
[3]

Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543
[4]

Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045
[5]

Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977
[6]

Rehana Naz, Fazal M Mahomed, Azam Chaudhry. First integrals of Hamiltonian systems: The inverse problem. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2829-2840. doi: 10.3934/dcdss.2020121
[7]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[8]

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

Jason Murphy, Fabio Pusateri. Almost global existence for cubic nonlinear Schrödinger equations in one space dimension. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2077-2102. doi: 10.3934/dcds.2017089
[10]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591
[11]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
[12]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405
[13]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[14]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[15]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
[16]

Pablo Aguirre, Eusebius J. Doedel, Bernd Krauskopf, Hinke M. Osinga. Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1309-1344. doi: 10.3934/dcds.2011.29.1309
[17]

Frol Zapolsky. On almost Poisson commutativity in dimension two. Electronic Research Announcements, 2010, 17: 155-160. doi: 10.3934/era.2010.17.155
[18]

BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85
[19]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063
[20]

Richard A. Norton, David I. McLaren, G. R. W. Quispel, Ari Stern, Antonella Zanna. Projection methods and discrete gradient methods for preserving first integrals of ODEs. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2079-2098. doi: 10.3934/dcds.2015.35.2079

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy