
Previous Article
Rigorous highdimensional shadowing using containment: The general case
 DCDS Home
 This Issue

Next Article
Covering relations and nonautonomous perturbations of ODEs
Topological methods in the instability problem of Hamiltonian systems
1.  Department of Mathematics, Northeastern Illinois University, Chicago, IL 60625, United States 
2.  Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States 
In these mechanisms, chains of heteroclinic connections between whiskered tori are constructed, based on the existence of a normally hyperbolic manifold $\Lambda$, so that: (a) the manifold $\Lambda$ is covered rather densely by transitive tori (possibly of different topology), (b) the manifolds $W^\s_\Lambda$, $W^\u_\Lambda$ intersect transversally, (c) the systems satisfies some explicit nondegeneracy assumptions, which hold generically.
In this paper we use the method of correctly aligned windows to show that, under the assumptions (a), (b), (c), there are orbits that move a significant amount.
As a matter of fact, the method presented here does not require that the tori are exactly invariant, only that they are approximately invariant. Hence, compared with the previous papers, we do not need to use KAM theory. This lowers the assumptions on differentiability.
Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.
[1] 
Jacky Cresson, Christophe Guillet. Periodic orbits and Arnold diffusion. Discrete & Continuous Dynamical Systems  A, 2003, 9 (2) : 451470. doi: 10.3934/dcds.2003.9.451 
[2] 
Ernest Fontich, Pau Martín. Arnold diffusion in perturbations of analytic integrable Hamiltonian systems. Discrete & Continuous Dynamical Systems  A, 2001, 7 (1) : 6184. doi: 10.3934/dcds.2001.7.61 
[3] 
Jacky Cresson. The transfer lemma for Graff tori and Arnold diffusion time. Discrete & Continuous Dynamical Systems  A, 2001, 7 (4) : 787800. doi: 10.3934/dcds.2001.7.787 
[4] 
Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete & Continuous Dynamical Systems  A, 2002, 8 (3) : 795811. doi: 10.3934/dcds.2002.8.795 
[5] 
Massimiliano Berti. Some remarks on a variational approach to Arnold's diffusion. Discrete & Continuous Dynamical Systems  A, 1996, 2 (3) : 307314. doi: 10.3934/dcds.1996.2.307 
[6] 
Claude Froeschlé, Massimiliano Guzzo, Elena Lega. First numerical evidence of global Arnold diffusion in quasiintegrable systems. Discrete & Continuous Dynamical Systems  B, 2005, 5 (3) : 687698. doi: 10.3934/dcdsb.2005.5.687 
[7] 
Amadeu Delshams, Rodrigo G. Schaefer. Arnold diffusion for a complete family of perturbations with two independent harmonics. Discrete & Continuous Dynamical Systems  A, 2018, 38 (12) : 60476072. doi: 10.3934/dcds.2018261 
[8] 
Takayoshi Ogawa, Hiroshi Wakui. Stability and instability of solutions to the driftdiffusion system. Evolution Equations & Control Theory, 2017, 6 (4) : 587597. doi: 10.3934/eect.2017029 
[9] 
Stephen Pankavich, Petronela Radu. Nonlinear instability of solutions in parabolic and hyperbolic diffusion. Evolution Equations & Control Theory, 2013, 2 (2) : 403422. doi: 10.3934/eect.2013.2.403 
[10] 
Masaharu Taniguchi. Instability of planar traveling waves in bistable reactiondiffusion systems. Discrete & Continuous Dynamical Systems  B, 2003, 3 (1) : 2144. doi: 10.3934/dcdsb.2003.3.21 
[11] 
Georg Hetzer, Anotida Madzvamuse, Wenxian Shen. Characterization of turing diffusiondriven instability on evolving domains. Discrete & Continuous Dynamical Systems  A, 2012, 32 (11) : 39754000. doi: 10.3934/dcds.2012.32.3975 
[12] 
Toshi Ogawa. Degenerate Hopf instability in oscillatory reactiondiffusion equations. Conference Publications, 2007, 2007 (Special) : 784793. doi: 10.3934/proc.2007.2007.784 
[13] 
Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete & Continuous Dynamical Systems  A, 2020, 40 (6) : 38133836. doi: 10.3934/dcds.2020160 
[14] 
Shiri ArtsteinAvidan, Dan Florentin, Vitali Milman. Order isomorphisms in windows. Electronic Research Announcements, 2011, 18: 112118. doi: 10.3934/era.2011.18.112 
[15] 
ShinIchiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multispot patterns in shadow systems of reactiondiffusion equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 717736. doi: 10.3934/cpaa.2015.14.717 
[16] 
Qing Li, Yaping Wu. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete & Continuous Dynamical Systems  A, 2020, 40 (6) : 36573682. doi: 10.3934/dcds.2020051 
[17] 
Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete & Continuous Dynamical Systems  S, 2008, 1 (1) : 107116. doi: 10.3934/dcdss.2008.1.107 
[18] 
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) : 9891011. doi: 10.3934/dcds.2008.20.989 
[19] 
Nurhadi Siswanto, Stefanus Eko Wiratno, Ahmad Rusdiansyah, Ruhul Sarker. Maritime inventory routing problem with multiple time windows. Journal of Industrial & Management Optimization, 2019, 15 (3) : 11851211. doi: 10.3934/jimo.2018091 
[20] 
Jacques Féjoz. On "Arnold's theorem" on the stability of the solar system. Discrete & Continuous Dynamical Systems  A, 2013, 33 (8) : 35553565. doi: 10.3934/dcds.2013.33.3555 
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]