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Iso-Manifold,Multiscale KAM Theory

Posted on:2024-03-18Degree:DoctorType:Dissertation
Country:ChinaCandidate:X F ZhaoFull Text:PDF
GTID:1520307064475844Subject:Applied Mathematics
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This thesis mainly studies the dynamical behavior of integrable systems with small perturbations,and obtains the persistence of KAM invariant tori under different circumstances.involving the classical Hamiltonian system,the generalized Hamiltonian system,the nearly integrable twist mapping system,and the n dimensional volumepreserving system.Main research in this thesis is divided into the following four parts:"Iso-manifold" property of Hamiltonian system,KAM persistence of generalized Hamiltonian system with multiscales,KAM persistence of nearly integrable twist mapping system with multiscales,the relationship of volume-preserving system and Hamiltonian system.This thesis is divided into five chapters,the main content is as follows:In Chapter 1,the research background of this thesis is introduced,including the development of classical KAM theory[39,2,55],KAM theory of generalized Hamiltonian system[46],KAM theory of nearly integrable twist mapping[55,20,71,72],theory of volume-preserving system[54]and related research results,at the same time,we introduce the main work of this thesis and full text arrangement.In Chapter 2,we extend the the results of iso-energy KAM[1,17,66]to the case of iso-manifold,and give the related results and proof of the iso-manifold KAM.First,under certain non-degenerate conditions,we give the properties of iso-manifold and partial frequency persistence of KAM tori on general Riemann surfaces(d-1 dimension),Secondly,for the case the Riemann surfaces(d-l dimension)we considered is the intersection of l Riemann surfaces(d-1 dimension),we obtain similar persistence results.In particular,when the Riemann surface considered is obtained by intersecting the surfaces which are level sets of first integrals of unperturbed Hamiltonian system with other Riemann surface,we obtain the KAM-type theorem of iso-manifold-energy.Finally,we obtain iso-manifold Melnikov persistence under a weak form of Melnikov’s second nonresonance condition and another related nondegenerate condition,i.e.,there are properties of iso-manifold and partial frequency persistence of KAM tori on a d+2m-1 dimensional Riemann surface.In Chapter 3,we consider the persistence of invariant tori for generalized Hamiltonian system with multiscales.Han,Li and Yi[30],Qian,Li and Yang[65]gave the multi-scale KAM theory for the Hamiltonian systems on classical symplectic manifolds,and in 2002,Li and Yi[46]firstly gives the KAM theory of generalized Hamiltonian system on Poisson manifold.Here we consider the Hamiltonian system with multiscales on Poisson manifold,firstly,we use finite step iteration to iterate the disturbed term of generalized Hamiltonian system to a sufficiently high order term,then by using KAM iteration,we obtain the persistence of invariant tori of the multiscale generalized Hamiltonian system.In Chapter 4,for nearly integrable twist mapping system with multiscales,we get the persistence of invariant tori.After Moser[55]first defined and studied the near integrable twist mapping on the annulas,a large number of results on twist mapping were obtained,for example,Rüssmann[71,72],Cheng and Sun[15],Xia[80]and so on.Here,we consider the multiscale mapping with intersection property,which has different numbers of action and angular variables,and get the persistence of invariant tori.When only angular variables are involved in the mapping system,a multiscale Herman theorem can be obtained by using the results we proved.In Chapter 5,we discuss the relationship between the n dimensional volumepreserving system and the Hamiltonian system.Inspired by the work of Mezic and Wiggins[54],we prove that the n dimensional volume-preserving system can be converted into a Hamiltonian system with one degree freedom and n-2 one order ordinary differential equations,moreover we develop a coordinate system to transform the n dimensional volume-preserving system into a system with one action variable and n-1 angle variables.Further,we give some conditions such that it becomes a generalized Hamiltonian system,so as to explore its stability by using the results of Li and Yi[46].Finally,for the n dimensional volume-preserving system,we also give the construction of first integral.
Keywords/Search Tags:Hamiltonian system, Iso-manifold KAM theory, multiscale KAM theory, Riemann manifold, twist mapping, volume-preserving system
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