We prove that there is an invariant torus with the given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian pert...We prove that there is an invariant torus with the given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for application, we prove that a finite network of Duffing oscillators with periodic external forces possesses Lagrange stability for almost all initial data.展开更多
For reversible systems of infinite dimension we prove an infinitely dimensional KAM theorem with an application to the network of weakly coupled oscillators of friction. The KAM theorem shows that there are many invar...For reversible systems of infinite dimension we prove an infinitely dimensional KAM theorem with an application to the network of weakly coupled oscillators of friction. The KAM theorem shows that there are many invariant tori of infinite dimension, and thus many almost periodic solutions, for the reversible systems.展开更多
We introduce several KAM theorems for infinite-dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori.Especially,we introduce a KAM theorem by Y...We introduce several KAM theorems for infinite-dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori.Especially,we introduce a KAM theorem by Yuan published in CMP(2002),which shows that there are rich KAM tori for a class of Hamiltonian with short range and with linearized operator of pure point spectra.We also present several open problems.展开更多
In this paper, using the KAM theorem of reversible systems, we obtain the boundednessof solutions, the existence of quasi-periodic solutions and subharmonic solutions for the non-linear differential equations of the s...In this paper, using the KAM theorem of reversible systems, we obtain the boundednessof solutions, the existence of quasi-periodic solutions and subharmonic solutions for the non-linear differential equations of the second order which is neither conservative nor dissipative.展开更多
In this paper, we study the persistence of lower dimensional tori for random Hamiltonian systems, which shows that majority of the unperturbed tori persist as Cantor fragments of lower dimensional ones under small per...In this paper, we study the persistence of lower dimensional tori for random Hamiltonian systems, which shows that majority of the unperturbed tori persist as Cantor fragments of lower dimensional ones under small perturbation. Using this result, we can describe the stability of the non-autonomous dynamic systems.展开更多
We introduce an ultra high energy combined KAM-Rindler fractal spacetime quantum manifold, which increasingly resembles Einstein’s smooth relativity spacetime, with decreasing energy. That way we derive an effective ...We introduce an ultra high energy combined KAM-Rindler fractal spacetime quantum manifold, which increasingly resembles Einstein’s smooth relativity spacetime, with decreasing energy. That way we derive an effective quantum gravity energy-mass relation and compute a dark energy density in complete agreement with all cosmological measurements, specifically WMAP and type 1a supernova. In particular we find that ordinary measurable energy density is given by E1= mc2 /22 while the dark energy density of the vacuum is given by E2 = mc2 (21/22). The sum of both energies is equal to Einstein’s energy E = mc2. We conclude that E= mc2 makes no distinction between ordinary energy and dark energy. More generally we conclude that the geometry and topology of quantum entanglement create our classical spacetime and glue it together and conversely quantum entanglement is the logical consequence of KAM theorem and zero measure topology of quantum spacetime. Furthermore we show via our version of a Rindler hyperbolic spacetime that Hawking negative vacuum energy, Unruh temperature and dark energy are different sides of the same medal.展开更多
By the iteration of the KAM, the following second- order differential equation （ Фp （x′））′ ＋ F（x, x′, t） ＋ ω^PФp （x′） ＋α│x│^l ＋e（x, t） =0 is studied, where Фp（S） = │S│^p-2s, p 〉 1, α〉 0...By the iteration of the KAM, the following second- order differential equation （ Фp （x′））′ ＋ F（x, x′, t） ＋ ω^PФp （x′） ＋α│x│^l ＋e（x, t） =0 is studied, where Фp（S） = │S│^p-2s, p 〉 1, α〉 0 and ω 〉 0 are positive constants, and l satisfies - 1 〈ω 〈p ＋ 2. Under some assumptions on the parities of F（x, x′, t） and e （x, t）, by a small twist theorem of reversible mapping, the existence of quasi-periodic solutions and boundedness of all the solutions are obtained.展开更多
In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimen...In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graft and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on submanifolds.展开更多
In this paper, one-dimensional （1D） nonlinear beam equations of the form utt - uxx ＋ uxxxx ＋ mu = f （u） with Dirichlet boundary conditions are considered, where the nonlinearity f is an analytic, odd function an...In this paper, one-dimensional （1D） nonlinear beam equations of the form utt - uxx ＋ uxxxx ＋ mu = f （u） with Dirichlet boundary conditions are considered, where the nonlinearity f is an analytic, odd function and f（u） = O（u3）. It is proved that for all m ∈ （0, M＊] R （M＊ is a fixed large number）, but a set of small Lebesgue measure, the above equations admit small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theory and a partial Birkhoff normal form technique.展开更多
In the paper, by applying the method of main integration, we show the boundedness of the quasi-periodic second order differential equation x′′+ ax;-bx;+φ（x） = p（t）, where a≠b are two positive constants and φ...In the paper, by applying the method of main integration, we show the boundedness of the quasi-periodic second order differential equation x′′+ ax;-bx;+φ（x） = p（t）, where a≠b are two positive constants and φ（s）, p（t） are real analytic functions. Moreover, the p（t） is quasi-periodic coefficient, whose frequency vectors are Diophantine. The results we obtained also imply that, under some conditions,the quasi-periodic oscillator has the Lagrange stability.展开更多
The quasi-periodic pendulum type equations are considered. A sufficient and necessary condition of Lagrange stability for this kind of equations is obtained. The result obtained answers a problem proposed by Moser und...The quasi-periodic pendulum type equations are considered. A sufficient and necessary condition of Lagrange stability for this kind of equations is obtained. The result obtained answers a problem proposed by Moser under the quasi-periodic case.展开更多
In this letter, I outline the intimate connection between the fractal spectra of the exact solution of the hydrogen atom and the issue of the missing dark energy of the cosmos. A proposal for a dark energy reactor har...In this letter, I outline the intimate connection between the fractal spectra of the exact solution of the hydrogen atom and the issue of the missing dark energy of the cosmos. A proposal for a dark energy reactor harnessing the dark energy of the Schrodinger wave via a quantum wave nondemolition measurement is also presented.展开更多
基金supported by National Natural Science Foundation of China(Grant No.12071254)。
文摘We prove that there is an invariant torus with the given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for application, we prove that a finite network of Duffing oscillators with periodic external forces possesses Lagrange stability for almost all initial data.
基金Supported by NNSFC and NCET-04-0365in part by STCSM-06ZR14014
文摘For reversible systems of infinite dimension we prove an infinitely dimensional KAM theorem with an application to the network of weakly coupled oscillators of friction. The KAM theorem shows that there are many invariant tori of infinite dimension, and thus many almost periodic solutions, for the reversible systems.
基金supported by National Natural Science Foundation of China (Grant Nos.11271076 and 11121101)the National Basic Research Program of China (973 Program) (Grant No.2010CB327900)
文摘We introduce several KAM theorems for infinite-dimensional Hamiltonian with short range and discuss the relationship between spectra of linearized operator and invariant tori.Especially,we introduce a KAM theorem by Yuan published in CMP(2002),which shows that there are rich KAM tori for a class of Hamiltonian with short range and with linearized operator of pure point spectra.We also present several open problems.
文摘In this paper, using the KAM theorem of reversible systems, we obtain the boundednessof solutions, the existence of quasi-periodic solutions and subharmonic solutions for the non-linear differential equations of the second order which is neither conservative nor dissipative.
基金Partially supported by the SFC(10531050,10225107)of Chinathe SRFDP(20040183030)the 985 program of Jilin University
文摘In this paper, we study the persistence of lower dimensional tori for random Hamiltonian systems, which shows that majority of the unperturbed tori persist as Cantor fragments of lower dimensional ones under small perturbation. Using this result, we can describe the stability of the non-autonomous dynamic systems.
文摘We introduce an ultra high energy combined KAM-Rindler fractal spacetime quantum manifold, which increasingly resembles Einstein’s smooth relativity spacetime, with decreasing energy. That way we derive an effective quantum gravity energy-mass relation and compute a dark energy density in complete agreement with all cosmological measurements, specifically WMAP and type 1a supernova. In particular we find that ordinary measurable energy density is given by E1= mc2 /22 while the dark energy density of the vacuum is given by E2 = mc2 (21/22). The sum of both energies is equal to Einstein’s energy E = mc2. We conclude that E= mc2 makes no distinction between ordinary energy and dark energy. More generally we conclude that the geometry and topology of quantum entanglement create our classical spacetime and glue it together and conversely quantum entanglement is the logical consequence of KAM theorem and zero measure topology of quantum spacetime. Furthermore we show via our version of a Rindler hyperbolic spacetime that Hawking negative vacuum energy, Unruh temperature and dark energy are different sides of the same medal.
基金The National Natural Science Foundation of China(No. 11071038)the Natural Science Foundation of Jiangsu Province(No. BK2010420)
文摘By the iteration of the KAM, the following second- order differential equation （ Фp （x′））′ ＋ F（x, x′, t） ＋ ω^PФp （x′） ＋α│x│^l ＋e（x, t） =0 is studied, where Фp（S） = │S│^p-2s, p 〉 1, α〉 0 and ω 〉 0 are positive constants, and l satisfies - 1 〈ω 〈p ＋ 2. Under some assumptions on the parities of F（x, x′, t） and e （x, t）, by a small twist theorem of reversible mapping, the existence of quasi-periodic solutions and boundedness of all the solutions are obtained.
文摘In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graft and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on submanifolds.
基金The NSF (11001042) of Chinathe SRFDP Grant (20100043120001)FRFCU Grant(09QNJJ002)
文摘In this paper, one-dimensional （1D） nonlinear beam equations of the form utt - uxx ＋ uxxxx ＋ mu = f （u） with Dirichlet boundary conditions are considered, where the nonlinearity f is an analytic, odd function and f（u） = O（u3）. It is proved that for all m ∈ （0, M＊] R （M＊ is a fixed large number）, but a set of small Lebesgue measure, the above equations admit small-amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theory and a partial Birkhoff normal form technique.
文摘In the paper, by applying the method of main integration, we show the boundedness of the quasi-periodic second order differential equation x′′+ ax;-bx;+φ（x） = p（t）, where a≠b are two positive constants and φ（s）, p（t） are real analytic functions. Moreover, the p（t） is quasi-periodic coefficient, whose frequency vectors are Diophantine. The results we obtained also imply that, under some conditions,the quasi-periodic oscillator has the Lagrange stability.
基金Partially supported by the NSF (10871203, 10601019) of Chinathe NCET (07-0386)of China
文摘The quasi-periodic pendulum type equations are considered. A sufficient and necessary condition of Lagrange stability for this kind of equations is obtained. The result obtained answers a problem proposed by Moser under the quasi-periodic case.
文摘In this letter, I outline the intimate connection between the fractal spectra of the exact solution of the hydrogen atom and the issue of the missing dark energy of the cosmos. A proposal for a dark energy reactor harnessing the dark energy of the Schrodinger wave via a quantum wave nondemolition measurement is also presented.
