scholarly journals Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation

2021 ◽  
pp. 1-22
Author(s):  
RAFAEL DE LA LLAVE ◽  
MARIA SAPRYKINA
Keyword(s):  
Analytic Function ◽  
Hamiltonian System ◽  
Normal Form ◽  
Power Series ◽  
Canonical Transformation ◽  
Invariant Torus ◽  
Birkhoff Normal Form ◽  
Formal Power ◽  
Analytic Invariant ◽  
Zero Frequency

Abstract Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation—not just a formal power series—bringing the Hamiltonian into its Birkhoff normal form.

Birkhoff normalization and superintegrability of Hamiltonian systems

2009 ◽  
Vol 29 (6) ◽  
pp. 1853-1880 ◽  
Author(s):  
HIDEKAZU ITO
Keyword(s):  
Hamiltonian System ◽  
Normal Form ◽  
Power Series ◽  
Holomorphic Function ◽  
Equilibrium Point ◽  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Birkhoff Normal Form ◽  
Convergent Power Series ◽  
Liouville Integrable

AbstractWe study Birkhoff normalization in connection with superintegrability of ann-degree-of-freedom Hamiltonian systemXHwith holomorphic HamiltonianH. Without assuming any Poisson commuting relation among integrals, we prove that, if the system XHhasn+qholomorphic integrals near an equilibrium point of resonance degreeq≥0, there exists a holomorphic Birkhoff transformation φ such thatH∘φ becomes a holomorphic function ofn−qvariables and thatXH∘φcan be solved explicitly. Furthermore, the Birkhoff normal formH∘φ is determined uniquely, independently of the choice of φ, as convergent power series. We also show that the systemXHis superintegrable in the sense of Mischenko–Fomenko as well as Liouville-integrable near the equilibrium point.

Semi-classical Integrability, Hyperbolic Flows and the Birkhoff Normal Form

2004 ◽  
Vol 56 (5) ◽  
pp. 1034-1067 ◽  
Author(s):  
Michel Rouleux
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Canonical Transformation ◽  
Fundamental Matrix ◽  
Normal Forms ◽  
Complex Case ◽  
Birkhoff Normal Form ◽  
Degenerate Critical Point ◽  
Hyperbolic Fixed Point ◽  
Hyperbolic Flows

AbstractWe prove that a Hamiltonianp∈C∞(T*Rn) is locally integrable near a non-degenerate critical point ρ0of the energy, provided that the fundamental matrix at ρ0has rationally independent eigenvalues, none purely imaginary. This is done by using Birkhoff normal forms, which turn out to be convergent in theC∞sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the complex case, showing that whenpis holomorphic near ρ0∈T*Cn, then Repbecomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge,i.e., pmay not be integrable. These normal forms also hold in the semi-classical frame.

scholarly journals Non-integrability of the 1:1:2–resonance

1984 ◽  
Vol 4 (4) ◽  
pp. 553-568 ◽  
Author(s):  
J. J. Duistermaat
Keyword(s):  
Hamiltonian System ◽  
Power Series ◽  
Equilibrium Point ◽  
Formal Power Series ◽  
Open Subset ◽  
Taylor Expansion ◽  
Degrees Of Freedom ◽  
Third Order ◽  
Formal Power ◽  
F System

AbstractA Hamiltonian system of n degrees of freedom, defined by the function F, with an equilibrium point at the origin, is called formally integrable if there exist A A formal power series , functionally independent, in involution, and such that the Taylor expansion of F is a formal power series in the .Take n = 3, , F(k) homogeneous of degree k, F(2) > 0 and the eigenfrequencies in ratio 1:1:2. If F(3) avoids a certain hypersurface of ‘symmetric’ third order terms, then the F system is not formally integrable. If F(3) is symmetric but F(4) is in a non-void open subset, then homoclinic intersection with Devaney spiralling occurs; the angle decays of order 1 when approaching the origin.

Criteria for Algebraicity of Analytic Functions

2019 ◽  
Author(s):  
Jacek Bochnak ◽  
Janusz Gwoździewicz ◽  
Wojciech Kucharz
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Formal Power Series ◽  
Open Subset ◽  
Analytic Functions ◽  
Algebraic Curves ◽  
Polynomial Function ◽  
Algebraic Set ◽  
Formal Power

Abstract We consider functions defined on an open subset of a nonsingular, either real or complex, algebraic set. We give criteria for an analytic function to be a Nash (resp. regular, resp. polynomial) function. Our criteria depend only on the behavior of such a function along irreducible nonsingular algebraic curves passing through a given point. In the proofs we use results on algebraicity of formal power series, which are also established in this paper.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

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