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Abstract: In this paper we first summarize our results published in recent years and their sketch proofs on local integrability,which are on the characterization of local integrability and on the existence of analytic normalization of analytically integrable differential systems.Then we present a new result on the equivalent characterization of the existence of the first integrals of an analytic differential systems near a nonhyperbolic singularity.Finally we pose some open problems on this subject.
Key words: formal diffeomorphism; embedding vector field; normal form
CLC number: O 175.11, O 175.14, O 193 Document code: A
Article ID: 1000-5137(2014)05-0476-12
2010 Mathematics Subject Classification: 34A34, 34C20, 34C41, 37G05
Received date: 2014-04-17
Foundation item: Partially supported by the NNSF of China Grant 11271252；the RFDP of Higher Education of China grant 20110073110054; the FP7-PEOPLE-2012-IRSES-316338 of Europe
Corresponding author: Xiang Zhang,Professor,E-mail:xzhang@mail.sjtu.edu.cn1 Introduction
The study of the integrability of differential systems can be traced back to Newton.The theory of integrability is very useful in the dynamical analysis of differential systems,because lots of physical and mechanical systems do have invariants［1-7］.How to recognize and find them is a difficult task.This paper mainly introduces some progress in recent years on local integrability.In section 2 we survey the results on the existence of local first integrals via resonance,and the equivalent characterization on local analytic integrability and the existence of analytic normalization of analytically integrable differential systems to their normal forms.In section 3 we present a new result on the existence of local first integrals for differential systems around nonhyperbolic singularities with a pair of pure imaginary eigenvalues.In section 4 we pose some open problems related to the local integrability of differential systems for further study.
2 A survey on the local integrability of analytic differential systems
This section will be separated into two parts.The first one is on the existence and the number of functionally independent local first integrals of analytic differential systems around a singularity.The second one is on the existence of analytic normalization of analytical integrable differential systems around a singularity or a periodic orbit.
2.1 Number of functionally independent local first integrals (1)the existence of first integrals can be used to reduce the dimension of the system.And so it becomes possibly easier to study the dynamics of the system.A function H(x) is a first integral of system(1) if
it is continuous and is defined in a full Lebesgue measure subset Ω1 of Ω,
it is not locally constant on any positive Lebesgue measure subset of Ω1,and
H(x) is constant along each orbit of system(1) in Ω1.
System(1) is Cr completely integrable,if it has n-1 functionally independent Cr first integrals in Ω with 1≤r≤k,where k∈
Local integrable differential systems and their normal forms (a) For n＞2, system(1) has a formal first integral in a neighborhood of x=0 if and only if the singularity x=0 is not isolated.Specially,if the singularity x=0 is isolated, system(1) has no analytic first integrals in any neighborhood of x=0.
(b) For n=2,system(1) has an analytic first integral in a neighborhood of x=0 if and only if the singularity x=0 is not isolated.
Theorem 1 was further extended to the next result (see［10,Theorem 3］).
Theorem 2 Assume that system(1) has 0 This last result was extended by Zhang (［11,Theorem 1］) to the next one(see also Kwek,Li and Shi［12,Theorem A］,which provided a proof of Theorem 3 with the restriction that A is diagonalizable).
Theorem 3 Assume that the analytic differential system(1) has 0 Here the functional independence of H1,…,Hk is in the sense that their gradients at the origin is linearly independent.
In 2008 Chen,Yi and Zhang［13］ provided an upper bound on the number of functionally independent analytic first integrals for quasiperiodic differential systems.Consider the quasiperiodic vector
2.2 Analytic normalization of analytical integrable differential systems
The study on the existence of analytic normalizations for analytic integrable vector fields to their normal forms was started from Poincaré［8］.
The Poincaré normal form theorem showed that any analytic or formal differential system(1) is formally equivalent to its Poincaré normal form.To characterize when a normalization is convergent,it is a difficult problem.One of the Poincaré classical results characterizes the planar analytic nondegenerate center via the existence of analytic normalization.
Poincaré theorem Ⅱ［8］ A planar analytic differential system(1) has the origin as a nondegenerate center if and only if it is analytically equivalent(via probably complex transformation of variables and time rescaling) tox ?=x(1+q(xy)), y ?=-y(1+q(xy)), (5)where q(u) is an analytic function in u starting from the terms of degree no less than 1.
By the Poincaré theorem Ⅱ it follows that
Theorem 9 Assume that the origin of a planar analytic differential system(1) is nondegenerate.Then the origin is an isochronous center if and only if it is analytically equivalent tou ?=-cv, v ?=cu,with c being a nonzero constant.
This last theorem indicates that a degenerate center of a planar analytic differential system cannot be isochronous.
For planar Hamiltonian systems, Moser［18］ obtained a similar result.He showed that a planar real analytic Hamiltonian system having the origin as a hyperbolic saddle can be reduced to system(5) by a real analytic area-preserving transformation of variables.
Poincaré and Moser′s results on planar analytic differential systems was extended to higher dimensional differential systems by Zhang［19-20］.
(10)with Φ(y,t)=O(|y|2) a formal series in y and periodic in t,which sends (9) to the Poincaré-Dulac normal form y ?=Ay+G(y,t).
(11)If Φ(y,t) contains only non-resonant pseudomonomials,we say that system(9) is analytically equivalent to its distinguished normal form.A pseudomonomial in a transformation xleiktej is nonresonant if ik+〈l,λ〉≠0.The transformation(10) is called distinguished normalization.We should take care of the difference between the resonances of the pseudomonomials in a vector field and in a transformation.
The main results in［21］ is the following one.
Theorem 11 Assume that system(7) is analytic and has a periodic orbit.If system(7) is analytically integrable in a neighborhood of the periodic orbit, then the system is analytically equivalent to its distinguished normal form in a neighborhood of the periodic orbit.
The proof of Theorem 11 explored the essential properties that the characteristic exponents of periodic orbits of analytically integrable differential system satisfy.
4 Open questions on local integrability
In the local theory of integrability there remains lots of unsolved open problems.We list part of them for readers′s further consideration.
In Theorem 11 we obtained the existence of analytic normalization of analytic integrable differential system near a periodic orbit,but we cannot get the exact expressions.
Open problem 1 What is the simplest normal form of analytic integrable differential system near a periodic orbit?
In all the previous results there is a basic assumption that the eigenvalues of A are not all zero. Open problem 2 If all eigenvalues of A vanish,what kinds of analytically integrable differential systems are locally analytically equivalent to its normal form?
Of course,the normal form mentioned in open problem 2 is not in the Poincaré normal form because it is trivial now.How to present a new notion and how to deform the original systems such that we can consider its Poincaré normal norm.
All the above results are on a single vector field.If we have a set of linearly independent vector fields,how to study their simultaneously normal forms and the convergence of the normalization.Zung［24］ studied this problem by a geometric method.His results are summarized in the following theorem.
References:
［1］ A. Goriely.Integrability,partial integrability,and nonintegrability for systems of ordinary differential equations［J］.J.Math.Phys.,1996,37:1871-1893.
［2］ J. Llibre.Integrability of polynomial differential systems,Handbook of Differential Equations,Ordinary Differential Equations,Eds. A. Caada,P. Drabek and A. Fonda［M］.Elsevier,2004:437-533.
［3］ M. J. Prelle,M. F. Singer.Elementary first integrals of differential equations［J］.Trans. Amer. Math. Soc.,1983,279:215-229.
［4］ M. F. Singer.Liouvillian first integrals of differential equations［J］.Trans. Amer. Math. Soc.,1992,333:673-688.
［5］ H. Yoshida.Necessary condition for the existence of algebraic first integrals.I Kowalevski′s exponents［J］.Celestial Mechanics,1983,31:363-379.
［6］ H. Yoshida.Necessary condition for the existence of algebraic first integrals.II:condition for algebraic integrability［J］.Celestial Mechanics,1983,31:381-399.
［7］ S. L. Ziglin.Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics I［J］.Functional Anal. Appl.,1983,16:181-189.
［8］ H. Poincaré.Sur l′intégration des équations différentielles du premier order et du premier degré Ⅰ and Ⅱ［J］.Rendiconti del circolo matematico di Palermo,1891,5:161-191; 1897,11:193-239.
［9］ S. D. Furta.On non-integrability of general systems of differential equations［J］.Z. angew Math. Phys.,1996,47:112-131.
［10］ W. Li,J. Llibre,X. Zhang.Local first integrals of differential systems and diffeomorphisms［J］.Z. angew Math. Phys.,2003,54:235-255.
［11］ X. Zhang.Local first integrals for systems of differential equations［J］.J. Phys. A,2003,36:12243-12253.
［12］ K. H. Kwek,Y. Li,S. Shi.Partial integrability for general nonlinear systems［J］.Z. Angew Math. Phys.,2003,54:26-47. ［13］ J. Chen,Y. Yi,X. Zhang.First integrals and normal forms for germs of analytic vector fields［J］.J. Differential Equations,2008,245:1167-1184.
［14］ S. Shi.On the nonexistence of rational first integrals for nonlinear systems and semiquasihomogeneous systems［J］.J. Math. Anal. Appl.,2007,335:125-134.
［15］ W. Cong,J. Llibre J.,X. Zhang.Generalized rational first integrals of analytic differential systems［J］.J. Differential Equations,2011,251:2770-2788.
［16］ A. Baider,R. C. Churchill,D. L. Rod,et al.On the infinitesimal geometry of integrable systems［R］.Rhode Island:Fields Institute Communications 7,American Mathematical Society,Providence,1996.
［17］ W. Fulton.Algebraic Curves:An Introduction to Algebraic Geometry［M］.London:The Benjamin/Cummings Publishing Com Inc,1978.
［18］ J. Moser.The analytic invariants of an area-preserving mapping near a hyperbolic fixed point［J］.Comm. Pure Appl. Math.,1956,9:673-692.
［19］ X. Zhang.Analytic normalization of analytic integrable systems and the embedding Flows［J］.Differential Equations,2008,244:1080-1092.
［20］ X. Zhang.Analytic integrable systems:Analytic normalization and embedding flows［J］.Differential Equations,2013,254:3000-3022.
［21］ K. S. Wu,X. Zhang.Analytic normalization of analytically integrable differential systems near a periodic orbit［J］.Differential Equations,2014,256:3552-3567.
［22］ J. Guckenheimer,P. Holmes.Nonlinear Oscillations,Dynamical Systems,and Bifurcations of Vector Fields,2nd Ed［M］.New York:Springer-Verlag,2002.
［23］ G. R. Belitskii.Smooth equivalence of germs of vector fields with one zero or a pair of purely imaginary eigenvalues［J］.Funct. Anal. Appl.,1986,20(4):253-259.
［24］ N. T. Zung.Convergence versus integrability in Poincaré-Dulac normal form［J］.Math. Res. Lett.,2002,9:217-228.
(Zhenzhen Feng,Haoran Gu)
Key words: formal diffeomorphism; embedding vector field; normal form
CLC number: O 175.11, O 175.14, O 193 Document code: A
Article ID: 1000-5137(2014)05-0476-12
2010 Mathematics Subject Classification: 34A34, 34C20, 34C41, 37G05
Received date: 2014-04-17
Foundation item: Partially supported by the NNSF of China Grant 11271252；the RFDP of Higher Education of China grant 20110073110054; the FP7-PEOPLE-2012-IRSES-316338 of Europe
Corresponding author: Xiang Zhang,Professor,E-mail:xzhang@mail.sjtu.edu.cn1 Introduction
The study of the integrability of differential systems can be traced back to Newton.The theory of integrability is very useful in the dynamical analysis of differential systems,because lots of physical and mechanical systems do have invariants［1-7］.How to recognize and find them is a difficult task.This paper mainly introduces some progress in recent years on local integrability.In section 2 we survey the results on the existence of local first integrals via resonance,and the equivalent characterization on local analytic integrability and the existence of analytic normalization of analytically integrable differential systems to their normal forms.In section 3 we present a new result on the existence of local first integrals for differential systems around nonhyperbolic singularities with a pair of pure imaginary eigenvalues.In section 4 we pose some open problems related to the local integrability of differential systems for further study.
2 A survey on the local integrability of analytic differential systems
This section will be separated into two parts.The first one is on the existence and the number of functionally independent local first integrals of analytic differential systems around a singularity.The second one is on the existence of analytic normalization of analytical integrable differential systems around a singularity or a periodic orbit.
2.1 Number of functionally independent local first integrals (1)the existence of first integrals can be used to reduce the dimension of the system.And so it becomes possibly easier to study the dynamics of the system.A function H(x) is a first integral of system(1) if
it is continuous and is defined in a full Lebesgue measure subset Ω1 of Ω,
it is not locally constant on any positive Lebesgue measure subset of Ω1,and
H(x) is constant along each orbit of system(1) in Ω1.
System(1) is Cr completely integrable,if it has n-1 functionally independent Cr first integrals in Ω with 1≤r≤k,where k∈
Local integrable differential systems and their normal forms (a) For n＞2, system(1) has a formal first integral in a neighborhood of x=0 if and only if the singularity x=0 is not isolated.Specially,if the singularity x=0 is isolated, system(1) has no analytic first integrals in any neighborhood of x=0.
(b) For n=2,system(1) has an analytic first integral in a neighborhood of x=0 if and only if the singularity x=0 is not isolated.
Theorem 1 was further extended to the next result (see［10,Theorem 3］).
Theorem 2 Assume that system(1) has 0
Theorem 3 Assume that the analytic differential system(1) has 0
In 2008 Chen,Yi and Zhang［13］ provided an upper bound on the number of functionally independent analytic first integrals for quasiperiodic differential systems.Consider the quasiperiodic vector
2.2 Analytic normalization of analytical integrable differential systems
The study on the existence of analytic normalizations for analytic integrable vector fields to their normal forms was started from Poincaré［8］.
The Poincaré normal form theorem showed that any analytic or formal differential system(1) is formally equivalent to its Poincaré normal form.To characterize when a normalization is convergent,it is a difficult problem.One of the Poincaré classical results characterizes the planar analytic nondegenerate center via the existence of analytic normalization.
Poincaré theorem Ⅱ［8］ A planar analytic differential system(1) has the origin as a nondegenerate center if and only if it is analytically equivalent(via probably complex transformation of variables and time rescaling) tox ?=x(1+q(xy)), y ?=-y(1+q(xy)), (5)where q(u) is an analytic function in u starting from the terms of degree no less than 1.
By the Poincaré theorem Ⅱ it follows that
Theorem 9 Assume that the origin of a planar analytic differential system(1) is nondegenerate.Then the origin is an isochronous center if and only if it is analytically equivalent tou ?=-cv, v ?=cu,with c being a nonzero constant.
This last theorem indicates that a degenerate center of a planar analytic differential system cannot be isochronous.
For planar Hamiltonian systems, Moser［18］ obtained a similar result.He showed that a planar real analytic Hamiltonian system having the origin as a hyperbolic saddle can be reduced to system(5) by a real analytic area-preserving transformation of variables.
Poincaré and Moser′s results on planar analytic differential systems was extended to higher dimensional differential systems by Zhang［19-20］.
(10)with Φ(y,t)=O(|y|2) a formal series in y and periodic in t,which sends (9) to the Poincaré-Dulac normal form y ?=Ay+G(y,t).
(11)If Φ(y,t) contains only non-resonant pseudomonomials,we say that system(9) is analytically equivalent to its distinguished normal form.A pseudomonomial in a transformation xleiktej is nonresonant if ik+〈l,λ〉≠0.The transformation(10) is called distinguished normalization.We should take care of the difference between the resonances of the pseudomonomials in a vector field and in a transformation.
The main results in［21］ is the following one.
Theorem 11 Assume that system(7) is analytic and has a periodic orbit.If system(7) is analytically integrable in a neighborhood of the periodic orbit, then the system is analytically equivalent to its distinguished normal form in a neighborhood of the periodic orbit.
The proof of Theorem 11 explored the essential properties that the characteristic exponents of periodic orbits of analytically integrable differential system satisfy.
4 Open questions on local integrability
In the local theory of integrability there remains lots of unsolved open problems.We list part of them for readers′s further consideration.
In Theorem 11 we obtained the existence of analytic normalization of analytic integrable differential system near a periodic orbit,but we cannot get the exact expressions.
Open problem 1 What is the simplest normal form of analytic integrable differential system near a periodic orbit?
In all the previous results there is a basic assumption that the eigenvalues of A are not all zero. Open problem 2 If all eigenvalues of A vanish,what kinds of analytically integrable differential systems are locally analytically equivalent to its normal form?
Of course,the normal form mentioned in open problem 2 is not in the Poincaré normal form because it is trivial now.How to present a new notion and how to deform the original systems such that we can consider its Poincaré normal norm.
All the above results are on a single vector field.If we have a set of linearly independent vector fields,how to study their simultaneously normal forms and the convergence of the normalization.Zung［24］ studied this problem by a geometric method.His results are summarized in the following theorem.
References:
［1］ A. Goriely.Integrability,partial integrability,and nonintegrability for systems of ordinary differential equations［J］.J.Math.Phys.,1996,37:1871-1893.
［2］ J. Llibre.Integrability of polynomial differential systems,Handbook of Differential Equations,Ordinary Differential Equations,Eds. A. Caada,P. Drabek and A. Fonda［M］.Elsevier,2004:437-533.
［3］ M. J. Prelle,M. F. Singer.Elementary first integrals of differential equations［J］.Trans. Amer. Math. Soc.,1983,279:215-229.
［4］ M. F. Singer.Liouvillian first integrals of differential equations［J］.Trans. Amer. Math. Soc.,1992,333:673-688.
［5］ H. Yoshida.Necessary condition for the existence of algebraic first integrals.I Kowalevski′s exponents［J］.Celestial Mechanics,1983,31:363-379.
［6］ H. Yoshida.Necessary condition for the existence of algebraic first integrals.II:condition for algebraic integrability［J］.Celestial Mechanics,1983,31:381-399.
［7］ S. L. Ziglin.Branching of solutions and nonexistence of first integrals in Hamiltonian mechanics I［J］.Functional Anal. Appl.,1983,16:181-189.
［8］ H. Poincaré.Sur l′intégration des équations différentielles du premier order et du premier degré Ⅰ and Ⅱ［J］.Rendiconti del circolo matematico di Palermo,1891,5:161-191; 1897,11:193-239.
［9］ S. D. Furta.On non-integrability of general systems of differential equations［J］.Z. angew Math. Phys.,1996,47:112-131.
［10］ W. Li,J. Llibre,X. Zhang.Local first integrals of differential systems and diffeomorphisms［J］.Z. angew Math. Phys.,2003,54:235-255.
［11］ X. Zhang.Local first integrals for systems of differential equations［J］.J. Phys. A,2003,36:12243-12253.
［12］ K. H. Kwek,Y. Li,S. Shi.Partial integrability for general nonlinear systems［J］.Z. Angew Math. Phys.,2003,54:26-47. ［13］ J. Chen,Y. Yi,X. Zhang.First integrals and normal forms for germs of analytic vector fields［J］.J. Differential Equations,2008,245:1167-1184.
［14］ S. Shi.On the nonexistence of rational first integrals for nonlinear systems and semiquasihomogeneous systems［J］.J. Math. Anal. Appl.,2007,335:125-134.
［15］ W. Cong,J. Llibre J.,X. Zhang.Generalized rational first integrals of analytic differential systems［J］.J. Differential Equations,2011,251:2770-2788.
［16］ A. Baider,R. C. Churchill,D. L. Rod,et al.On the infinitesimal geometry of integrable systems［R］.Rhode Island:Fields Institute Communications 7,American Mathematical Society,Providence,1996.
［17］ W. Fulton.Algebraic Curves:An Introduction to Algebraic Geometry［M］.London:The Benjamin/Cummings Publishing Com Inc,1978.
［18］ J. Moser.The analytic invariants of an area-preserving mapping near a hyperbolic fixed point［J］.Comm. Pure Appl. Math.,1956,9:673-692.
［19］ X. Zhang.Analytic normalization of analytic integrable systems and the embedding Flows［J］.Differential Equations,2008,244:1080-1092.
［20］ X. Zhang.Analytic integrable systems:Analytic normalization and embedding flows［J］.Differential Equations,2013,254:3000-3022.
［21］ K. S. Wu,X. Zhang.Analytic normalization of analytically integrable differential systems near a periodic orbit［J］.Differential Equations,2014,256:3552-3567.
［22］ J. Guckenheimer,P. Holmes.Nonlinear Oscillations,Dynamical Systems,and Bifurcations of Vector Fields,2nd Ed［M］.New York:Springer-Verlag,2002.
［23］ G. R. Belitskii.Smooth equivalence of germs of vector fields with one zero or a pair of purely imaginary eigenvalues［J］.Funct. Anal. Appl.,1986,20(4):253-259.
［24］ N. T. Zung.Convergence versus integrability in Poincaré-Dulac normal form［J］.Math. Res. Lett.,2002,9:217-228.
(Zhenzhen Feng,Haoran Gu)