The Euler-Jacobi-Lie integrability theorem

This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of n differential equations is proved, which admits n ? 2 independent symmetry fields and an invariant volume n-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.     

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n₁+n₂ into the pure-slow regular matrix differential Riccati equation of dimension n₁ and the pure-fast stiff matrix differential Riccati equation of dimension n₂. A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.     

Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation

A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension (2n+2) with an n-dimensional abelian symmetry group. Examples for n=1,2,3 are presented to illustrate application of the theory.     

On the number of zeros of Abelian integrals for a polynomial Hamiltonian irregular at infinity

Up to now, most of the results on the tangential Hilbert 16th problem have been concerned with the Hamiltonian regular at infinity, i.e., its principal homogeneous part is a product of the pairwise different linear forms. In this paper, we study a polynomial Hamiltonian which is not regular at infinity. It is shown that the space of Abelian integral for this Hamiltonian is finitely generated as a R[h] module by several basic integrals which satisfy the Picard-Fuchs system of linear differential equations. Applying the bound meandering principle, an upper bound for the number of complex isolated zeros of Abelian integrals is obtained on a positive distance from critical locus. This result is a partial solution of tangential Hilbert 16th problem for this Hamiltonian. As a consequence, we get an upper bound of the number of limit cycles produced by the period annulus of the non-Hamiltonian integrable quadratic systems whose almost all orbits are algebraic curves of degree k+n, under polynomial perturbation of arbitrary degree.     

On the dichotomy of linear autonomous systems of functional-differential equations

We study the exponential dichotomy of solutions to the Cauchy problem for a subclass of linear autonomous systems of functional-differential equations of neutral type by reducing this problem to the same problem for a difference equation of the form x _n = Γx _n?1 with a compact operator Γ, in Banach space. The spectrum of Γ is described. The dichotomy of the solutions of the Cauchy problem is a consequence of the dichotomy of the spectrum. In a particular case, the result obtained provides a criterion for the dichotomy for a linear autonomous system of functional-differential equations of retarded type of general form.     

Noncommutative Integrability from Classical to Quantum Mechanics

We propose in this work a definition of integrable quantum system, which is based upon the correspondence with the concept of noncommutative integrability for classical mechanical systems. We then determine sufficient conditions under which, given an integrable classical system, it is possible to construct an integrable quantum system by means of a quantization procedure based on the symmetrized product of operators. As a first example of application of such an approach, we will consider the possible cases of noncommutative integrability for systems with rotational symmetry in an n-dimensional Euclidean configuration space.     

Non-integrability of some Painlevé VI-equations and dilogarithms

The paper studies the Painlevé VI^e equations from the point of view of Hamiltonian nonintegrability. For certain infinite number of points in the parameter space we prove that the equations are not integrable. Our approach uses recent advance in Hamiltonian integrability reducing the problem to higher differential Galois groups as well as the monodromy of dilogarithic functions.      

Aspects of Poincaré’s Program for Dynamical Systems and Mathematical Physics

This article is mainly historical, except for the discussion of integrability and characteristic exponents in Sect.?2. After summarising the achievements of Henri Poincaré, we discuss his theory of critical exponents. The theory is applied to the case of three degrees-of-freedom Hamiltonian systems in (1:2:n)-resonance (n>4). In addition we discuss Poincaré??s mathematical physics, in particular the theory of partial differential equations, rotating fluid masses and relativity. Attention is given to the priority question of Special Relativity.     

Integrable Systems on Phase Spaces with a Nonflat Metric

We study the integrability problem for evolution systems on phase spaces with a nonflat metric. We show that if the phase space is a sphere, the Hamiltonian systems are generated by the action of the Hamiltonian operators on the variations of the phase-space geodesics and the integrability problem for the evolution systems reduces to the integrability problem for the equations of motion for the frames on the phase space. We relate the bi-Hamiltonian representation of the evolution systems to the differential-geometric properties of the phase space.     

Automorphism groups of second order differential equations

The automorphism group of a system ofn second order differential equations is shown to be a Lie group of dimension at mostn ²+n in the autonomous case, and of dimension at most (n+1)² in the heteronomous case. The equations whose automorphism groups have this maximum dimension are classified.     

Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients

We show that the structure of the Lie symmetry algebra of a system of n linear second-order ordinary differential equations with constant coefficients depends on at most n-1 parameters. The tools used are Jordan canonical forms and appropriate scaling transformations. We put our approach to test by presenting a simple proof of the fact that the dimension of the symmetry Lie algebra of a system of two linear second-order ordinary differential with constant coefficients is either 7, 8 or 15. Also, we establish for the first time that the dimension of the symmetry Lie algebra of a system of three linear second-order ordinary differential equations with constant coefficients is 10, 12, 13 or 24.     

Method for Solving the Multidimensional n‐Wave Resonant Equations and Geometry of Generalized Darboux‐Manakov‐Zakharov Systems

The intrinsic geometric properties of generalized Darboux‐Manakov‐Zakharov systems of semilinear partial differential equations (1) for a real‐valued function u(x₁, …, x_n) are studied with particular reference to the linear systems in this equation class. System (1) is overdetermined and will not generally be involutive in the sense of Cartan: its coefficients will be constrained by complicated nonlinear integrability conditions. We derive tools for explicitly constructing involutive systems of the form (1) , essentially solving the integrability conditions. Specializing to the linear case provides us with a novel way of viewing and solving the multidimensional n‐wave resonant interaction system and its modified version. For each integer n≥ 3 and nonnegative integer k, our procedure constructs solutions of the n‐wave resonant interaction system depending on at least k arbitrary functions each of one variable. The construction of these solutions relies only on differentiation, linear algebra, and the solution of ordinary differential equations.     

On stability of autonomous systems with internal resonance : PMM vol. 39, n 6, 1975, pp. 974–984

When investigating the stability of the trivial solution of an autonomous system of ordinary differential equations in the critical case of n pairs of pure imaginary roots an essential role can be played by the presence of integral linear dependences between the system's frequencies or, in other words, by the internal resonance. Various special cases of this problem were examined in [1–6]. Our aims are: to obtain a special (normal) form of the differential equation system with internal resonance of most general form in it; to ascertain the conditions under which the presence of internal resonance does not permit the application stability investigation methods developed for resonance-free systems; to solve the stability problem in one of the most important cases of odd-order internal resonance, generalizing the preceding investigations. In the solution of the last problem the necessary and sufficient conditions are given for the stability of the model (simplified) system. Using Chetaev's theorem we show that as a rule the instability of the original system follows from the Instability of the model system. Cases of structurally-unstable instability (^*) for which the model system does not resolve the problem of stability are outlined. The results obtained are extended, in particular, to Hamiltonian systems.     

Indefinite Boundary Eigenvalue Problems in a Pontrjagin Space Setting

We study eigenvalue problems Fy = λ Gy consisting of Hamiltonian systems of ordinary differential equations on a compact interval with symmetric λ-linear boundary conditions. The problems we are interested in are non-definite: neither left-nor right-definite. Instead of this, we give some weak condition on one coefficient of the Hamiltonian system which ensures that a hermitian form associated with the operator F has at most finitely many negative squares. This enables us to study the problem by the help of a compact self-adjoint operator in a Pontrjagin space and we obtain as a main result uniformly convergent eigenfunction expansions. In the final section, applications to formally self-adjoint differential equations of higher order are given.     

The complete symmetry group of the generalised hyperladder problem

We further consider the n-dimensional ladder system, that is the homogeneous quadratic system of first-order differential equations of the form , i=1,n, where (a_ij)=(i+1−j), i,j=1,n introduced by Imai and Hirata (nlin.SI/0212007). We establish the most general system of first-order ordinary differential equations invariant under the algebra which characterises the ladder system of Imai and Hirata and the algebra of minimal dimension required to specify completely this most general system. We provide the complete symmetry group of the generalised hyperladder system and discuss its integrability.     

First integrals and normal forms for germs of analytic vector fields

For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré's classical results [H. Poincaré, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rend. Circ. Mat. Palermo 5 (1891) 161-191; 11 (1897) 193-239] on autonomous systems and Theorem 5 of [Weigu Li, J. Llibre, Xiang Zhang, Local first integrals of differential systems and diffeomorphism, Z. Angew. Math. Phys. 54 (2003) 235-255] on periodic systems. Then in the space of analytic autonomous systems in C2n with exactly n resonances and n functionally independent first integrals, our results are related to the convergence and generic divergence of the normalizations. Lastly for a planar Hamiltonian system it is well known that the system has an isochronous center if and only if it can be linearizable in a neighborhood of the center. Using the Euler-Lagrange equation we provide a new approach to its proof.     

Conditions implying the vanishing of the Hamiltonian at infinity in optimal control problems

In an infinite horizon optimal control problem the Hamiltonian vanishes at infinity when the differential equation is autonomous and the integrand in the criterion satisfies some weak integrability conditions. A generalization of Michel’s result (in Econometrica 50:975–985, 1982) is obtained.