Morales Ramis Theory Research Articles

Semiclassical perturbations of single-degree-of-freedom Hamiltonian systems II: Nonintegrability

Continuing from Paper I [Ohsawa and Yagasaki, J. Math. Phys. 65, 102706 (2024)], we study semiclassical perturbations of single-degree-of-freedom analytic Hamiltonian systems and provide a sufficient condition for its meromorphic nonintegrability such that the first integrals depend on the small parameter meromorphically. Our approach is based on a generalization due to Ayoul and Zung of the Morales-Ramis theory, which enables us to show the meromorphic nonintegrability of dynamical systems by using the differential Galois theory. We remark that standard systems of Hagedorn and Heller for the semiclassical Gaussian wave packet dynamics are analytically integrable as well as the corresponding classical systems. We illustrate our theory for a bounded potential.

Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales–Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable. We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homoclinic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.

Nonintegrability of the SEIR epidemic model

We prove the nonintegrability of the susceptible–exposed–infected–removed (SEIR) epidemic model in the Bogoyavlenskij sense. This property of the SEIR model is different from the more fundamental susceptible–infected–removed (SIR) model, which is Bogoyavlenskij-integrable. Our basic tool for the proof is an extension of the Morales–Ramis theory due to Ayoul and Zung. Moreover, we introduce three new state variables and extend the system to a six-dimensional system to treat transcendental first integrals and commutative vector fields. We also use the fact that the incomplete gamma function Γ(α,x) is not elementary for α⁄∈N, of which a proof is included.

Differential Galoisian approach to Jacobi integrability of general analytic dynamical systems and its application

The Morales-Ramis theory provides an effective and powerful non-integrability criterion for complex analytic Hamiltonian systems via the differential Galoisian obstruction. In this paper, we give a new Morales-Ramis type theorem on the meromorphic Jacobi non-integrability of general analytic dynamical systems. The key point is to show that the existence of Jacobian multipliers of a nonlinear system implies the existence of common Jacobian multipliers of Lie algebra associated with the identity component. In addition, we apply our results to the polynomial integrability of Karabut systems for stationary gravity waves in finite depth.

Non-integrability of a three-dimensional generalized Hénon-Heiles system

In recent paper Fakkousy et al. show that the 3D Hénon-Heiles system with Hamiltonian $$ H = \frac{1}{2} (p_1 ^2 + p_2 ^2 + p_3 ^2) +\frac{1}{2} (A q_1 ^2 + C q_2 ^2 + B q_3 ^2) + (\alpha q_1 ^2 + \gamma q_2 ^2)q_3 + \frac{\beta }{3}q_3 ^3 $$ is integrable in sense of Liouville when $$\alpha = \gamma , \frac{\alpha }{\beta } = 1, A = B = C$$ ; or $$\alpha = \gamma , \frac{\alpha }{\beta } = \frac{1}{6}, A = C$$ , B-arbitrary; or $$\alpha = \gamma , \frac{\alpha }{\beta } = \frac{1}{16}, A = C, \frac{A}{B} = \frac{1}{16}$$ (and of course, when $$\alpha =\gamma =0$$ , in which case the Hamiltonian is separable). It is known that the second case remains integrable for A, C, B arbitrary. Using Morales-Ramis theory, we prove that there are no other cases of integrability for this system.

On the integrability of Hamiltonian 1:2:2 resonance

We study the integrability of the Hamiltonian normal form of 1:2:2 resonance. It is known that this normal form truncated to order three is integrable. The truncated to order four normal form contains many parameters. For a generic choice of parameters in the normal form up to order four, we carry on non-integrability analysis, based on the Morales–Ramis theory using only first variational equations along certain particular solutions. The non-integrability obtained by algebraic proofs produces dynamics illustrated by some numerical experiments.We also isolate a non-trivial case of integrability.

Nonintegrability of the unfoldings of codimension-two bifurcations

Codimension-two bifurcations are fundamental and interesting phenomena in dynamical systems. Fold-Hopf and double-Hopf bifurcations are the most important among them. We study the unfoldings of these two codimension-two bifurcations, and obtain sufficient conditions for their nonintegrability in the meaning of Bogoyavlenskij. We reduce the problems of the unfoldings to those of planar polynomial vector fields and analyze the nonintegrability of the planar vector fields, based on Ayoul and Zung’s version of the Morales–Ramis theory. New useful criteria for nonintegrability of planar polynomial vector fields are also obtained. The approaches used here are applicable to many problems including circular symmetric systems.

Integrability analysis of the Shimizu–Morioka system

The aim of this paper is to give some new insights into the Shimizu–Morioka systemx˙=y,y˙=x−λy−xz,z˙=−αz+x2,from the integrability point of view. Firstly, we propose a linear scaling in time and coordinates which converts the Shimizu–Morioka system into a special case of the Rucklidge system when α ≠ 0 and discuss the relationship between Shimizu–Morioka system and Rucklidge system. Based on this observation, Darboux integrability of the Shimizu–Morioka system with α ≠ 0 is trivially derived from the corresponding results on the Rucklidge system. When α=0, we investigate Darboux integrability of the Shimizu–Morioka system by the Gröbner basis in algebraic geometry. Secondly, we use the stability of the singular points and periodic orbits to study the nonexistence of global C1 first integrals of the Shimizu–Morioka system. Finally, in the case α ≠ 0, we prove it is not rationally integrable for almost all parameter values by an extended Morales-Ramis theory, and in the case α=0, we show that it is not algebraically integrable by quasi-homogeneous decompositions and Kowalevski exponents. Our results are in accord with the fact that this system admits chaotic behaviors for a large range of its parameters.

Some tastings in Morales-Ramis theory

In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian systems through Liouville Arnold theorem and integrability of linear differential equations through differential Galois theory. As contribution, we obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system.

Nonintegrability of Parametrically Forced Nonlinear Oscillators

We discuss nonintegrability of parametrically forced nonlinear oscillators which are represented by second-order homogeneous differential equations with trigonometric coefficients and contain the Duffing and van der Pol oscillators as special cases. Specifically, we give sufficient conditions for their rational nonintegrability in the meaning of Bogoyavlenskij, using the Kovacic algorithm as well as an extension of the Morales–Ramis theory due to Ayoul and Zung. In application of the extended Morales–Ramis theory, for the associated variational equations, the identity components of their differential Galois groups are shown to be not commutative even if the differential Galois groups are triangularizable, i. e., they can be solved by quadratures. The obtained results are very general and reveal their rational nonintegrability for the wide class of parametrically forced nonlinear oscillators. We also give two examples for the van der Pol and Duffing oscillators to demonstrate our results.

Non-integrability of the spacial n-center problem

We prove the non-integrability of the spacial n-center problem. In order to prove it, we focus on the singularity of the differential equations extended to the complex space and then apply the Morales–Ramis theory to it. We also show the non-integrability of the spacial restricted n+1-body problem.

Nonintegrability of the Armbruster--Guckenheimer--Kim Quartic Hamiltonian Through Morales--Ramis Theory

We show the non-integrability of the three-parameter Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory, with the exception of the three already known integrable cases. We use Poincar\'e sections to illustrate the breakdown of regular motion for some parameter values.

DIFFERENTIAL GALOIS GROUPS AND REPRESENTATION OF QUIVERS FOR SEISMIC MODELS WITH CONSTANT HESSIAN OF SQUARE OF SLOWNESS

The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point view of the theory of Differential Algebra. In particular, by Morales-Ramis theory it is possible to analyze integrable Hamiltonian systems through the abelian structure of their variational equations. In this paper we obtain the abelian differential Galois groups and the representation of the quiver, that allow us to obtain such abelian differential Galois groups, for some seismic models with constant Hessian of square of slowness, proposed in Cerveny 2001, which are equivalent to linear Hamiltonian systems with three uncoupled harmonic oscillators.

Non-integrability of a class of Painlevé IV equations as Hamiltonian systems

In this paper, we will prove the rational non-integrability of a class of Hamiltonian systems associated with Painlevé IV equation by using Morales-Ramis theory and Kovacic's algorithm, which, to some extent, also implies the non-integrability of the fourth Painlevé equation itself.

A note on algebraic potentials and Morales–Ramis theory

We present various properties of algebraic potentials, and then prove that some Morales-Ramis theorems readily apply for such potentials even if they are not in general meromorphic potentials. This allows in particular to precise some non-integrability proofs in celestial mechanics, where the mutual distances between the bodies appear in the potentials, and thus making this analysis unavoidable.

Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum

In this paper we analyze the non-integrability of the Wilbeforce spring-pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.

Galoisian obstruction to the integrability of general dynamical systems

The Galoisian approach to study the integrability of classical Hamiltonian systems, the so-called Morales–Ramis theory, has been proved to be useful and powerful by many applications. Here, two analogous forms of the Morales–Ramis theory for general dynamical systems both in vector field and mapping forms are given. Galois groups of the corresponding variational equations are studied, and some necessary conditions of the system to possess a certain number of integrals are presented. Several applications are given at last to illustrate our results.

Non-integrability of a class of Hamiltonian systems

In this paper, using the Morales-Ramis theory, we will give some new non-integrability results for a class of famous Hamiltonian models.

Non-Integrability of Some Few Body Problems in Two Degrees of Freedom

The basic theory of Differential Galois and in particular Morales–Ramis theory is reviewed with focus in analyzing the non-integrability of various problems of few bodies in Celestial Mechanics. The main theoretical tools are: Morales–Ramis theorem, the algebrization method and Kovacic’s algorithm. Morales–Ramis states that if a Hamiltonian system is completely integrable with meromorphic first integrals in involution in a neighborhood of a specific solution, then the differential Galois group of the normal variational equations is abelian. In case of two degrees of freedom, completely integrable means to have an additional first integral in involution with the Hamiltonian. The algebrization method permits under general conditions to recast the variational equation in a form suitable for its analysis by means of Kovacic’s algorithm. We apply these tools to various examples of few body problems in Celestial Mechanics: (a) the elliptic restricted three body in the plane with collision of the primaries; (b) a general Hamiltonian system of two degrees of freedom with homogeneous potential of degree −1; here we perform McGehee’s blow up and obtain the normal variational equation in the form of an hypergeometric equation. We recover Yoshida’s criterion for non-integrability. Then we contrast two methods to compute the Galois group: the well known, based in the Schwartz–Kimura table, and the lesser based in Kovacic’s algorithm. We apply these methodology to three problems: the rectangular four body problem, the anisotropic Kepler problem and two uncoupled Kepler problems in the line; the last two depend on a mass parameter, but while in the anisotropic problem it is integrable for only two values of the parameter, the two uncoupled Kepler problems is completely integrable for all values of the masses.

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases

In this paper the problem of classification of integrable natural Hamiltonian systems with $n$ degrees of freedom given by a Hamilton function which is the sum of the standard kinetic energy and a homogeneous polynomial potential $V$ of degree $k>2$ is investigated. It is assumed that the potential is not generic. Except for some particular cases a potential $V$ is not generic, if it admits a nonzero solution of equation $V'(\vd)=0$. The existence of such solution gives very strong integrability obstructions obtained in the frame of the Morales-Ramis theory. This theory gives also additional integrability obstructions which have the form of restrictions imposed on the eigenvalues $(\lambda_1,...,\lambda_n)$ of the Hessian matrix $V''(\vd)$ calculated at a non-zero $\vd\in\C^n$ satisfying $V'(\vd)=\vd$. Furthermore, we show that similarly to the generic case also for nongeneric potentials some universal relations between $(\lambda_1,...,\lambda_{n})$ calculated at various solutions of $V'(\vd)=\vd$ exist. We derive them for case $n=k=3$ applying the multivariable residue calculus. We demonstrate the strength of the obtained results analysing in details the nongeneric cases for $n=k=3$. Our analysis cover all the possibilities and we distinguish those cases where known methods are too weak to decide if the potential is integrable or not. Moreover, for $n=k=3$ thanks to this analysis a three-parameter family of potentials integrable or super-integrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.