Existence Of First Integrals Research Articles

Invariants of polynomial vector fields

We characterize the existence of first integrals and invariants (first integrals depending on the time) for the polynomial vector fields which are invariant under an involution.

On Integral Conditions for the Existence of First Integrals for Analytic Deformations of Complex Saddle Singularities

On Integral Conditions for the Existence of First Integrals for Analytic Deformations of Complex Saddle Singularities

The problem of motion of a rigid body with a fixed point in a flow of particles

The problem of motion of a rigid body with a fixed point in a free molecular flow of particles is considered. It is shown that the equations of motion of the body generalize the classical Euler - Poisson equations of motion of a heavy rigid body with a fixed point and they are represented in the form of the classical Euler - Poisson equations in the case, when the surface of the body in a flow of particles is a sphere. Problems of the existence of first integrals in the considered system are discussed.

Local-Activity and Simultaneous Zero-Hopf Bifurcations Leading to Multistability in a Memristive Circuit

In this paper, we consider a memristive circuit consisting of three elements: a passive linear inductor, a passive linear capacitor and an active memristive device. The circuit is described by a four-parameter system of ordinary differential equations. We study in detail the role of parameters in the dynamics of the system. Using the existence of first integrals, we show that the circuit may present a continuum of stable periodic orbits, which arise due to the occurrence of infinitely many simultaneous zero-Hopf bifurcations on a line of equilibria located in the region where the memristance is negative and, consequently, the memristive device is locally-active. These bifurcations lead to multistability, which is a difficult and interesting problem in applied models, since the final state of a solution depends crucially on its initial condition. We also study the control of multistability by varying a parameter related to the state variable of the memristive device. All analytical results obtained were corroborated by numerical simulations.

Integrability and Limit Cycles via First Integrals

In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems.

On the integrability problem for the Hopf-zero singularity and its relation with the inverse Jacobi multiplier

In this paper we use the orbital normal form of the nondegenerate Hopf-zero singularity to obtain necessary conditions for the existence of first integrals for such singularity. Also, we analyze the relation between the existence of first integrals and of inverse Jacobi multipliers. Some algorithmic procedures for determining the existence of first integrals are presented, and they are applied to some families of vector fields.

Irreducible holonomy groups and first integrals for holomorphic foliations

Abstract We study groups of germs of complex diffeomorphisms having a property called irreducibility. The notion is motivated by the similar property of the fundamental group of the complement of an irreducible hypersurface in the complex projective space. Natural examples of such groups of germ maps are given by holonomy groups and monodromy groups of integrable systems (foliations) under certain conditions. We prove some finiteness results for these groups extending previous results in [D. Cerveau and F. Loray, Un théorème de Frobenius singulier via l’arithmétique élémentaire, J. Number Theory 68 1998, 2, 217–228]. Applications are given to the framework of germs of holomorphic foliations. We prove the existence of first integrals under certain irreducibility or more general conditions on the tangent cone of the foliation after a punctual blow-up.

Integrability of reversible and equivariant quadratic polynomial differential systems in the plane

We study the existence of first integrals for the class of reversible and equivariant quadratic polynomial differential systems in the plane. We put special emphasis in the study of the analytic first integrals.

On the overfly algorithm in deep learning of neural networks

In this paper, we investigate the supervised backpropagation training of multilayer neural networks from a dynamical systems point of view. We discuss some links with the qualitative theory of differential equations and introduce the overfly algorithm to tackle the local minima problem. Our approach is based on the existence of first integrals of the generalised gradient system with build-in dissipation.

Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere

We consider the multi-dimensional generalisation of the problem of a sphere, with axi-symmetric mass distribution, that rolls without slipping or spinning over a plane. Using recent results from Garc?a-Naranjo [21] and Garc?a-Naranjo and Marrero [22], we show that the reduced equations of motion possess an invariant measure and may be represented in Hamiltonian form by Chaplygin?s reducing multiplier method. We also prove a general result on the existence of first integrals for certain Hamiltonisable Chaplygin systems with internal symmetries that is used to determine conserved quantities of the problem.

Quadratic three-dimensional differential systems having invariant planes with total multiplicity nine

In this paper we consider all the quadratic polynomial differential systems in $$\mathbb {R}^3$$ having exactly nine invariant planes taking into account their multiplicities. This is the maximum number of invariant planes that these kind of systems can have, without taking into account the infinite plane. We prove that there exist thirty possible configurations for these invariant planes, and we study the realization and the existence of first integrals for each one of these configurations. We show that at least twenty three of these configurations are realizable and provide explicit examples for each one of them.

Dicritical nilpotent holomorphic foliations

We study in this paper several properties concerning singularities of foliations in \begin{document}$ {\left( {{\mathbb{C}}^{3}}\rm{,}\bf{0} \right)}$\end{document} that are pull-back of dicritical foliations in \begin{document}$ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$\end{document} . Particularly, we will investigate the existence of first integrals (holomorphic and meromorphic) and the dicriticalness of such a foliation. In the study of meromorphic first integrals we follow the same method used by R. Meziani and P. Sad in dimension two. While the foliations we study are pull-back of foliations in \begin{document}$ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$\end{document} , the adaptations are not straightforward.

First Integrals and Periodic Solutions of a System with Power Nonlinearities

Under consideration is some system of ordinary differential equations with power nonlinearities. These systems are widely used in mathematical biology and chemical kinetics, and can also occur by reduction of more sophisticatedmodels. We formulate conditions on the system parameters which guarantee the existence of first integrals defined by the combinations of power and logarithmic functions of the phase variables. Using the first integrals, we construct periodic solutions for the three-variable systems. A few examples are given illustrating the results.

PREY-PREDATOR TRIDIAGONAL LOTKA-VOLTERRA MODELS

The prey-predator Lotka-Volterra models are some of the mostpopular mathematical models in biology and chemistry. Usually, these mod-els are the first tool used to analyze cooperativity, oscillatory behavior, andspaces synchronization at large scale of biochemistry, bio-molecular, and med-ical interaction models. These properties are in relationship with existence offirst integrals and stability behavior of the systems. Especially, and maybe themost essential results are related to existence (or non-existence) of boundedand periodic orbits. In this paper we determine a family of independent first integrals, criteria forexistence of bounded orbits, and stability criteria for a family of n-dimensionalLotka-Volterra systems, generated by periodic tridiagonal matrices.

Transcendental First Integrals of Dynamical Systems on the Tangent Bundle to the Sphere

In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.

Symmetry analysis and conservation laws of some third-order difference equations

We adopt a procedure for determining the continuous (Lie) symmetries for third-order difference equations and utilize these to reduce the order of the equations – this reduction leads to some solutions of the equations under investigation. We further investigate the existence of first integrals of the third order equations using a, now, established, procedure.

Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom

We consider the problem of the existence of first integrals that are polynomial in momenta for Hamiltonian systems with two degrees of freedom on a fixed energy level (conditional Birkhoff integrals). It is assumed that the potential has several singular points. We show that in the presence of conditional polynomial integrals, the sum of degrees of the singularities does not exceed twice the Euler characteristic of the configuration space. The proof is based on introducing a complex structure on the configuration space and estimating the degree of the divisor corresponding to the leading term of the integral with respect to the momentum. We also prove that the topological entropy is positive under certain conditions.

First Integrals, Liouville Theorem, and Dirac Brackets

In this paper, we discuss the conditions for the existence of first integrals of movement and the Liouville theorem on integrable systems. We revise the core results of the Hamilton-Jacobi theory and discuss the extension of the formalism to encompass constrained systems using Dirac brackets, originally developed in the context of the canonical quantization of constrained systems. As an application, we analyze a Hamiltonian that represents the classical limit of a Fermionic system of oscillators.

Integrability and Dynamics of Quadratic Three-Dimensional Differential Systems Having an Invariant Paraboloid

Invariant algebraic surfaces are commonly observed in differential systems arising in mathematical modeling of natural phenomena. In this paper, we study the integrability and dynamics of quadratic polynomial differential systems defined in [Formula: see text] having an elliptic paraboloid as an invariant algebraic surface. We obtain the normal form for these kind of systems and, by using the invariant paraboloid, we prove the existence of first integrals, exponential factors, Darboux invariants and inverse Jacobi multipliers, for suitable choices of parameter values. We characterize all the possible configurations of invariant parallels and invariant meridians on the invariant paraboloid and give necessary conditions for the invariant parallel to be a limit cycle and for the invariant meridian to have two orbits heteroclinic to a point at infinity. We also study the dynamics of a particular class of the quadratic polynomial differential systems having an invariant paraboloid, giving information about localization and local stability of finite singular points and, by using the Poincaré compactification, we study their dynamics on the Poincaré sphere (at infinity). Finally, we study the well-known Rabinovich system in the case of invariant paraboloids, performing a detailed study of its dynamics restricted to these invariant algebraic surfaces.

Polynomial conservation laws for the Lorentz gas and the Boltzmann–Gibbs gas

The problem of conditions ensuring the existence of first integrals that are polynomials in the momenta (velocities) is considered for certain multidimensional billiard systems which play an important role in non-equilibrium statistical mechanics. These are the Lorentz gas, a particle in a Euclidean space with (not necessarily convex) scattering domains, and the Boltzmann–Gibbs gas, a system of small identical balls in a rectangular box which collide elastically with one another and the walls of the box. The ergodic properties of such systems are only partially understood: some problems are still waiting for solution, and in certain cases (for instance, when the scatterers are non-convex) the system is known not to be ergodic. An approach to showing the absence of a non-trivial polynomial first integral with continuously differentiable coefficients is developed. The known first integrals for integrable problems in dynamics are mostly polynomials in the momenta (or functions of polynomials). The investigation of multidimensional billiards with non-compact configuration space, when there is no hope for ergodic behaviour, is of particular interest. Applications of the general results on the absence of non-trivial polynomial integrals to problems in statistical mechanics are discussed. Bibliography: 62 titles.