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Non-existence criteria for Laurent polynomial first integrals [PDF]
Electronic Journal of Qualitative Theory of Differential Equations, 2003 In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial first integrals for a general nonlinear systems of ordinary differential equations $\dot x = f(x)$, $x \in \mathbb{R}^n$ with $f(0) = 0$. We show
Shaoyun Shi, Yuzhu Han
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Darboux polynomials and first integrals of natural polynomial Hamiltonian systems [PDF]
Physics Letters A, 2004 We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function.
Maciejewski, Andrzej J. +1 more
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Some metrics admitting nonpolynomial first integrals of the geodesic equation
Physics Letters B, 2021 It is commonly known that Killing vectors and tensors are in one–to–one correspondence with polynomial first integrals of the geodesic equation. In this work, metrics admitting nonpolynomial first integrals of the geodesic equation are constructed, each ...
Anton Galajinsky
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Covariants, Invariant Subsets, and First Integrals [PDF]
Épijournal de Géométrie Algébrique, 2020 Let $k$ be an algebraically closed field of characteristic 0, and let $V$ be a finite-dimensional vector space. Let $End(V)$ be the semigroup of all polynomial endomorphisms of $V$.
Frank Grosshans, Hanspeter Kraft
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First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator [PDF]
Symmetry, Integrability and Geometry: Methods and Applications, 2011 We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians H obtained as one-dimensional extensions of natural (geodesic) n-dimensional Hamiltonians L. The Liouville integrability of
Giovanni Rastelli +2 more
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Liouvillian first integrals of second order polynomial differential equations
Electronic Journal of Differential Equations, 1999 The author proves the following theorem: If the system \[ \dot x= P(x,y),\quad \dot y= Q(x,y)\quad (P,\;Q\text{ polynomials})\tag{1} \] has a Liouvillian integrating factor of the form: \[ \exp\{\int Udx+ Vdy\},\quad U_y= V_x,\tag{2} \] where \(U\), \(V\) are rational functions of \(x\) and \(y\), then there exists a Darbouxian integrating factor of ...
Colin Christopher
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Polynomial and rational first integrals for planar homogeneous polynomial differential systems [PDF]
Publicacions Matemàtiques, 2021 In this paper we find necessary and suficient conditions in order that a planar homogeneous polynomial differential system has a polynomial or rational first integral. We apply these conditions to linear and quadratic homogeneous polynomial differential systems.
Giné, Jaume, Grau, Maite, Llibre, Jaume
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Limit Cycles of Polynomially Integrable Piecewise Differential Systems
Axioms, 2023 In this paper, we study how many algebraic limit cycles have the discontinuous piecewise linear differential systems separated by a straight line, with polynomial first integrals on both sides.
Belén García +3 more
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Darboux and rational first integrals for a family of cubic three dimensional systems
Zanco Journal of Pure and Applied Sciences, 2021 In this paper, we investigate the first integrals of the following system exam.PNG exam1.PNG This kind of system is a particular case of the jerk cubic three dimensional polynomial differential systems.
Sarbast Hussein Mikaeel, Azad I. Amen
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POLYNOMIAL FIRST INTEGRALS FOR THE CHEN AND LÜ SYSTEMS [PDF]
International Journal of Bifurcation and Chaos, 2012 We characterize all the values of the parameters for which the Chen and Lü systems have polynomial first integrals by using weight homogeneous polynomials and the method of characteristics for solving partial differential equations. We improve previous results which were not complete.
Llibre, Jaume, Valls, Clàudia
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