Integrability of Planar Polynomial Differential Systems through Linear Differential Equations

Giacomini, Héctor; Giné, Jaume; Grau, Maite

doi:10.1216/rmjm/1181069462

Cited by 19 publications

(54 citation statements)

References 24 publications

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“…The transformation method we shall apply is derived in [6]. For the sake of completeness, we summarize it here and give some remarks.…”

Section: Statement Of the Methodsmentioning

confidence: 99%

“…Comparing with (6) we see that the coefficient of y 4 must be b k4 x k , which determines c as follows:…”

Section: Application To Kukles Systemsmentioning

confidence: 99%

“…In the present note we want to do a step towards finding non-liouvillian first integrals. We consider a transformation method introduced very recently [6] that relates planar first-order differential systems to second order equations. The latter equations are sometimes called "special function equations" as they are very often solved by special (non-liouvillian) functions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A simple method for constructing non-liouvillian first integrals of autonomous planar systems

Schulze-Halberg

2006

Czech Math J

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“…The transformation method we shall apply is derived in [6]. For the sake of completeness, we summarize it here and give some remarks.…”

Section: Statement Of the Methodsmentioning

confidence: 99%

“…Comparing with (6) we see that the coefficient of y 4 must be b k4 x k , which determines c as follows:…”

Section: Application To Kukles Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A simple method for constructing non-liouvillian first integrals of autonomous planar systems

Schulze-Halberg

2006

Czech Math J

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“…This observation permits the generalization of the DarbouxÕs integrability theory given in [59,60] where a new kind of first integrals, not only the Liouvillian ones, is described. In [63] it is showed that a transformation method relating planar differential systems to second order linear equations is an effective tool for finding non-Liouvillian first integrals. The problem of finding a first integral for system (1) and the functional class that it belongs to is what we call the integrability problem.…”

Section: Invariant Algebraic Curvesmentioning

confidence: 99%

On some open problems in planar differential systems and Hilbert’s 16th problem

Giné

2007

Chaos, Solitons & Fractals

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“…Many examples of invariants with a quasipolynomial cofactor are given in [9] as well as a method to find first integrals, which are non-Liouvillian in general, for certain families of systems.…”

Section: ·1 Particular Algebraic Solutionsmentioning

confidence: 99%

The role of algebraic solutions in planar polynomial differential systems

Giacomini¹,

Giné²,

Grau³

2007

Math. Proc. Camb. Phil. Soc.

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International audienceWe study a planar polynomial differential system, given by $\dot{x}=P(x,y)$, $\dot{y}=Q(x,y)$. We consider a function $I(x,y)=\exp\!\{h_2(x) A_1(x,y) \diagup A_0(x,y) \}$ $ h_1(x)\prod_{i=1}^{\ell} (y-g_i(x))^{\alpha_i}$, where gi(x) are algebraic functions of $x$, $A_1(x,y)=\prod_{k=1}^r (y-a_k(x))$, $A_0(x,y)=\prod_{j=1}^s (y-\tilde{g}_j(x))$ with ak(x) and $\tilde{g}_j(x)$ algebraic functions, A0(x,y) and A1(x,y) do not share any common factor, h2(x) is a rational function, h(x) and h1(x) are functions of x with a rational logarithmic derivative and $\alpha_i \in \mathbb{C}$. We show that if I(x,y) is a first integral or an integrating factor, then I(x,y) is a Darboux function. A Darboux function is a function of the form $f_1^{\lambda_1} \cdots f_p^{\lambda_p} \exp\{h/f_0\}$, where fi and h are polynomials in $\mathbb{C}[x,y]$ and the λi's are complex numbers. In order to prove this result, we show that if g(x) is an algebraic particular solution, that is, if there exists an irreducible polynomial f(x,y) such that f(x,g(x)) ≡ 0, then f(x,y) = 0 is an invariant algebraic curve of the system. In relation with this fact, we give some characteristics related to particular solutions and functions of the form I(x,y) such as the structure of their cofactor.Moreover, we consider A0(x,y), A1(x,y) and h2(x) as before and a function of the form $\Phi(x,y):= \exp \{h_2(x)\, A_1(x,y)/A_0 (x,y) \}$. We show that if the derivative of Φ(x,y) with respect to the flow is well defined over {(x,y): A0(x,y) = 0} then Φ(x,y) gives rise to an exponential factor. This exponential factor has the form exp {R(x,y)} where $R=h_2 A_1/A_0 + B_1/B_0$ and with B1/B0 a function of the same form as h2 A1/A0. Hence, exp {R(x,y)} factorizes as the product Φ(x,y) Ψ(x,y), for Ψ(x,y): = exp {B1/B0

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Integrability of Planar Polynomial Differential Systems through Linear Differential Equations

Cited by 19 publications

References 24 publications

A simple method for constructing non-liouvillian first integrals of autonomous planar systems

A simple method for constructing non-liouvillian first integrals of autonomous planar systems

On some open problems in planar differential systems and Hilbert’s 16th problem

The role of algebraic solutions in planar polynomial differential systems

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