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Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides

Abstract

The explicit solution \(x_{n}\left (t\right ) ,\) n = 1,2, of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODEs with second-degree polynomial right-hand sides, hence featuring 12 a priori arbitrary (time-independent) coefficients:

$$ \dot{x}_{n}=c_{n1}\left( x_{1}\right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}\left( x_{2}\right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. $$

The solution is explicitly provided if the 12 coefficients cnj (n = 1,2; j = 1,2,3,4,5,6) are expressed by explicitly provided formulas in terms of 10 a priori arbitrary parameters; the inverse problem to express these 10 parameters in terms of the 12 coefficients cnj is also explicitly solved, but it is found to imply—as it were, a posteriori—that the 12 coefficients cnj must then satisfy 4 algebraic constraints, which are explicitly exhibited. Special subcases are also identified the general solutions of which are completely periodic with a period independent of the initial data (“isochrony”), or are characterized by additional restrictions on the coefficients cnj which identify particularly interesting models.

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Acknowledgements

The results reported in this paper have been mainly obtained by a collaboration at a distance among its two authors (essentially via e-mails). We would like to acknowledge with thanks 2 grants, facilitating our future collaboration by allowing FP to visit (hopefully more than once) in 2021 the Department of Physics of the University of Rome “La Sapienza”: one granted by that University, and one granted jointly by the Istituto Nazionale di Alta Matematica (INdAM) of that University and by the International Institute of Theoretical Physics (ICTP) in Trieste in the framework of the ICTP-INdAM “Research in Pairs” Programme. Finally, we gratefully acknowledge a special contribution by François Leyvraz, who pointed out a serious flaw in a preliminary version of this paper, the elimination of which also entailed a substantial simplification of its presentation.

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Correspondence to Francesco Calogero.

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Communicated by:Frank Nijhoff

Appendices

Appendix : A

In this Appendix A we tersely demonstrate the following elementary fact, which clearly implies the result (6): that the solution of the initial-values problem for the ODE

$$ \dot{y}\left( t\right) =a_{2}\left[ y\left( t\right) \right]^{2}+a_{1}y\left( t\right) +a_{0}~, $$
(74a)

is provided by the formula

$$ y\left( t\right) =\frac{y_{+}\left[ y\left( 0\right) -y_{-}\right] -y_{-} \left[ y\left( 0\right) -y_{+}\right] \exp \left( \beta t\right) }{y\left( 0\right) -y_{-}-\left[ y\left( 0\right) -y_{+}\right] \exp \left( \beta t\right) }~, $$
(74b)

with y± defined as follows:

$$ y_{\pm }=\left( -a_{1}\pm \beta \right) /\left( 2a_{2}\right) ~,~~~\beta = \sqrt{\left( a_{1}\right)^{2}-4a_{0}a_{2}}~. $$
(74c)

Indeed the ODE (74a) can clearly be reformulated as follows:

$$ \dot{y}=a_{2}\left( y-y_{+}\right) \left( y-y_{-}\right) $$
(75a)

with y± defined by (74c); and then (again, via (74c)) this ODE can be rewritten as follows:

$$ \dot{y}\left[ \left( y-y_{+}\right)^{-1}-\left( y-y_{-}\right)^{-1}\right] =\beta ~. $$
(75b)

The integration of this ODE for the dependent variable \(y\left (t^{\prime }\right ) \) over the independent variable \(t^{\prime }\)—from \(t^{\prime }=0\) to \(t^{\prime }=t\)—clearly yields

$$ \ln \left[ \frac{y\left( t\right) -y_{+}}{y\left( 0\right) -y_{+}}\right] -\ln \left[ \frac{y\left( t\right) -y_{-}}{y\left( 0\right) -y_{-}}\right] =\beta t~, $$
(76)

which coincides—after exponentiation—with (74b). Q. E. D.

Finally let us emphasize that the results reported above are valid for generic values of the parameters; the interested reader shall have no difficulty to figure out the results in special cases, for instance those with a2 = 0 or β = 0.

Appendix : B

In this Appendix ?? we tersely outline the derivation of the expressions (??) and (??) of the 12 parameters cnj in terms of the 10 parameters Anm and an.

The first step is to invert the relations (??), getting

$$ y_{1}\left( t\right) =\left[ A_{22}x_{1}\left( t\right) -A_{12}x_{2}\left( t\right) \right] /D~, $$
(77a)
$$ y_{2}\left( t\right) =\left[ A_{11}x_{2}\left( t\right) -A_{21}x_{1}\left( t\right) \right] /D~, $$
(77b)

where the quantity D is defined as above, see (??).

The second step is to note that the relations (??) imply

$$ \dot{x}_{n}=A_{n1}\dot{y}_{1}+A_{n2}\dot{y}_{2}~,~~~n=1,2~, $$
(78a)

hence, via the ODEs (??),

$$ \begin{array}{@{}rcl@{}} \dot{x}_{n}=A_{n1}\left[ a_{12}\left( y_{1}\right)^{2}+a_{11}y_{1}+a_{10} \right] && \\ +A_{n2}\left[ a_{22}\left( y_{2}\right)^{2}+a_{21}y_{2}+a_{20}\right] ~,~~~n=1,2~. && \end{array} $$
(78b)

The third and last step is to insert the expressions (??) of y1 and y2 in terms of x1 and x2 in the right-hand sides of these ODEs. Then, via a bit of trivial if tedious algebra, there obtains the system (??) with the definitions (??) and (??) of the 12 coefficients cnj. Q. E. D.

Appendix : C

In this Appendix ?? we show that the solutions \(x_{n}\left (t\right ) \) of the dynamical system (??)—as treated above, see Sections ?? and ??—do not depend on the free parameters λn introduced via the positions (??).

To this end we insert the expressions (??) of the parameters Anm in terms of the free parameters λn in the formulas (??) and (??) expressing the 6 parameters ank; in order to display the very simple dependence of these 8 parameters from the 2 free parameters λn. We thus easily find the following formulas:

$$ a_{n\ell }\equiv \left( \lambda_{n}\right)^{\ell -1}\alpha_{n\ell }\left( C\right) ~,~~~n=1,2~,~~~\ell =0,1,2~, $$
(79a)

where the notation C indicates—above and hereafter—the set of the 12 parameters cnj, and the 6 functions \(\alpha _{n\ell }\left (C\right ) \) are explicitly displayed in Section ??, see (??); of course to this end we also used the definitions (??) of the 2 auxiliary parameters zn in terms of the 4 coefficients cnm (n = 1, 2; m = 1, 2).

The next step is to insert the formulas (79a) in the expressions (??), getting thereby

$$ y_{n\pm }\equiv \left( \lambda_{n}\right)^{-1}w_{n\pm }\left( C\right) ~,~~~\beta_{n}\equiv \beta_{n}\left( C\right) ~,~~~n=1,2~, $$
(79b)

again with the functions \(w_{n\pm }\left (C\right ) \) and \(\beta _{n}\left (C\right ) \) explicitly displayed in Section ??, see (??) and (??).

The insertion of these formulas in the expressions (??) of the solutions \(y_{n}\left (t\right ) \) of the auxiliary dynamical system (??) evidences the following, very simple, dependence of these functions from the free parameters λn:

$$ y_{n}\left( t\right) \equiv \left( \lambda_{n}\right)^{-1}w_{n}\left( C,t\right) ~,~~~n=1,2~, $$
(80)

where again the 2 functions \(w_{n}\left (C,t\right ) \) are explicitly displayed in Section ??, see (??).

And via the insertion in the expressions (??) of \(x_{n}\left (t\right ) \) of these formulas (80), together with the expressions (??) of Anm, we conclude that the solutions \(x_{n}\left (t\right ) \) are independent of the free parameters λn; as indeed displayed in Section 4, see the set of eqs. from (39) to (44). Q. E. D.

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Calogero, F., Payandeh, F. Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides. Math Phys Anal Geom 24, 29 (2021). https://doi.org/10.1007/s11040-021-09400-7

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Keywords

  • Solvable dynamical systems
  • Solvable systems of nonlinearly-coupled ordinary differential equations

Mathematics Subject Classification (2010)

  • 34A34