Abstract
In a recent work, Calogero and Payandeh have identified a class of solvable two coupled first order nonlinear ordinary differential equations by connecting the roots of a monic polynomial of degree four with the coefficients of the polynomial. In this paper, we apply one of their earlier works to these first order nonlinear differential equations and enumerate the evolution equations in Newtonian form. Further, we demonstrate that second order evolution equations of the roots also follow the dynamics of the coefficients of the polynomial.
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References
Calogero, F.: Isochronous Systems. Oxford University Press, Oxford (2008)
Calogero, F.: Zeros of Polynomials and Solvable Nonlinear Evolution Equations. Cambridge University Press, Cambridge (2018)
Bolsinov, A., Morales-Ruiz, J.J., Zung, N.T., Miranda, E., Matveev, V.: Geometry and Dynamics of Integrable Systems. Springer, Switzerland (2016)
Euler, N., Nucci, M.C.: Nonlinear Systems and Their Remarkable Mathematical Structures, vol. 2. CRC Press, Boca Raton (2019)
Dumortier, F., Llibre, J., Artés, J.C.: Qualitative Theory of Planar Differential Systems. Springer, Berlin (2006)
Giné, J., Llibre, J.: Invariant algebraic curves of generalized Li\(\acute{e}\)nard polynomial differential systems. Mathematics 10, 209 (2022)
Demina, M.V., Valls, C.: On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations. Proc. Roy. Soc. Edinb. Sect. A Math. 150, 3231 (2020)
Demina, M.V., Giné, J., Valls, C.: Puiseux integrability of differential equations. Qual. Theory Dyn. Syst. 21, 35 (2022)
Ferčec, B., Giné, J.: Formal Weierstrass integrability for a Liénard differential system. J. Math. Anal. Appl. 499, 125016 (2021)
Llibre, J., Valls, C.: Phase portraits of uniform isochronous centers with homogeneous nonlinearities. J. Dyn. Control Syst. 28, 319–332 (2022)
Giné, J., Llibre, J.: A characterization of the generalized Liénard polynomial differential systems having invariant algebraic curves. Chaos, Solitons & Fractals 158, 112075 (2022)
Demina, M.V., Sinelshchikov, D.I.: Darboux first integrals and linearizability of quadratic-quintic Duffing-van der Pol oscillators. J. Geom. Phys. 165, 104215 (2021)
Christopher, C., Llibre, J., Pantazi, C., Walcher, S.: On planar polynomial vector fields with elementary first integrals. J. Differ. Eqs. 267, 4572 (2019)
Giné, J., Grau, M.: Characterization of isochronous foci for planar analytic differential systems. Proc. R. Soc. Edinb. Sect. A Math. 135(5), 985–998 (2005)
Chouikha, A.R.: On isochronous analytic motions and the quantum spectrum. Phys. Scr. 94, 125220 (2019)
Algaba, A., Freire, E., Gamero, E.: Isochronicity via normal form. Qual. Theory Dyn. Syst. 1, 133–156 (2000)
Iacono, R., Russo, F.: Class of solvable nonlinear oscillators with isochronous orbits. Phys. Rev. E 83, 027601 (2011)
Mohanasubha, R., Shakila, M.I.S., Senthilvelan, M.: On the linearization of isochronous centre of a modified Emden equation with linear external forcing. Commun. Nonlinear Sci. Numer. Simul. 19, 799 (2014)
Parkavi, J.R., Mohanasubha, R., Chandrasekar, V.K., Senthilvelan, M., Lakshmanan, M.: A class of isochronous and non-isochronous nonlinear oscillators. Eur. Phys. J. Spec. Top. 231, 2387–2399 (2022)
Rañada, M.F.: Bi-Hamiltonian structure of the bi-dimensional superintegrable nonlinear isotonic oscillator. J. Math. Phys. 57, 052703 (2016)
Calogero, F.: New solvable dynamical systems. J. Nonlinear Math. Phys. 23, 486–493 (2016)
Calogero, F.: Novel isochronous N-body problems featuring N arbitrary rational coupling constants. J. Math. Phys. 57, 072901 (2016)
Ghose-Choudhury, A., Guha, P.: Isochronicity conditions and Lagrangian formulations of the Hirota type oscillator equations. Qual. Theory Dyn. Syst. 21, 144 (2022)
Bihun, O., Calogero, F.: Time-dependent polynomials with one double root, and related new solvable systems of nonlinear evolution equations. Qual. Theory Dyn. Syst. 18, 153 (2019)
Bihun, O.: Time-dependent polynomials with one multiple root and new solvable dynamical systems. J. Math. Phys. 60, 103503 (2019)
Calogero, F., Payandeh, F.: Polynomials with multiple zeros and solvable dynamical systems including models in the plane with polynomial interactions. J. Math. Phys. 60, 082701 (2019)
Acknowledgements
RMS is funded by the Center for Computational Modeling, Chennai Institute of Technology, India, vide funding number CIT/CCM/2022/RP-006. MS wishes to thank the National Board for Higher Mathematics, Government of India for their financial support by the research project under the Grant No. 02011/20/2018NBHM(R.P)/R &D 24II/15064.
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Mohanasubha, R., Senthilvelan, M. A Class of New Solvable Nonlinear Isochronous Systems and Their Classical Dynamics. Qual. Theory Dyn. Syst. 22, 40 (2023). https://doi.org/10.1007/s12346-023-00744-9
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DOI: https://doi.org/10.1007/s12346-023-00744-9