PaperNo Access

The Extended 16th Hilbert Problem for Discontinuous Piecewise Systems Formed by Linear Centers and Linear Hamiltonian Saddles Separated by a Nonregular Line

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Catalonia, Spain

E-mail Address: jaume.llibre@uab.cat

Search for more papers by this author

and

Claudia Valls

https://orcid.org/0000-0001-8279-1229

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal

E-mail Address: cvalls@math.ist.utl.pt

Search for more papers by this author

https://doi.org/10.1142/S0218127423501961Cited by:2 (Source: Crossref)

Abstract

We study discontinuous piecewise linear differential systems formed by linear centers and/or linear Hamiltonian saddles and separated by a nonregular straight line. There are two classes of limit cycles: the ones that intersect the separation line at two points and the ones that intersect the separation line in four points, named limit cycles of type ${I I}_{2}$ ${I I}_{2}$ and limit cycles of type ${I I}_{4}$ ${I I}_{4}$ , respectively. We prove that the maximum numbers of limit cycles of types ${I I}_{2}$ ${I I}_{2}$ and ${I I}_{4}$ ${I I}_{4}$ are two and one, respectively. We show that all these upper bounds are reached providing explicit examples.

Keywords:

References

Belousov, B. P. [1959] “ A periodic reaction and its mechanism,” Collection of Short Papers on Radiation Medicine for 1958 (Med. Publ., Moscow) (in Russian). Google Scholar
Braga, D. C. & Mello, L. F. [2013] “ Limit cycles in a family of discontinuous piecewise linear differential systems with two zones in the plane,” Nonlin. Dyn. 73, 1283–1288. Crossref, Web of Science, Google Scholar
Buzzi, C., Pessoa, C. & Torregrosa, J. [2013] “ Piecewise linear perturbations of a linear center,” Discr. Contin. Dyn. Syst. 33, 3915–3936. Crossref, Web of Science, Google Scholar
di Bernardo, M., Budd, C. J., Champneys, A. R. & Kowalczyk, P. [2008] Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci., Vol. 163 (Springer-Verlag, London). Google Scholar
Esteban, M., Llibre, J. & Valls, C. [2021] “ The extended 16th problem for discontinuous piecewise linear centers separated by a non-regular line,” Int. J. Bifurcation and Chaos 31, 2150225-1–15. Link, Web of Science, Google Scholar
Filippov, A. F. [1988] Differential Equations with Discontinuous Right-Hand Sides, Translated from Russian. Mathematics and Its Applications (Soviet Series), Vol. 18 (Kluwer Academic Publishers Group, Dordrecht). Crossref, Google Scholar
Freire, E., Ponce, E. & Torres, F. [2014] “ A general mechanism to generate three limit cycles in planar Filippov systems with two zones,” Nonlin. Dyn. 78, 251–263. Crossref, Web of Science, Google Scholar
Giannakopoulos, F. & Pliete, K. [2014] “ Planar systems of piecewise linear differential equations with a line of discontinuity,” Nonlinearity 14, 1611–1632. Crossref, Web of Science, Google Scholar
Han, M. & Zhang, W. [2010] “ On Hopf bifurcation in non-smooth planar systems,” J. Diff. Eqs. 248, 2399–2416. Crossref, Web of Science, Google Scholar
Huan, S. M. & Yang, X. S. [2012] “ On the number of limit cycles in general planar piecewise systems,” Discr. Cont. Dyn. Syst., Series A 32, 2147–2164. Crossref, Web of Science, Google Scholar
Huan, S. M. & Yang, X. S. [2019] “ Limit cycles in a family of planar piecewise linear differential systems with a nonregular separation line,” Int. J. Bifurcation and Chaos 29, 1950109-1–22. Link, Web of Science, Google Scholar
Li, L. [2014] “ Three crossing limit cycles in planar piecewise linear systems with saddle-focus type,” Electron. J. Qual. Th. Diff. Eqs. 70, 14. Google Scholar
Li, Z. & Liu, X. [2022] “ Limit cycles in discontinuous piecewise linear planar Hamiltonian systems without equilibrium points,” Int. J. Bifurcation and Chaos 32, 2250153-1–20. Link, Web of Science, Google Scholar
Llibre, J. & Ponce, E. [2012] “ Three nested limit cycles in discontinuous piecewise linear differential systems with two zones,” Dyn. Cont. Disc. Impul. Syst., Series B 19, 325–335. Google Scholar
Llibre, J., Novaes, D. D. & Teixeira, M. A. [2015] “ Maximum number of limit cycles for certain piecewise linear dynamical systems,” Nonlin. Dyn. 82, 1159–1175. Crossref, Web of Science, Google Scholar
Llibre, J. & Teixeira, M. A. [2018] “ Piecewise linear differential systems with only centers can create limit cycles? Nonlin. Dyn. 91, 249–255. Crossref, Web of Science, Google Scholar
Llibre, J. & Valls, C. [2021] “ Limit cycles of piecewise differential systems with linear Hamiltonian saddles,” Symmetry 13, 1128-1–10. Crossref, Web of Science, Google Scholar
Llibre, J. & Valls, C. [2022] “ Limit cycles of piecewise differential systems with linear Hamiltonian saddles and linear centers,” Dyn. Syst. 37, 262–279. Crossref, Web of Science, Google Scholar
Makarenkov, O. & Lamb, J. S. W. [2012] “ Dynamics and bifurcations of nonsmooth systems: A survey,” Physica D 241, 1826–1844. Crossref, Web of Science, Google Scholar
Simpson, D. J. W. [2010] Bifurcations in Piecewise-Smooth Continuous Systems, World Sci. Ser. Nonlinear Sci. Ser. A, Vol. 69 (World Scientific, Singapore). Link, Google Scholar
van der Pol, B. [1920] “ A theory of the amplitude of free and forced triode vibrations,” Radio Review (Later Wireless World) 1, 701–710. Google Scholar
van der Pol, B. [1926] “ On relaxation-oscillations,” The London, Edinburgh and Dublin Phil. Mag. J. Sci. 2, 978–992. Crossref, Google Scholar
Zhabotinsky, A. M. [1964] “ Periodical oxidation of malonic acid in solution (a study of the Belousov reaction kinetics),” Biofizika 9, 306–311. Google Scholar
Zhao, Q., Wang, C. & Yu, J. [2021] “ Limit cycles in discontinuous planar piecewise linear systems separated by a nonregular line of center-center type,” Int. J. Bifurcation and Chaos 31, 2150136-1–17. Link, Web of Science, Google Scholar

Vol. 33, No. 16

Metrics

Downloaded 40 times

History

Received 13 September 2023

Revised 12 October 2023

Keywords

PDF download

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

The Extended 16th Hilbert Problem for Discontinuous Piecewise Systems Formed by Linear Centers and Linear Hamiltonian Saddles Separated by a Nonregular Line

Abstract

References

Limit Cycles of Planar Discontinuous Piecewise Linear Hamiltonian Systems Without Equilibria Separated by Nonregular Curves

Limit Cycles in Discontinuous Piecewise Linear Planar Hamiltonian Systems Without Equilibrium Points

The Extended 16th Hilbert Problem for Discontinuous Piecewise Linear Centers Separated by a Nonregular Line

HORSESHOES NEAR HOMOCLINIC ORBITS FOR PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ³

Limit Cycles in a Family of Planar Piecewise Linear Differential Systems with a Nonregular Separation Line

On Hilbert’s 16th Problem for Some Discontinuous Piecewise Differential Systems

Limit Cycles of Some Families of Discontinuous Piecewise Differential Systems Separated by a Straight Line

Limit Cycles of the Discontinuous Piecewise Differential Systems Separated by a Nonregular Line and Formed by a Linear Center and a Quadratic One

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

The Extended 16th Hilbert Problem for Discontinuous Piecewise Systems Formed by Linear Centers and Linear Hamiltonian Saddles Separated by a Nonregular Line

Abstract

Recommended

Limit Cycles of Planar Discontinuous Piecewise Linear Hamiltonian Systems Without Equilibria Separated by Nonregular Curves

Limit Cycles in Discontinuous Piecewise Linear Planar Hamiltonian Systems Without Equilibrium Points

The Extended 16th Hilbert Problem for Discontinuous Piecewise Linear Centers Separated by a Nonregular Line

HORSESHOES NEAR HOMOCLINIC ORBITS FOR PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ3

Limit Cycles in a Family of Planar Piecewise Linear Differential Systems with a Nonregular Separation Line

On Hilbert’s 16th Problem for Some Discontinuous Piecewise Differential Systems

Limit Cycles of Some Families of Discontinuous Piecewise Differential Systems Separated by a Straight Line

Limit Cycles of the Discontinuous Piecewise Differential Systems Separated by a Nonregular Line and Formed by a Linear Center and a Quadratic One

HORSESHOES NEAR HOMOCLINIC ORBITS FOR PIECEWISE LINEAR DIFFERENTIAL SYSTEMS IN ℝ³