A periodic bifurcation problem depending on a random variable

Mikhail I. Kamenskiĭ, Paolo Nistri, Paul Raynaud de Fitte

DOI: http://dx.doi.org/10.12775/TMNA.2019.043


We consider an abstract bifurcation equation $P(x)+\varepsilon Q(x,\varepsilon, \omega)=0$, where $P$ and $Q$ are operators, $\varepsilon$ is the bifurcation parameter, $\omega \in \Omega$, is the random variable and $(\Omega, \mathcal{F})$ is a measurable space. The aim of the paper is to provide conditions on $P$ and $Q$ to ensure the existence, for any $\omega \in \Omega$, of a branch of solutions originating from the zeros of the operator $P$. We show that the considered abstract bifurcation is the model of a random autonomous periodically perturbed differential equation having the property that the unperturbed equation corresponding to $\varepsilon = 0$ has a limit cycle. As a consequence we obtain the existence, for any $\omega \in \Omega$, of a branch of periodic solutions of the perturbed equation emanating from the limit cycle.


Random variable; Malkin bifurcation function; limit cycle; periodic perturbation

Full Text:



A.T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc 82 (1976), 641–657. https://DOI.org/10.1090/S0002-9904-1976-14091-8

O. Blasco and I. Garcı́a-Bayona, Remarks on measurability of operator-valued functions, Mediterr. J. Math. 13 (2016), 5147–5162. https://DOI.org/10.1007/s00009-0160798-1

I.I. Blekhman, Synchronization of dynamical systems, Izdat. Nauka, Moscow, 1971.

C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer–Verlag, Berlin, NewYork, 1977.

J.-F. Couchouron, M. Kamenskiı̆ and P. Nistri, An infinite dimensional bifurcation problem with application to a class of functional differential equations of neutral type, Commun. Pure Appl. Anal. 12, (2013), 1845–1859. DOI: 103934/cpaa.2013.12

J.-F. Couchouron, M. Kamenskiı̆, B. Mikhaylenko and P. Nistri, Periodic bifurcation problems for fully nonlinear neutral functional differential equations via an integral operator approach: the multidimensional case, Topol. Methods Nonlinear Anal. 46 (2015), 631–663. DOI: 10.12775/TMNA.2015.062

B.C. Dhage, Nonlinear functional random differential equations in Banach algebras, Tamkang J. Math. 38, (2007), 57–73.

M.I. Freidlin and A.D. Wentzell, Random perturbations of dynamical systems, Grundlehren der Mathematischen Wissenschaften, vol. 260, Ed. Springer Science & Business Media, 2012.

I. Gohberg, S. Goldberg and M.A. Kaashoek, Classes of linear operators, Vol. I, Operator Theory: Advances and Applications, vol. 49, Birkhäuser Verlag, Basel, 1990. https://DOI.org/10.1007/978-3-0348-7509-7

P. Hartman, Ordinary differential equations, Classics in Applied Mathematics, vol. 38, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002, https://DOI.org/10.1137/1.9780898719222; Corrected reprint of the second (1982) edition [Birkhäuser, Boston, MA; MR0658490 (83e:34002)], with a foreword by Peter Bates.

M. Henrard and F. Zanolin, Bifurcation from a periodic orbit in perturbed planar Hamiltonian systems, J. Math. Appl. 277 (2003), 79–103.

M. Kamenskiı̆, O. Makarenkov and P. Nistri, A continuation principle for a class of periodically perturbed autonomous systems, Math. Nachr. 281, (2008), 42–61. https://DOI:10.1002/mana.200610586

M. Kamenskiı̆, O. Makarenkov and P. Nistri, An alternative approach to study bifurcation from a limit cycle in periodically perturbed autonomous systems, J. Dynam. Differential Equations 23, (2011), 425–435. https://DOI: 10.1007/s10884-011-9207-4

M. Kamenskiı̆, B. Mikhaylenko and P. Nistri, Nonsmooth bifurcation problems in finite dimensional spaces via scaling of variables, Differ. Equ. Dyn. Syst. 10 (2012), 191–205. DOI: 10.1007/s12591-011-0102-6

M. Kamenskiı̆, B. Mikhaylenko and P. Nistri, A bifurcation problem for a class of periodically perturbed autonomous parabolic equations, Bound. Value Probl. 2013:11 (2013), DOI: 10.1186/1687-2770-2013-101.

Y. Kifer, Random Perturbations of Dynamical Systems, Progress in Probability, vol. 16, Birkhäuser Basel, 1988.

M.A. Krasnosel’skiı̆, The operator of translation along the trajectories of differential equations, Translations of Mathematical Monographs, Vol. 19., American Mathematical Society, Providence, R.I., 1968.

W.S. Loud, Periodic solutions of a pertubed autonomous system, Ann. Math. 70 (1959), 490–529.

O. Makarenkov and P. Nistri, Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations, J. Math. Anal. Appl. 338 (2008), 1401–1417. DOI: 10.1016/j.jmaa.2007.05.086

O. Makarenkov and P. Nistri, On the rate of convergence of periodic solutions in perturbed autonomous systems as the perturbation vanishes, Commun. Pure Appl. Anal. 7, (2008), 49–61.

I.G. Malkin, Some Problems of the Theory of Nonlinear Oscillations, Gosudarstv. Isdat. Techn. Teor. Lit., Moscow, 1956. (in Russian)

D.S. Palimkar, Perturbed random differential equations, Journal of Global Research in Mathematical Archives 1, (2013), 53–58. https://www.jgrma.info.

L. Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973, Tata Institute of Fundamental Research Studies in Mathematics, No. 6.

A.D. Ventsel and M.I. Freidlin, On small random perturbations of dynamical systems, Russian Math. Surveys 25 (1970), 1–55.


  • There are currently no refbacks.

Partnerzy platformy czasopism