Hopf bifurcation and stability analysis for a delayed equation with φ-Laplacian
DOI:
https://doi.org/10.12775/TMNA.2024.015Keywords
Functional-delay equations, Lyapunov-Krasovskii functional, stability, Hopf bifurcation, periodic solutionsAbstract
A formal framework for the analysis of Hopf bifurcations for a kind of delayed equation with $\varphi$-Laplacian and with a discrete time delay is presented, thus generalizing known results for the sunflower equation given by Somolinos in 1978. Also, under appropriate assumptions we prove the gradient-like behavior of the equation which, in turn, implies the non-existence of nonconstant periodic solutions. Our conditions improve previous results known in the literature for the standard case $\varphi(x)=x$.References
P. Amster, M.P. Kuna and D.P. Santos, Stability, existence and non-existence of T periodic solutions of nonlinear delayed differential equations with ϕ-Laplacian, Comm. Pure Appl. Anal. 21 (2022), no. 8, 2723–2737, DOI: 10.3934/cpaa.2022070.
C. Bereanu and J. Mawhin, Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian, J. Differential Equations. 243 (2007), 536–557.
T.A. Burton, Stability and Periodic Solution of Ordinary and Functional Differential Equations, Academic Press, Orland, FL, 1985.
T.A. Burton and L. Hatvani, Stability Theorems for non autonomous functional differential equations by Liapunov functionals, Tohoku Math. J. 41 (1989), 65–104.
J. Hale, Theory of Functional Differential Equations, Springer–Verlag, New York, Heidelberg, Berlin, 1977.
X. Huang and Z. Xiang, On existence of 2π-periodic solutions for delay Duffing equation x00 + g(t, x(t − τ (t))) = p(t), Chinese Sci. Bull. 39 (1994), 201–203.
X. Liu, M. Tang and R. Martin, Periodic solutions for a kind of Liénard equation, J. Comput. Appl. Math. 219 (2008), 263–275.
S. Lu and W. Ge, Periodic solutions for a kind of second order differential equation with multiple deviating arguments, Appl. Math. Comput. 146 (2003), 195–209.
S. Lu and W. Ge, Sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument, Appl. Math. Comput. 308 (2005), 393–419.
V. Smirnova, A. Proskurnikov and I. Zgoda, The sunflower equation: novel stability criteria IFAC-PapersOnLine 54 (2021), issue 17, 135–140.
A. Somolinos, Periodic solutions of the sunflower equation: ẍ + (a/r)ẋ + (b/r) sin x(t − r) = 0, Quart. Appl. Math. 35 (1978), 465–478.
A. Somolinos, Forced oscillations for the sunflower equation, entrainment, Nonlinear Anal. 4 (1982), 397–414.
G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, John Wiley, New York, 1989.
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