# American Institute of Mathematical Sciences

October  2001, 7(4): 747-762. doi: 10.3934/dcds.2001.7.747

## On subharmonics bifurcation in equations with homogeneous nonlinearities

 1 Institute for Information Transmission Problems, Russian Academy of Sciences, 19 Bolshoi Karetny Lane, Moscow 101447, Russian Federation 2 Institute for Nonlinear Science, Department of Physics, National University of Ireland, University College, Cork, Ireland

Received  January 2001 Published  July 2001

The bifurcation of subharmonics for resonant nonautonomous equations of the second order is studied. The set of subharmonics is defined by principal homogeneous parts of the nonlinearities provided these parts are not polynomials. Analogous statements are proved for bifurcations of $p$-periodic orbits of a planar dynamical system. The analysis is based on topological methods and harmonic linearization.
Citation: Alexander Krasnosel'skii, Alexei Pokrovskii. On subharmonics bifurcation in equations with homogeneous nonlinearities. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 747-762. doi: 10.3934/dcds.2001.7.747
 [1] Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247 [2] Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703 [3] Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623 [4] Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393 [5] Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563 [6] Andrus Giraldo, Bernd Krauskopf, Hinke M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. Journal of Computational Dynamics, 2020, 7 (2) : 489-510. doi: 10.3934/jcd.2020020 [7] Hong Li. Bifurcation of limit cycles from a Li$\acute{E}$nard system with asymmetric figure eight-loop case. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022033 [8] Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208 [9] Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions. Communications on Pure and Applied Analysis, 2005, 4 (3) : 665-682. doi: 10.3934/cpaa.2005.4.665 [10] Johanna D. García-Saldaña, Armengol Gasull, Hector Giacomini. Bifurcation values for a family of planar vector fields of degree five. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 669-701. doi: 10.3934/dcds.2015.35.669 [11] José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078 [12] Tian Hou, Yi Wang, Xizhuang Xie. Instability and bifurcation of a cooperative system with periodic coefficients. Electronic Research Archive, 2021, 29 (5) : 3069-3079. doi: 10.3934/era.2021026 [13] Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357 [14] Victor S. Kozyakin, Alexander M. Krasnosel’skii, Dmitrii I. Rachinskii. Arnold tongues for bifurcation from infinity. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 107-116. doi: 10.3934/dcdss.2008.1.107 [15] Jongmin Han, Chun-Hsiung Hsia. Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2431-2449. doi: 10.3934/dcdsb.2012.17.2431 [16] Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165 [17] Carlos Garca-Azpeitia, Jorge Ize. Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 975-983. doi: 10.3934/dcdss.2013.6.975 [18] Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 [19] Samuel Walsh. Steady stratified periodic gravity waves with surface tension II: Global bifurcation. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3287-3315. doi: 10.3934/dcds.2014.34.3287 [20] Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209

2020 Impact Factor: 1.392