Article Contents
Article Contents

# Periodic forcing on degenerate Hopf bifurcation

• * Corresponding author: renjl@zzu.edu.cn
• This paper is devoted to the effect of periodic forcing on a system exhibiting a degenerate Hopf bifurcation. Two methods are employed to investigate bifurcations of periodic solution for the periodically forced system. It is obtained by averaging method that the system undergoes fold bifurcation, transcritical bifurcation, and even degenerate Hopf bifurcation of periodic solution. On the other hand, it is also shown by Poincaré map that the system will undergo fold bifurcation, transcritical bifurcation, Neimark-Sacker bifurcation and flip bifurcation. Finally, we make a comparison between these two methods.

Mathematics Subject Classification: Primary: 34C23, 37G15, 37G10.

 Citation:

• Figure 1.  (a) Phase portrait near a degenerate Hopf bifurcation for $r_{1} = 0.3$, $r_{2} = 0.6$, $a_{1} = 0.42$, $a_{2} = 0.6$, $b_{1} = 1.0857$, $b_{2} = 0.25$. (b) Phase portrait near a degenerate Hopf bifurcation for $r_{1} = 0.3$, $r_{2} = 0.6$, $a_{1} = 0.447$, $a_{2} = 0.6$, $b_{1} = 1.13$, $b_{2} = 0.25$

Figure 2.  Phase portrait of the averaged system, the red points are the equilibria. (a) For $\mu = 0.06$, $\nu = 3$, $\alpha = 1$, $\beta = 2$. (b) For $\mu = 0$, $\nu = 3$, $\alpha = 1$, $\beta = 2$

Figure 3.  (b) Bifurcation diagram of the forced system (8) in $(a_{1},\epsilon)$-plane for $r_{1} = 0.3,b_{1} = 1.13,r_{2} = 0.6,a_{2} = 0.6,b_{2} = 0.25$. (c) and (d) Partial enlargements of (b). The solutions of system (8) are as follows, region 1-unstable period-one solution and stable quasiperiodic solution; region 2-unstable period-one solution and unstable period-two solution; region 3-unstable period-one solution, unstable period-two solutions, and period-four solution; region 4-stable and unstable period-one solutions; region 5-stable and unstable period-one solutions

Figure 4.  (a) Bifurcation diagram of the forced system (8) in $(a_{1},\epsilon)$-plane for $r_{1} = 0.3,b_{1} = 0.85,r_{2} = 0.4,a_{2} = 0.6,b_{2} = 0.4$. (b), (c) Partial enlargements of (a). The solutions of system (8) are as follows, region 1-unstable period-one solutions, unstable period-two solutions, and period-four solution; region 2-stable and unstable period-one solutions; region 3-unstable period-one solutions and stable quasiperiodic solution; region 4-no periodic solution; region 5-unstable period-one solutions and unstable period-two solutions; region 6-stable and unstable period-one solutions

Figure 5.  (a) Bifurcation diagram of the forced system (26) in $(b_{1},\epsilon)$-plane for $r_{1} = 0.3,a_{1} = 0.2,r_{2} = 0.4,a_{2} = 0.4,b_{2} = 0.162$. (b) Partial enlargements of (a). The solutions of system (26) are as follows, region 1-stable and unstable period-one solutions, unstable quasiperiodic solution; region 2- unstable period-one solutions; region 3-unstable period-one solutions, unstable period-two solutions, and period-four solution; region 4-unstable period-one solutions, stable period-two solutions; region 5-stable and unstable period-one solutions

Figure 6.  (a) Bifurcation diagram of the forced system (26) in $(b_{1},\epsilon)$-plane for $r_{1} = 0.3,b_{1} = 0.6,r_{2} = 0.4,a_{2} = 0.6,b_{2} = 0.4$. (b) bifurcation curves of a period-one saddle. (c) and (d) Partial enlargements of (a). The solutions of system (26) are as follows, region 1-stable and unstable period-one solutions; region 2- unstable period-one solutions and stable quasiperiodic solution; region 3-unstable period-one solutions, unstable period-two solutions, and period-four solution; region 4-stable period-one solutions; region 5-unstable period-one solutions, unstable period-two solutions; region 6-stable and unstable period-one solutions

Figure 7.  (a) Bifurcation diagram of the forced system (27) in $(a_{2},\epsilon)$-plane for $r_{1} = 0.3, a_{1} = 0.2, b_{1} = 0.5,r_{2} = 0.4, b_{2} = 0.162$. (b) Bifurcation diagram of the forced system (27) in $(a_{2},\epsilon)$-plane for $r_{1} = 0.3, a_{1} = 0.6, b_{1} = 0.83, r_{2} = 0.4, b_{2} = 0.4$

Figure 8.  Bifurcation diagram of the forced system (28) in $(b_{2},\epsilon)$-plane for the case $r_{1} = 0.3,a_{1} = 0.2,b_{1} = 0.83,r_{2} = 0.4,a_{2} = 0.4$

Figure 9.  (a) Bifurcation diagram of the forced system (28) in $(b_{2},\epsilon)$-plane for $r_{1} = 0.3,a_{1} = 0.6,b_{1} = 0.85,r_{2} = 0.4,a_{2} = 0.6$. (b), (c) and (d) Partial enlargements of (a). The solutions of system (28) are as follows, region 1-unstable period-one solution, unstable period-two solutions, and period-four solution or chaos in some subregion; region 2-stable and unstable period-one solution; region 3-unstable period-one solution and stable quasiperiodic solution; region 4-unstable period-one solutions and unstable period-two solutions; region 5-stable and unstable period-one solutions; region 6-unstable period-one solution, unstable period-two solutions, and period-four solution

Figure 10.  Phase portraits and Poincaré section of the periodically forced system. (a) A stable period-two orbit in system (27) for $r_{1} = 0.3$, $r_{2} = 0.4$, $a_{1} = 0.6$, $a_{2} = 0.4$, $b_{1} = 0.83$, $b_{2} = 0.4$, $\epsilon = 0.337$. (b) A stable period-four orbit in system (27) for $r_{1} = 0.3$, $r_{2} = 0.4$, $a_{1} = 0.6$, $a_{2} = 0.4$, $b_{1} = 0.83$, $b_{2} = 0.4$, $\epsilon = 0.277$. (c) Torus in system (8) for $r_{1} = 0.3$, $r_{2} = 0.6$, $a_{1} = 0.45183$, $a_{2} = 0.6$, $b_{1} = 1.13$, $b_{2} = 0.25$, $\epsilon = 0.277$. (d) Poincaré section of the torus

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