| Citation: |
Huimiao Dong, Tiansi Zhang, Xingbo Liu. BIFURCATIONS OF DOUBLE HETERODIMENSIONAL CYCLES WITH THREE SADDLE POINTS[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2143-2162. doi: 10.11948/20210082
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BIFURCATIONS OF DOUBLE HETERODIMENSIONAL CYCLES WITH THREE SADDLE POINTS
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Abstract
In this paper, bifurcations of double heterodimensional cycles of an "$\infty$" shape consisting of two saddles of (1,2) type and one saddle of (2,1) type are studied in three dimensional vector field. We discuss the gaps between returning points in transverse sections by establishing a local active coordinate system in the tubular neighborhood of unperturbed double heterodimensional cycles, through which the preservation of "$\infty$"-shape double heterodimensional cycles is proved. We then get the existence of a new heteroclinic cycle consisting of two saddles of (1,2) type and one saddle of (2,1) type, which is composed of one big orbit linking $p_1$, $p_3$ and two orbits linking $p_3$, $p_2$ and $p_2$, $p_1$ respectively, and another heterodimensional cycle consisting of one saddle $p_1$ of (2,1) type and one saddle $p_2$ of (1,2) type, which is composed of one orbit starting from $p_1$ to $p_2$ and another orbit starting from $p_2$ to $p_1$. Moreover, the 1-fold and 2-fold large 1-heteroclinic cycle consisting of two saddles $p_1$ and $p_3$ of (1,2) type is also presented. As well as the coexistence of a 1-fold large 1-heteroclinic cycle and the "$\infty$"-shape double heterodimensional cycles and the coexistence conditions are also given in the parameter space.
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References
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Huimiao Dong, Tiansi Zhang, Xingbo Liu. BIFURCATIONS OF DOUBLE HETERODIMENSIONAL CYCLES WITH THREE SADDLE POINTS[J]. Journal of Applied Analysis & Computation, 2022, 12(6): 2143-2162. doi: 10.11948/20210082
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Figure 1. Double heterodimensional cycles of three saddle points
$ p_i $ with four orbits$ \gamma_k(t) $ . -
Figure 2. The gap
$ \left\|\gamma_k^+(-T_k+\tilde{T}_k, \mu)-\gamma_k^-(T_k, \mu)\right\|\neq0(k=2, 3) $ in the figure, the original double heterodimensional cycles do not exist. -
Figure 3. The gap
$ ||{\gamma _1}( - {T_1} + {\tilde T}_1 + {\tau _2} + {\tilde T}_3, \mu ) - \gamma _3^ - ({T_3}, \mu )||\neq0 $ ,$ \left\|\gamma_2^-(T_2, \mu)-\gamma_2^+(-T_2+\tilde T_2, \mu)\right\|\neq0 $ in the figure, there is not the second heterodimensional cycle in general, which consists of two saddles of (1,2) type and one saddle of (2,1) type and is composed of one big orbit linking$ p_1, p_3 $ and two orbits linking$ p_3, p_2 $ and$ p_2, p_1 $ respectively. -
Figure 4. The gap
$ \left\|{\gamma _1}({-T_1} + {\tilde T}_1 + {\tau _2} + {\tilde T}_3, \mu ) - \gamma _3^ - ( {T_3}, \mu )\right\| \neq0 $ ,$ \left\|\gamma_4(-T_4+\tilde T_4+\tau_4+\tilde T_2, \mu)-\right. $ $ \left.\gamma_2^-(T_2, \mu)\right\|\ne 0 $ in the figure, there is not a large 1-heteroclinic cycle in general.