Abstract
In this paper, using the local moving frame approach, we investigate bifurcations of nongeneric heteroclinic loop with a nonhyperbolic equilibrium $p_1$ and a hyperbolic saddle $p_2$, where $p_1$ is assumed to undergo a transcritical bifurcation. Firstly, we establish the persistence of a nongeneric heteroclinic loop, the existence of a homoclinic loop and a periodic orbit when the transcritical bifurcation does not occur. Secondly, bifurcations of a nongeneric heteroclinic loop accompanied with a transcritical bifurcation are discussed. We obtain the existence of heteroclinic orbits, a homoclinic loop, a heteroclinic loop and a periodic orbit. Some bifurcation patterns different from the case of the generic heteroclinic loop accompanied with transcritical bifurcation are revealed. The results achieved here can be extended to higher dimensional systems.
Highlights
As we know, homoclinic or heteroclinic orbits always lead to complex dynamic behaviors
If we eliminate s2 and λ, it is clear that the two-parameter family of (l − 3)-dimensional surfaces {Σ1(λp, 0, 0, s2, λ)|0 < s2, λ
The heteroclinic orbits with p01 and p2 in Figure 4 (b), (c) and the homoclinic orbit joining p01 in Figure 5 (a) cannot be bifurcated from the generic heteroclinic loop accompanied with transcritical bifurcation
Summary
Homoclinic or heteroclinic orbits always lead to complex dynamic behaviors. Poincare map, transcritical bifurcation, nongeneric heteroclinic loop, nonhyperbolic equilibrium. It follows that for M21μ = 0, there exists an i ∈ {1, 2, .
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