Abstract
This paper mainly concerns the derivation of the normal forms of the Bogdanov–Takens (BT) and triple zero bifurcations for differential systems with m discrete delays. The feasible algorithms to determine the existence of the corresponding bifurcations of the system at the origin are given. By using center manifold reduction and normal form theory, the coefficient formulas of normal forms are derived and some examples are presented to illustrate our main results.
Highlights
Many researchers have studied some kinds of codimension bifurcation phenomena for some delayed differential systems, these bifurcation phonmena include saddle-node bifurcation, Hopf bifurcation, Hopf-zero bifurcation, double Hopf bifurcation and so on
There are some results about the Bogdanov–Takens (BT) bifurcation and triple zero bifurcation for general differential systems with one delay, one can see, for example, [4, 6, 16, 20, 21], and their results can be used to study the codimension bifurcation of some predator–prey systems, neural networks models, Van der Pol’s oscillator, etc
We will generalize and apply these methods used in [16, 20] to deduce the normal forms of BT and triple zero bifurcations of the following system with m delays: m x = A(P)x(t) + Bl(P)x(t − τl) + F x(t), x(t − τ1), . . . , x(t − τm), P . (1)
Summary
Many researchers have studied some kinds of codimension bifurcation phenomena for some delayed differential systems, these bifurcation phonmena include saddle-node bifurcation, Hopf bifurcation, Hopf-zero bifurcation, double Hopf bifurcation and so on. The general coefficient formulas of normal forms corresponding to BT and triple zero bifurcations for general differential systems with many discrete delays have not been given except for some special differential systems as discussed in [1, 13, 17, 22]. We will generalize and apply these methods used in [16, 20] to deduce the normal forms of BT and triple zero bifurcations of the following system with m delays:.
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