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Bi-center conditions and bifurcation of limit cycles in a class of $Z_2$-equivariant cubic switching systems with two nilpotent points

Ting Chen, Feng Li, Yun Tian, Pei Yu

Abstract

In this paper, we generalize the Poincaré-Lyapunov method for systems with linear type centers to study nilpotent centers in switching polynomial systems and use it to investigate the bi-center problem of planar $Z_2$-equivariant cubic switching systems associated with two symmetric nilpotent singular points. With a properly designed perturbation, 6 explicit bi-center conditions for such polynomial systems are derived. Then, based on the $6$ center conditions, by using Bogdanov-Takens bifurcation theory with general perturbations, we prove that there exist at least $20$ small-amplitude limit cycles around the nilpotent bi-center for a class of $Z_2$-equivariant cubic switching systems. This is a new lower bound of cyclicity for such cubic polynomial systems, increased from $12$ to $20$.

Bi-center conditions and bifurcation of limit cycles in a class of $Z_2$-equivariant cubic switching systems with two nilpotent points

Abstract

In this paper, we generalize the Poincaré-Lyapunov method for systems with linear type centers to study nilpotent centers in switching polynomial systems and use it to investigate the bi-center problem of planar -equivariant cubic switching systems associated with two symmetric nilpotent singular points. With a properly designed perturbation, 6 explicit bi-center conditions for such polynomial systems are derived. Then, based on the center conditions, by using Bogdanov-Takens bifurcation theory with general perturbations, we prove that there exist at least small-amplitude limit cycles around the nilpotent bi-center for a class of -equivariant cubic switching systems. This is a new lower bound of cyclicity for such cubic polynomial systems, increased from to .
Paper Structure (8 sections, 4 theorems, 101 equations, 5 figures, 2 tables)

This paper contains 8 sections, 4 theorems, 101 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Simplification of system \ref{['Eqn2']} and the main results
  3. The generalized Poincaré-Lyapunov method
  4. The proof of Theorem \ref{['Thm2.2']}
  5. The $3$rd-order critical point $(1,0)$ of the upper system in \ref{['Eqn9']}
  6. The $2$nd-order critical point $(1,0)$ of the upper system in \ref{['Eqn9']}
  7. The proof of Theorem \ref{['Thm2.3']}
  8. Conclusion

Key Result

Proposition 2.1

The multiplicity of the nilpotent singular point $(1,0)$ of the upper system $($or $(-1,0)$ of the lower system$)$ of Eqn9 is at most 6.

Figures (5)

  • Figure 1: Illustration of the idea of pseudo-Hopf bifurcation.
  • Figure 2: The phase portrait of system \ref{['Eqn9']} with $a_{02} \!=\! b_{12} \!=\! 1$, $a_{21} \!=\! a_{12} \!=\! -1$,$a_{03} \!=\! -4$, $b_{02} \!=\! b_{21}=0$ and $b_{03} \!=\! \frac{1}{3}$, showing that the singular points $(\pm1,0)$ are two cusps.
  • Figure 3: The phase portraits of system \ref{['Eqn9']} showing bi-center at $(\pm 1, 0)$ for (a) Condition I: $a_{21}=1,\, a_{12}=3,\,a_{03}=-3,\, b_{02}=b_{21}=0,\, a_{02}=b_{12}=b_{03}=-1$; (b) Condition II: $a_{21}=a_{03}=-1,\, a_{02}=a_{12}=b_{02}=b_{21}=b_{12}=b_{03}=0$; (c) Condition III: $a_{21}=1,\,a_{03}=-4,\, a_{02}=a_{12}=b_{02}=0,\, b_{21}=\frac{3\sqrt{5}}{10},\, b_{12}=-1, \,b_{03}=- \frac{\sqrt{5}}{5}$; (d) Condition IV: $a_{21}=-\frac{3}{2},\,a_{03}=-3,\, a_{02}=a_{12}=b_{02}=0,\, b_{21}=-2,\, b_{12}=-2, \,b_{03}=5$.
  • Figure 4: The phase portrait of system \ref{['Eqn9']} with $a_{02}=2$, $a_{12}=-3$ and $a_{03}=a_{21}=b_{02}=b_{12}=b_{21}=b_{03}=1$, showing two cusps at $(\pm 1, 0)$.
  • Figure 5: The phase portraits of system \ref{['Eqn9']} showing bi-center at $(\pm 1, 0)$ for (a) Condition V: $a_{02}=-1,\, a_{12}=b_{03}=b_{02}=b_{21}=0,\, a_{21}=a_{03}=b_{12}=1$; and (b) Condition VI: $a_{02}=-4,\,a_{21}=b_{03}=-1,\, a_{12}=3,\, a_{03}=b_{12}=1,\, b_{02}=b_{21}=0$.

Theorems & Definitions (10)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1: TIAN
  • Example 4.1
  • Example 4.2
  • Example 4.3
  • Example 4.4
  • Remark 5.1