Computation of universal unfolding of the double zero bifurcation in the Z₂-symmetric system

Abstract

We present explicit formulae of universal unfolding of the double zero bifurcation for any autonomous ordinary differential equation systems (ODEs) with Z₂-symmetry. These formulae only need to use coefficients of the Taylor expansion of the vector field at its equilibria and are equally suitable for both numerical and symbolic computations, and they construct the relationship between original parameters and topological structures for the double zero bifurcation analysis in any autonomous ODEs. They can be directly used to check whether there exist the original parameters such that a system can exhibit the versal double zero bifurcation. Moreover, we can use them to compute the corresponding bifurcation curves with high precision. We take Chua's circuit as an example to demonstrate the advantages of these formulae.

Keywords:

• Z₂-symmetry
• double zero bifurcation
• normal form
• universal unfolding
• centre manifold

2010 AMS Subject Classifications:

• 34C20
• 34C23
• 37G05
• 37G10
• 37G40

Acknowledgements

The work is supported by the Natural Science Foundation of Guangdong Province (Grant no. s2012040006688), the Science and Technology Innovation Project of Education Commission of Guangdong Province (Grant no. 2012KJCX0073), the National Natural Science Foundation of China (Grant no. 10872119), the Shanghai Leading Academic Discipline Project (Grant no. S30104) and the Key Program of Shanghai Municipal Education Commission (Grant no. 12ZZ084).