Two codimension-two bifurcations of a second-order difference equation from macroeconomics

Abstract

In this paper we mainly investigate two codimension-two bifurcations of a second-order difference equation from macroeconomics. Applying the center manifold theorem and the normal form analysis, we firstly give the parameter conditions for the generalized flip bifurcation, and prove that the system does not produce a strong resonance. Then, we compute the normal forms to obtain the parameter conditions for the Neimark-Sacker bifurcation, from which we present the conditions for the Chenciner bifurcation. In order to verify the correctness of our results, we also numerically simulate a half stable invariant circle and two invariant circles, one stable and one unstable, arising from the Chenciner bifurcation.

Another Case of Degenerated Discrete Chenciner Dynamic System and Economics

“…For example, in [31], the Chenciner bifurcation was observed when a potential mechanism from bifurcation analyses was used for studying the occurrence of modulated oscillations in synchronous machine nonlinear dynamics, being reported for the first time in power engineering for this bifurcation. Other authors have analyzed the normal forms to provide the parameter conditions for the Chenciner bifurcation [32] or the conditions to obtain a Chenciner bifurcation in macroeconomics [33].…”