Hopf-zero bifurcation of Oregonator oscillator with delay

Abstract

In this paper, we study the Hopf-zero bifurcation of Oregonator oscillator with delay. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, we get the normal form by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, we obtain a critical value to predict the bifurcation diagrams and phase portraits. Under some conditions, saddle-node bifurcation and pitchfork bifurcation occur along M and N, respectively; Hopf bifurcation and heteroclinic bifurcation occur along H and S, respectively. Finally, we use numerical simulations to support theoretical analysis.

Spatiotemporal dynamics of a delayed IS-LM model with interest rate and general investment function

“…Using the Routh-Hurwitz criterion and the analysis of the roots of the characteristic equation of the linearized system, we have established the local asymptotic stability of the economic equilibrium and the existence of Hopf bifurcation. Furthermore, the business cycle models and the IS-LM macroeconomic models presented in [1,2,13,20] are improved and extended by considering other economic factors.…”

“…It is important to note that our IS-LM model formulated by system (1.1) improves and generalizes various economic models existing in the literature. For example, when we neglect the diffusion effect, we get the model introduced in [20] that includes the temporal models proposed in [1,2]. Further, the spatiotemporal model presented by Hu et al [13] is a special case of our model (1.1) when the interest rate is absent and the investment function has a particular form.…”

“…Gabisch and Lorenz [3] considered an augmented IS-LM business cycle model with three ODEs. Based on the ideas given in [3,14], Cai [2] proposed a delayed IS-LM model by assuming that the investment, the saving and the demand for money functions depend linearly on their arguments. The delay in [2] denotes the time lag between the decision of investment and his implementation.…”

“…Based on the ideas given in [3,14], Cai [2] proposed a delayed IS-LM model by assuming that the investment, the saving and the demand for money functions depend linearly on their arguments. The delay in [2] denotes the time lag between the decision of investment and his implementation. Abta et al [1] extended the model of Cai [2] to a nonlinear case of investment function of the form given in [17].…”

“…The delay in [2] denotes the time lag between the decision of investment and his implementation. Abta et al [1] extended the model of Cai [2] to a nonlinear case of investment function of the form given in [17]. A recent IS-LM model was introduced in [20] in order to improve and generalize the above models.…”