Control of Hopf and Bautin bifurcation in a modified Goodwin model of growth cycle

Abstract

In this thesis, we conduct a comprehensive analysis of regulating key bifurcation features, such as the stability of the equilibrium and the stability and orientation of the limit cycles, in Desai et al.'s modified Goodwin model of growth cycle, which elucidates the dynamics of class struggle in controlling economic systems. The study systematically manipulates model parameters to position the economy within the desired regions of both the Hopf and Bautin bifurcation diagrams. The original Goodwin model comprises two dynamic equations representing the share of labour in national income and the proportion of labour force employment. Solutions of the system exhibit clockwise closed cycles, each centered at the singular point. Values of this point may exceed one. It implies that the singular point may extend beyond the unit area, leading to trajectories that partially or entirely exist outside that area. Even if the singular point remains within the unit box, trajectories could extend partially outside its boundaries. As a result, values surpassing one produce impractical outcomes for both the labour share and the employment rate. The modified Goodwin model proposed by Desai et al. satisfies this requirement, and all trajectories lie inside the unit square. The Jacobian matrix of the modified system at the origin has a pair of purely imaginary eigenvalues. An equilibrium with a pair of complex conjugate eigenvalues may lose its stability at a parameter exceeding a threshold value and transition to a small amplitude limit cycle called Hopf bifurcation. In the supercritical Hopf bifurcation, the limit cycle emerges at the bifurcation parameter greater than its critical value. In the subcritical Hopf bifurcation, the limit cycle is observed at the parameter values less than its critical value. The sign of the first Lyapunov coefficient at the critical value of the bifurcation parameter determines the type of the Hopf bifurcation. The second Lyapunov coefficient is evaluated where the first Lyapunov coefficient vanishes. The system undergoes another bifurcation, incorporating families of stable and unstable limit cycles coexisting, with one nested within the other, colliding, and eventually disappearing through a saddle-node bifurcation. It is called Bautin bifurcation, and it influences the stability of the equilibrium and the orientation of the resulting limit cycle. Understanding the behaviour of such systems is crucial, as they often exhibit fascinating bifurcation diagrams. Recently, a range of control laws were proposed to manipulate the bifurcation features. In this study, we will utilise the control law offered by Braga et al. in 2010 to generate controllable Hopf and Bautin bifurcation. The control law depends on two bifurcation parameters and four control parameters. If we evaluate Desai et al.'s modified system with this control law, the system's Jacobian matrix at the origin gives a pair of complex conjugate eigenvalues with a non-zero real part. The transversality condition of the Hopf bifurcation indicates that the derivative of the eigenvalue's real part with respect to the Hopf bifurcation parameter is non-zero. Considering that the Hopf bifurcation necessitates a pair of purely imaginary eigenvalues and the transversality condition, we fix the Hopf bifurcation parameter to zero when evaluating the Lyapunov coefficients. We calculate the first and second Lyapunov coefficients as presented by Kuznetsov et al. and then determine the critical values of the control parameters where the Lyapunov coefficients vanish. Following Braga et al.'s control law, the control parameter values and the initial conditions are carefully selected to regulate the stability and the direction of the limit cycles emerging near the origin. For various values of the bifurcation parameters, all possible Hopf and Bautin bifurcation diagrams are plotted using NDSolve command of Mathematica, with a specified accuracy and precision goal of 10 digits. We observe that minor alterations in parameter values lead to variations in the behaviour of the modified model, resulting in different types of bifurcations. Through this interdisciplinary analysis, we have advanced and expanded the findings regarding the Goodwin model's controllability and aimed to bridge theoretical insights with practical applications, thereby offering valuable contributions to policy decisions and strategic interventions to navigate the complexities of economic management. This thesis is organised as follows. Chapter 1 includes the purpose of the study, literature review, and hypothesis research questions. In Chapter 2, bifurcation theory and stability in nonlinear dynamical systems are explained. The theorem and accompanying proof of both Hopf and Bautin bifurcation are given in Section 2.1 and Section 2.2, respectively. Section 2.3 introduces Braga et al.'s control law, and Section 2.4 presents the original Goodwin model and Desai et al.'s modified version. In Chapter 3, we perform the bifurcation analysis of the modified model for Hopf in Section 3.2 and Bautin in Section 3.3. Numerical simulations are also presented in this chapter. The last chapter is devoted to some concluding remarks and potential future studies.