Chaotic Behavior in Happiness Model with Fuzzy External Force

In this paper, we propose a new happiness model based on fuzzy theory when there exists an external force. Because perception and happiness represent a human mind, they cannot be measured precisely. In addition, it is not appropriate to express the degree of an external force using a precise value. Therefore, we apply fuzzy theory to the external force to measure the vagueness of its degree. The happiness model comprises a new non-autonomous quadratic system that can be constructed equivalent to forms of the Duffing equation. Adding a nonlinear term to the proposed happiness equation and using a sinusoidal external force (SEF) with a triangular fuzzy number (TFN) for happiness, we confirm that there are typical routes of chaotic behaviors, including periodic doubling, chaotic motion, and periodic windows. For these, we applied two methods. First, we fixed the parameters c, d, beta, omega, and A and varied the parameter s from 0 to - 1. Second, we fixed c, d, beta, omega, and s, and varied the parameter A from 0.1 to 1. Finally, we show the route of chaotic behaviors by using time series, phase portraits, Poincare map, bifurcation diagram and Lyapunov exponent.