Resilient emotional dynamics in a fuzzy happiness model: Cognitive and noncognitive perspectives

The happiness model employs a differential equation that expresses how a person feels happy when external forces change. When a human feels a specific emotion, it is difficult to express it with a real number to describe its intensity because human emotions are ambiguous. It is more reasonable to express the degree of happiness with fuzzy numbers that can handle this ambiguity mathematically, and trapezoidal fuzzy numbers are the most suitable for this. This paper proposes an exponential function for resilience by applying the fuzzy theory. In considering a real environment, a sinusoidal function and Gaussian noise are applied when considering a real environment. In differential equations, such as the happiness model, external forces often decrease over time rather than have a constant periodicity. Thus, it may be appropriate to express the model as a resilience function that approaches zero over time. In this paper, the function representing the overall shape of the external force is defined as the forced power function expressed as an exponential function that decreases over time. The sinusoidal function, a wave function, was added as an auxiliary function, and a Gaussian noise term was added for randomness, irregularity, or noise terms. We observe the nonlinear phenomenon by expressing differential equations representing human emotions in cognitive and noncognitive states using complex fuzzy numbers and Euler’s formula. When people feel happy or sad, they cognitively know why, but in some cases, when faced with a certain situation, they may feel happier or sadder than they expected. A person’s emotions can be considered the result of a complex effect of the cognitive part, of which the person is aware, and the noncognitive state, of which the person is not aware. Complex numbers are used to represent cognitive and noncognitive states. The real part of a complex number represents the cognitive state, and the imaginary part represents the noncognitive state. Complex numbers are expressed using Euler’s formula to describe the mixed results of cognitive and noncognitive states. This paper uses a fuzzy dynamic happiness model to observe how mixed results occur in human emotions, especially nonlinear behavior. Finally, nonlinear behavior is observed through phase portraits and bifurcation diagrams for the proposed fuzzy dynamic happiness model.