Group analysis of nonlinear dynamical systems

Abstract

Differential equations play a vital role in modeling a wide range of natural phenomena in diverse disciplines, including biology, physics, engineering, mathematics, and economics. Despite the availability of numerous solution methods, extracting meaningful interpretations from the obtained solutions and integrating them across disciplines remains a significant challenge. It is noteworthy that nature often displays nonlinear dynamics, with linear models forming a smaller subset. As a result, extensive research is dedicated to unraveling the complexities presented by these nonlinear differential equations. Among various emerging methods, Lie group theory stands out for its effectiveness and power. Developed in the 19th century by Sophus Lie, this theory originated from his investigations into systems of differential equations. The 20th century witnessed a notable increase in interest in continuous one-parameter transformation groups, resulting in significant contributions to the field. Lie group theory finds extended application in nonlinear dynamical systems through the artificial Hamiltonian approach. This method provides a means to compute exact solutions for various coupled ordinary differential equations (ODEs). Notably, all first-order ODE systems can be represented as artificial Hamiltonian systems. By employing the partial Hamiltonian approach, one can determine partial Hamiltonian operators and subsequently derive corresponding first integrals and analytical solutions. Obtaining first integrals for ODEs holds immense significance as they facilitate the derivation of closed-form solutions. However, systematically finding such integrals remains a challenge, particularly for nonlinear dynamical systems where closed-form solutions are often elusive. Consequently, developing methods for obtaining first integrals holds a prominent position in relevant research. In the literature, mathematical modeling of epidemic diseases is of utmost importance for understanding the general characteristics of such diseases. The mathematical models used correspond to non-linear dynamic systems. These types of models have been derived as coupled ordinary non-linear differential equation systems in two, three, four, six, and twelve dimensions. The doctoral thesis, coinciding with the pandemic period, focuses on these systems as non-linear dynamic systems, and extensive work has been conducted on well-known mathematical epidemic models associated with Covid-19 in the literature. Analytical and applied results have been sought concerning both the mathematical and physical characteristics of such epidemic diseases. Motivated by the quest to understand the behavior of dynamical systems and obtain their first integrals, this thesis delves into exploring the effectiveness of the artificial Hamiltonian method across various contexts. Initially, we applied this method to several systems involving two and four-dimensional nonlinear coupled systems of ordinary differential equations (ODEs). By imposing specific parameter constraints, we successfully derived the first integrals necessary for obtaining their exact analytical solutions. These findings served as the foundation for our second published paper. In this research, we address biological population-related models, specifically the two-dimensional Easter Island, Verhulst, and Lotka-Volterra, as well as the four-dimensional MSEIR (M: Populations with passive immunity, S: Suspected, E: Under Supervision, I: Infectious, R: Recovered) and SIRD (S: Suspected, I: Infectious, R: Recovered, D: Death) models. Our primary objective was to examine the nonlinear dynamical system following the successful application of the artificial Hamiltonian method to these models. In the context of a pandemic, with numerous new and unresolved nonlinear dynamical models emerging, our focus shifted to addressing COVID-19 nonlinear models in our subsequent two studies. Building upon this experience, we subsequently demonstrated the complete integrability of the SIRV COVID-19 model. Two different cases are considered with respect to the model parameters. Additionally, the integrability properties and associated approximate and exact analytical solutions of the SIRV (Susceptible-Infected-Recovered-Vaccinated) model are analyzed and investigated by considering two different phase spaces. In the special case $b=k$, two nontrivial first integrals are obtained, demonstrating that these two first integrals make the system completely integrable. Next, analytical solutions are derived, and novel exact analytical solutions for the initial-value problem are introduced using the associated initial conditions of the model. Furthermore, graphics illustrating the evaluations of the solutions are presented, showcasing compatibility with the expected results. Comparing the results obtained by numerical methods with the analytical results from Lie group analysis in this study reveals that the analytical solutions are consistent with the results obtained by numerical methods. Moreover, the results in this study also represent actual physical situations in the real world. These explorations constitute the key themes of our first published research study. Additionally, graphical representations of susceptible, infected, recovered, and vaccinated population fractions evolving with time for the sub-case are introduced and discussed. Building upon previous successes, the third published study delves into the integrability properties and analytical solutions of a fourth-order, first-order coupled system of nonlinear ordinary differential equations (ODEs): the SIRD-CAAP model with a constant alive population. This model serves as a real-world application of COVID-19 dynamics, enabling us to leverage the analytical results obtained. Utilizing the partial Hamiltonian method, we investigated the first integrals and associated exact analytical solutions of the SIRD-CAAP model, uncovering algebraic relations among its parameters. Subsequently, we analyzed the dynamical behavior of the model based on its analytical solutions for both cases, showcasing and comparing the graphical representations of the closed-form solutions. Notably, we demonstrate the decoupling of the SIRD-CAAP model based on its first integrals, a significant finding from a mathematical perspective. With the decoupled equations and solutions at hand, we were able to classify the solution types and derive characteristic features of the solutions, such as their period, location, amplitudes of their extremes, and stationary values if they exist. Such analytical expressions are particularly useful for fitting existing data with the SIRD-CAAP model. We provided an example of how to obtain the SIRD-CAAP parameters from real COVID-19 data. The study further explores the periodicity properties and classification of solution regimes with respect to parameter constraints. Finally, we provide COVID-19 applications based in data from various countries. Based on available COVID-19 data for the number of newly infected and deceased populations in different countries during pandemic waves, we demonstrated that the SIRD-CAAP model is able to capture their behavior quantitatively and simultaneously, using an identical set of parameters for each region. This allowed us to predict the unreported $R(t)$, $I(t)$, and $S(t)$. While the SIRD-CAAP model exhibits oscillatory (periodic) type regimes, giving rise to $D(t)$ curves that are qualitatively similar to the measured data over a period of several years. Successfully applying the Artificial Hamiltonian approach to dynamical systems has been a transformative endeavor, leading to the discovery of unique first integrals and analytical solutions for these systems. Through the utilization of this method, we conducted a comprehensive study of the dynamics and behavior of the models. Notably, in the investigation of the SIRD-CAAP and SIRV models, a groundbreaking achievement was realized as, for the first time in the literature, these models were proven to be fully integrable without imposing any constraints on the parameters. This breakthrough not only expands our understanding of these epidemiological models but also opens avenues for novel applications and insights in the realm of dynamical systems