Order from chaos: How mathematical modeling reveals Nature's secrets
In our everyday life, we experience a number of natural changes around us. Such changes are governed by certain principles of nature and if these principles can be written in the form of mathematical equations, many natural phenomena can be explained or predicted in advance.
However, until recently we have had very little success in mathematically modeling any such natural phenomena. This is because almost all natural phenomena are nonlinear in nature and behave with their own independent laws.
No definite common principle has been discovered which explains nonlinear phenomenon. On the other hand, complexity may emerge in any nonlinear system.
Mathematical dynamical systems theory, as a tool to study complexity of the systems, had its inception with Newton. The subject seems to have begun systematically only after the invention of differential equations, and discovery of the Laws of Motion by Newton.
Newton has the distinction of solving many problems which opened the field of complexity. For instance, the problem of calculating the motion of the Earth around the Sun, given the inverse square law of gravitational attraction between them. This naturally led to the search for the solution of some more important problems.
This search was not in vain as it led to the development of a totally new approach. It was Henri Poincaré who emphasized the need for a qualitative rather than quantitative approach to problems. Poincaré was the first person to acknowledge the possibility of complexity and irregularity.
Chaos is one of the most important properties of some complex systems. The phrases "sensitivity with respect to small changes in initial conditions" and "strange attractor or chaos" were introduced in the work by Ruelle and Takens in 1971.
The discovery of chaos changes our understanding of the foundations of physics, and has many practical applications. Indeed, interest in chaos or more generally, nonlinearity and complexity has grown rapidly since 1963, when Lorenz published his numerical work on a simplified mode of convection and discussed its implications for weather prediction.
Chaotic motion is not a rare phenomenon. Indeed, in any dynamical system which can be described by a set of differential equations, several necessary conditions for chaotic motion are needed.
First, the system should have at least three independent dynamical variables, and second, the equations of motion should contain a nonlinear term that couples several of the variables.
Despite the fact that, many dynamical systems which described a real physical phenomenon and satisfy the above conditions may show chaotic motion, these conditions do not guarantee chaos, they do make its existence possible.
Chaos theory may provide new approaches to assessing cardiac risk and forecasting sudden cardiac death. Similar applications show promise in assessing other regulatory systems, such as human gait. Hence, elucidating the chaotic and nonlinear mechanisms involved in physiologic control and complex signaling networks is emerging as a major challenge in the field of human physiology.
In our QNRF-funded project we are interested in exploring the theory of bifurcation, chaos and synchronization of chaos in the form of Fractional Differential Equations (FDEs).
It is notable that FDEs are increasingly used to model problems in a number of research areas including dynamical systems, mechanical systems, signal processing, control, chaos, chaos synchronization and others.
The most important advantage of using FDEs in these and other applications is their non-local property. It is well known that the integer order differential operator is a local operator but the fractional order differential operator is non-local. This means that the next state of a system depends not only upon its current state but also upon all of its historical states.
This is more realistic situation and it is one reason why fractional calculus has become more and more popular. On the other hand, the integer order differential operator is indifferent to its history. Therefore considering chaotic systems in the form of FDEs preserves this crucial property and connects chaotic systems to their related natural phenomena.
Prof. G Hussein Erjaee