Theory of Chaos – Solutions for Typical Engineering Problems

Arvind Kumar Mishra

Mathematical phenomenon of chaos, has given solutions in the Astronomy, Meteorology, Population Biology, Economics and Social Psychology.

There are few casual mechanisms vis-à-vis diverse disciplines as above, have in common that is called Phenomenon Behavior of Chaos. The sensitivity to the tiniest changes in the initial conditions or seemingly random and unpredictable behavior that never the less follows precise rules which could be seem in many of the models of the disciplines mentioned above.

Most of the Engineering problems are solved through the fundamentals of physics and classical structural mechanics but there are typical/unique problems in the Engineering also whose solutions may be falling beyond the classical theories of structural mechanics.

Aristotle, a philosopher once commented “the least initial deviation from the truth is multiplied later on thousand fold”. French mathematician Jacques Hadamard developed the framework for Partial Differential Equations exhibiting both continuous and discontinuous dependence on a small change in initial conditions by the year 1922.

Later on, Lorenz’s pioneer work demonstrated that the sensitive dependence was not the matter of mathematical misdescription but there was something interesting in mathematical model exhibiting Chaos.

Sensitive Dependence on Initial Conditions (SDIC) is highly important and this very input decides the outcome of the Linear Analysis. For example, failure of slopes along the highways and sudden dislodging of the rockmass from the hill, have no answers in the classical structural mechanics but it is happening.

To begin, Chaos is typically understood as a mathematical property of a dynamical system. A dynamical system is a deterministic mathematical model, where time can be either a continuous or a discreet variable. In dynamical system of interest with respect to chaos studies then they are Non-Linear by nature.

 Kellert defines chaos theory as a qualitative study of unstable aperiodic behavior in deterministic Non-Liner Dynamical Systems. Kellert’s definition as above picks up two key features that are simultaneously present i.e. instability and aperiodicity. Unstable systems are those exhibiting SDIC but aperiodic behavior means that the system variables never repeat any values in any regular fashion.

In nutshell, many of the Engineering problems especially the failure of slopes and dislodging of the rockmass from the hill, are not solvable through Linear Analysis which is usually carried out by the Designers. However, all these problems are required to be simulated through theory of Chaos under Non-Linear Analysis where the basic assumption i.e. stress is proportional to strain (Linear Analysis) is overruled.

Therefore, at some point of time, the classical structural mechanics has no Engineering solutions for the typical problems as brought out above. In that situation, we have to look for Engineering solutions beyond structural mechanics. These days, it is possible to simulate the Geometry, Engineering properties and Reponses such as earthquake into the basic programme meant for the Non-Linear Analysis.

(The contributor is the Managing Director, Mangdechhu Hydroelectric Project, Trongsa)