This post is the first in the series “Nonlinear Dynamics”. (The subject of nonlinear dynamics builds up on Differential Equations and Classical Mechanics. You may want to brush up on those)

Dynamical Systems

Let me start with an example.

We take a very small population of bacteria and give them a favorable environment to grow. As time passes, they’ll increase. The question is, what is the rate of growth as time increases? You can argue that as long as they get enough “food” and “space” to grow, they’ll keep on doubling in number like crazy. One splits into two, two into four, four into eight and so on.

We can roughly say that the rate of change of increase in their number, r at a time t is proportional to their number N at time t:

    \[r(t)=\frac{\mathrm{d} N}{\mathrm{d} t}\]

    \[\frac{\mathrm{d} N}{\mathrm{d} t}\propto N(t)\]

In simple words, the more their number, more they grow.

If I were to plot this, I’d get something like:

 

Clearly, the graph isn’t “linear”, had it been, it’d have looked like a line(blue):

In the case above, there is a variable N (no. of bacteria) that varies with time. In one case the rate of change with time is constant(blue) while in the other it depends on N itself (orange), along with some other factors.

Nonlinear dynamics is the subject that deals with systems that evolve in time. That is, there is ‘something’ that changes with time. In the case above, this something is the number N of bacteria. We call such systems ‘dynamical systems’.

Representing a dynamical system

In modeling a system that evolves with time, we start by writing an equation of the form

    \[\dot{x}=f(x,y,a)\]

where x and y are variables, \dot{x} is the rate of change of x with time and a is some parameter.

If the relation only involves a single power of x, we call it linear, e.g.

    \[\dot{x} = 3x+4\]

If however, the right-hand side involves sine/cosine/exponential terms or powers of x, we call them Nonlinear, e.g.

    \[\dot{x} = sin(x)+x^{^{4}}\]

    \[\dot{x} = e^{x}+xy^{3}\]

etc.

Nonlinear systems behave fundamentally different from linear ones.

It is not always easy to obtain the solution to a system of nonlinear equations in closed form. That is, solutions of the form x(t) = 3sin(t)x = 6+4t^{2} etc. are not obtained in most cases. Most of the systems we encounter in daily life are nonlinear.

In future posts, we will look deeply into why the nonlinear systems behave differently and why studying them is difficult. But first, we shall build up vocabulary to describe nonlinear phenomena and analyze simple systems.

Reference

“Overview (Chapter 1).” Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering, by Strogatz, Westview Press, a Member of the Perseus Books Group, 2015.

 

Nonlinear Dynamics: An introduction
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