User:Sijothankam/Low-dimensional chaos

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Nonlinear dynamics and chaos is an emerging field with potential application. Since almost all the process is nonlinear, chaos and non-linearity is important in all the areas of science. When we think about chaos there is a general tendency to believe that it occurs only in complex systems. It appeared in physics as a theory in a higher dimensional system called three-body problem. But even in simple systems now we can see chaos. Chaos in simple sense is the result of sensitive dependence on initial conditions.

Examples

Low-dimensional chaos in population biology

Professor Robert May published an article about the population dynamics in biology. The equation that he presented is called logistic equation.

$X_{n+1}=aX_{n}(1-X_{n})\,$

This is one of the simplest system which shows chaos. Here X(n) represent the population in nth year. So from the equation we can see that the population in the next year is proportional to the initial population and the competition effect (1 − X(n)). For simplicity we limit X = [0, 1] in this range. Let us fix $a=4$ and take two initial conditions x₁ = 0.1000 and x₂ = 0.0001 which is much closer to each other. Our question is, what will be the population after 10 years If we do the computation one be one we get,

n	1	2	3	4	5	6	7	8	9	10
x1	0.1000	0.3600	0.9216	0.2890	0.8219	0.5854	0.9708	0.1133	0.4020	0.9616
x2	0.1001	0.3603	0.9220	0.2878	0.8199	0.5907	0.9671	0.1272	0.4441	0.9875

If we look the data carefully we can see that if we have a slight difference in the initial population value as the time progress the difference increases much faster. After ten years the population is 0.9616 in first case, but in the second case it is 0.9875 . But we started with a small difference of 0.0001 after then steps the difference became 0.0259. Two thousand times higher than the initial difference !!!

This sensitive dependence of initial condition is called chaos. So what will be the problem is we have this much dependence on initial condition ?? As we know every physical quantity that we know has an error. When you consider a weather predicting model you have to consider about the initial temperature. If I ask you a question, what is the temperature today ?? you may say 20 degree. But a meteorologist may say 20.000005 degree. So there is an error in our measurement always , the value of the error depends on the apparatus you use. Now the problem comes, if you make a little error in the initial data your prediction will be entirely different after sometime. So the sensitive dependence of initial condition make some restriction on your prediction because of error propagation. But you can make short term prediction because in short term the error will not ruin the data. This is called butterfly effect. Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

Low-dimensional chaos in physics

In physics also simple systems shows chaotic behavior. The knowledge about such system makes the theory of chaos more attractive beautiful.

Simple bouncing ball system

In which a ball is bouncing on top of an oscillating table under the action of gravity. This is an impact type system. For smaller frequency you cannot observe chaos in the system. Chaos happens when you increase the table frequency. So it is natural to study the system behavior as we change a parameter which is shown below bifurcation diagram.

Bouncing ball map

$x_{n+1}=x_{n}+y_{n}\,$

$y_{n+1}=\alpha y_{n}+\beta \cos(x_{n}+y_{n})\,$

here $x$ related to the time interval between two collisions of the ball with the table(dont confuse with table position in the figure), and $y$ is associated with the velocity of the ball. Here we can see the period doubling happens as we increase the table frequency $\beta$ One of the other simplest low dimensional chaotic system is called simple bouncing ball system.

Taylor–Couette flow

Category:Chaos theory

See also

• Low dimensional chaos in stellar pulsations