Chaos and Strange Attractors Archana Gupta

What is Chaos? Essence of Science is predictability, and almost everything can be predicted. Example Halley’s Comet. Most of the basic laws of nature are deterministic, i.e. they allow us to determine what will happen next from the knowledge of present conditions. Deterministic is not same as predictable. Example weather forecast. Unpredictable behavior of deterministic system is called Chaos. One of the pervasive features of chaos is “sensitivity to initial conditions”.

Chaos – Sensitivity to Initial Conditions To see it, we set r=3.99 and begin at x1=.3. The next graph shows the time series for 48 iterations of the logistic map. Now, suppose we alter the starting point a bit. The next figure compares the time series for x1=.3 (in black) with that for x1=.301 (in blue).

When was chaos first discovered? The first true experimenter in chaos was a meteorologist, Edward Lorenz, who in 1960 discovered it while working on the problem of weather prediction. However the term “Chaos” was introduced by Tien-Yien and James A. Yorke in a 1975 paper entitled “Period Three Implies Chaos”.

Chaos in Real World Some examples of Chaos in Real World Disease – An outbreak of a deadly disease which has no cure. Political Unrest – Can cause revolt, overthrow of government and vast war. War – Lives of many people can be ruined in no time. Stock Market Chemical Reactions

Bucks and Bugs $$$.... Xn+1 = R Xn (1) Xn - Money in the account during nth year R - (1 + Interest Rate) Xn+1 - Amount after n + 1 years Population of Bugs Above equation is also applicable to the population growth of bugs Where X is the original population and R is a constant This equation leads to exponential growth!!! But is that possible???

Bucks and Bugs contd… Exponential growth cannot go for ever whether bucks in bank or bugs in backyard. There is a need to modify equation 1. So modified equation is Xn+1 = R Xn (1 – Xn) This is the Logistic Equation This equation grows rapidly for first few steps and then levels off.

Bucks and Bugs contd… X = 0,1 are unstable fixed points, and are called Repellors. Chaos results when two or more repellors are present. 0 < X < 1 – results in bounded solution which is known as Basin of attraction. X < 0 and X > 1 – results in unbounded solution; attracts to infinity.

Bucks and Bugs contd… Bifurcation or Period Doubling Chaotic Behavior persists up to R = 4 For R > 4, unbounded solution; approaches -∞ For R < 3, Fixed Point Sol. For R > 3, Single sol. splits into pair of solutions. This is called Bifurcation or Period Doubling, which leads to Chaos At still larger values of R, we see chaotic region filled densely with points For R = 4 Solution occupy entire interval from X=0 to 1

The Butterfly Effect Extreme sensitivity to initial conditions is referred to as the Butterfly Effect, i.e. the flap of a butterfly's wings in Central Park could ultimately cause an earthquake in China. The Butterfly Effect was discovered by Edward Lorenz in 1960. In a paper in 1963 given to the New York Academy of Sciences he remarks: “One meteorologist remarked that if the theory were correct, one flap of a seagull's wings would be enough to alter the course of the weather forever”.

Attractors Attractors are the pinnacle and origin of chaos. Attractor is a set A of trajectories in phase space to which all neighboring trajectories converge. A is Invariant (If you happen to start in A you remain there). A has a "basin of attraction“.

Types of Attractors There are four different types of attractors The Point Attractors Limit Cycle Attractors Torus Attractors Strange Attractors

Point Attractors A point attractor is a fixed point that a system evolves towards, such as a falling book, a damped pendulum, or the halting state of a computer. In the figure, the arrows represent trajectories starting from different points but all converging in the same equilibrium state. Point Attractor

Point Attractors contd…

Limit Cycle Attractors A limit cycle attractor is a repeating loop of states. Example, a planet orbiting around a star, an undamped pendulum.

Torus Attractors A system which changes in detailed characteristics over time but does not change its form will have a trajectory which will produce a path looking like the doughnut shape of a torus Example, picture walking on a large doughnut, going over, under and around its outside surface area, circling, but never repeating exactly the same path you went before.  The torus attractor depicts processes that stay in a confined area but wander from place to place in that area.

Strange Attractors An attractor in phase space, where the points never repeat themselves, and orbits never intersect, but they stay within the same region of phase space. Unlike limit cycles or point attractors, strange attractors are non-periodic. The Strange Attractor can take an infinite number of different forms.  Rössler Attractor

Relationship with Chaos Theory Point, Limit Cycle and Torus attractors are not associated with Chaos theory, because they are fixed. Even though there is a high degree of irregularity and complexity in the pattern associated with Limit Cycle and Torus attractors, their pattern is finite and predictions can still be made. The Strange Attractors can take an infinite number of different forms. This is one of the most important properties of strange attractors and show their chaotic behavior. Two initial neighboring points will quickly drive apart and finally will not have the same behavior at all. For example, these two particles start at "almost" the same point (0.5, 0.1 and 0.501, 0.099) but rapidly diverge over time . This shows the sensitive dependence of Chaos on initial conditions.  

Lorentz Attractor In 1960’s Edward Lorentz while attempting to simulate the behavior of the atmosphere came up with this strange shape known as Lorentz attractor. Lorentz's model for atmospheric convection consisted of the following three ordinary differential equations:

Lorentz Attractor contd… A plot of the numerical values calculated from these equations using particular initial conditions can be seen from the picture. Starting from any initial condition the calculations will approach the paths displayed in the image, but the actual path is highly dependent on the initial conditions. The strange shape in the picture attracts points outside of it and as such is called an attractor. The lines in the picture have a dimension greater than two but less that three, a fractional dimension. This property define a shape called a fractal. All strange attractors are fractals and demonstrate infinite self similarity.

Rössler Attractor The Rössler attractor is the solution of these 3 coupled non-linear differential equations: dx/dt = -y -z dy/dt = x + ay dz/dt = b + z(x - c) where a=0.2, b=0.2, c=5.7 The series does not form limit cycles nor does it ever reach a steady state. Instead it is an example of deterministic chaos. As with other chaotic systems the Rossler system is sensitive to the initial conditions.

Henon’s Attractor In 1970's Michele Henon discovered a very simple iterated mapping that showed a chaotic attractor, now called Henon's attractor. It allowed him to make a direct connection between deterministic chaos and fractals. It consists of two X and Y equations that produce a fractal made up of strands. As the image on the right shows, two main strands, that form a rough arc, are the main part of the fractal. Each strand visible contains an infinite amount of smaller counterparts within. This trait is called self-similarity at all levels. The equations are : xn = yn-1 + 1 - (1.4*sqr(xn-1)) yn = 0.3 * xn-1

Pictures of Strange Attractors Ikeda Attractor KAM Islands Tinkerbell Attractor

Conclusion Chaos theory do give us the tools to approximate real systems by mathematics and to tell us how good those approximations are. For instance, by running weather models on a supercomputer using slightly different initial conditions forecasters can tell how predictable the weather will be. If the results diverge widely between the two sets of data, then the weather is in an unpredictable mode, otherwise they know their forecasts will be pretty good. Chaos Theory challenges old assumptions that most things are predictable by math and physics.  The theory covers many areas but primarily dynamic systems and how they operate.  It describes that almost all things in life follow a dynamic pattern, not a linear one, even though math, physics, engineering are based on linear systems.  The Chaos Theory implies that physics and math are the exceptions in life, not the rule. 

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