## Chaos

My first interaction with chaos was James Gleicks amazing book "Chaos" which in my opinion is the best starting point for a venture into the world of chaos. I was enthralled by this new world. All (well most) of the equations I had ever used to solve physics problems lay bare to sad reality that they did not truly describe our actual turbulent and non-ideal world. Though I guess the point particles and spherical cows should have given me a fair warning.

## Chaotic Magnetic Pendulums and Circuits

The first step was to see chaos happening around me. Luckily the physics lab course had two experiments based on investigating chaos. First up was a simple RLD circuit followed by a magnetic pendulum. I saw the systems go through period doubling before getting lost into beautiful luminescent chaotic patterns on the oscilloscope screen.

Overview: Chaos expresses itself in certain unique ways that can be observed through careful observation and measurement if one searches for these fine signs of chaos amidst the apparent noise and confusion. There are several ways a chaotic system can be differentiated from statistical indeterminacy. Several simple laboratory experiments have often been used to identify chaotic systems. Some famous examples are the double pendulum, the RLD circuit, pendulums with oscillating pivots and the magnetic pendulum. In each system varying a certain parameter controls the entry into chaos. Slight changes in this parameter take the systems from periodic to chaotic motion. As this control parameter is varied the system shows initial signs of chaos such as period doubling which can be observed using time series plots, phase plane portraits and Poincare sections. Chaos then also has its unique face in these various graphical representations. In this chase after chaos we used a driven magnetic pendulum and mapped its progress from a linear system to a chaotic one while carefully mapping its progress as it displayed clear signs of the onset of chaos. (NOTE: The absence of Poincare sections in the report makes in hard to differentiate doubled periods from chaotic orbits)

Report

Overview: Chaos expresses itself in certain unique ways that can be observed through careful observation and measurement if one searches for these fine signs of chaos amidst the apparent noise and confusion. There are several ways a chaotic system can be differentiated from statistical indeterminacy. Several simple laboratory experiments have often been used to identify chaotic systems. Some famous examples are the double pendulum, the RLD circuit, pendulums with oscillating pivots and the magnetic pendulum. In each system varying a certain parameter controls the entry into chaos. Slight changes in this parameter take the systems from periodic to chaotic motion. As this control parameter is varied the system shows initial signs of chaos such as period doubling which can be observed using time series plots, phase plane portraits and Poincare sections. Chaos then also has its unique face in these various graphical representations. In this chase after chaos we used a driven magnetic pendulum and mapped its progress from a linear system to a chaotic one while carefully mapping its progress as it displayed clear signs of the onset of chaos. (NOTE: The absence of Poincare sections in the report makes in hard to differentiate doubled periods from chaotic orbits)

Report

## Hamiltonian Chaos and Statistical Mechanics

Invariant Tori [1]

The foundation stone of all the statistical ensembles is the Ergodic Hypothesis, where one assumes that if a system is left for long enough it will go through all phase states possible. We first assume that the molecules are interacting i.e. there are non-linear cross terms in the Hamiltonian and due to these terms the particles can interact and share energy reaching equipartition as predicted

Chaos was assumed to be a sufficient condition to believe in the Ergodic Hypothesis. The Fermi Pasta Ulam Experiment showed otherwise. This course project for my Statistical Mechanics course, based on a partial review of "Chaos and Coarse Graining in Statistical Mechanics" by Patrizia Castiglione, repeated their simulations and reviewed the basis of the Ergodic Hypothesis.

Surprisingly, some chaotic systems do not reach equipartition and some states regain their energies even after loosing them to other states due to some some leftover invariant tori. I also presented these ideas at the departmental Brown Bag!

[1] http://www.pip.uni-bremen.de/index.php?option=com_content&task=view&id=56&Itemid=39

Report

Brown Bag Presentation

Chaos was assumed to be a sufficient condition to believe in the Ergodic Hypothesis. The Fermi Pasta Ulam Experiment showed otherwise. This course project for my Statistical Mechanics course, based on a partial review of "Chaos and Coarse Graining in Statistical Mechanics" by Patrizia Castiglione, repeated their simulations and reviewed the basis of the Ergodic Hypothesis.

Surprisingly, some chaotic systems do not reach equipartition and some states regain their energies even after loosing them to other states due to some some leftover invariant tori. I also presented these ideas at the departmental Brown Bag!

[1] http://www.pip.uni-bremen.de/index.php?option=com_content&task=view&id=56&Itemid=39

Report

Brown Bag Presentation

## Spatio - Temporal Chaos in a Plankton prey predator system

Though this new field gave a more realistic view on life the theory was still based on the activities of a single point that traveled through phase space as I learnt in my Non-Linear Dynamics course. All the attractors, repellers and limit cycles still weren't very real. As a course project I ventured into the realm of spatio-temporal chaos. A place where limit cycles were born and died in the middle of the system and oscillating systems became chaotic without warning. This project reproduced the first few sections of the results discussed in the paper titled "Spatiotemporal Complexity of Plankton and Fish Dynamics" by Medvinsky et. al.

This still just looks at a one dimensional case (for x going from 0 - 900) but plankton of course live in 3. If homogeneous initial conditions are applied the system just oscillates as a normal prey predator system. However if for example the phytoplankton population varies with x even just slightly the system spirals into chaos after some time. Diffusion is allowed and the boundaries follow Neuman zero flux conditions. Interestingly the space average populations of the plankton is more stable in the chaotic regime.

Report

M File

This still just looks at a one dimensional case (for x going from 0 - 900) but plankton of course live in 3. If homogeneous initial conditions are applied the system just oscillates as a normal prey predator system. However if for example the phytoplankton population varies with x even just slightly the system spirals into chaos after some time. Diffusion is allowed and the boundaries follow Neuman zero flux conditions. Interestingly the space average populations of the plankton is more stable in the chaotic regime.

Report

M File