If there is one aspect that is common to musical instruments, ocean waves, gravitational waves, antennae, and pendula, it is that all of these are related to oscillations. An important question to ask is what happens when several oscillators are coupled, i.e., they are put together in a situation where one influences another. A mathematical study by Mr Tejas Kotwal, from the Indian Institute of Technology Bombay, collaborating with Dr Xin Jiang at Beihang University in China and Prof. Daniel Abrams at the Northwestern University, has found a simple mathematical explanation of the origin of a special situation known as the Chimera State of these coupled oscillators.
Before the turn of the twenty-first century, it was widely believed that a collection of identical coupled oscillators, would either oscillate randomly or in complete synchrony with one another. However, it was later found that under some conditions some of them synchronise in little groups, while others swing at random, as though each of these groups has a different identity. This paradoxical behaviour, discovered in 2002, has been dubbed the “chimera state”, named after a fire-breathing monster, composed of parts of more than one animal, in Homer’s Iliad. In their study published in the journal Physical Review Letters, researchers have shown how we can understand the origin of the “chimera state” using known mathematical equations (model) of coupled oscillators.
The starting point of their analysis is the Kuramoto model. “The Kuramoto model has been around since the 1970s and has been used by thousands of researchers to understand why things in nature tend to synchronise with each other, such as fireflies may flash in unison, crickets may chirp in sync, cells in the heart contract together to pump blood, etc. While the model has the necessary complexity, it can also be solved using pen-and-paper”, explains Prof. Daniel Abrams from Northwestern University, who is an author of the study. Such a rare combination of “strongly nonlinear and exactly solvable” equations are a dream come true for theoretical mathematicians.”
The researchers studied how the Kuramoto model behaved when different parameters like the natural frequencies of the oscillators, the coupling strength between them and the phase lag between them were varied. "The key highlight of this paper is that the chimera state can be achieved via a pitchfork bifurcation off of the well-understood Kuramoto synchronised state. Our analysis reveals that the formation of the chimera state is indeed a case of symmetry breaking,” says lead author Tejas Kotwal from IIT Bombay.
The term ‘bifurcation’ is used in a mathematical sense to describe a sudden, dramatic change in a system when some controlling input is gradually changed. For example, if you slowly change the temperature of water, it will suddenly go from liquid to gas when that temperature passes 100 degrees Celsius. In a pitchfork bifurcation, as you slowly change the control input beyond the bifurcation threshold, the system goes from having one equilibrium state to having two equilibria, both different from the original one. The original equilibrium behaviour disappears (it is no longer stable). “In our coupled oscillator example, as we slowly change a control parameter, we go from a single equilibrium--the fully synchronised state--to having two different stable equilibria--two types of chimera states,” explains Prof. Abrams.
One can also understand ‘symmetry-breaking’ using a simple example. Imagine a pencil balanced on its tip. It is perfectly symmetrical but bound to topple one way or the other although the laws of nature do not prescribe a particular direction. But once it topples, it does so in a particular direction and we say that the symmetry has been broken. In previous works, it was unclear whether the formation of a chimera state is a case of symmetry breaking and there was no explicit connection found between the chimera state and the fully synchronised state. However, this study presents an intuitive understanding of chimera states and where they originate from.
Discussing the potential applications of this paper, Mr Kotwal says, “The formulation of this model can be used to describe the behaviour of systems of chemical and biological oscillators, lasers and mechanical pendula. It is also of widespread use in neuroscience and heart cell dynamics.”