Chaotic behaviour could emerge within systems where all motion dies out due to the effects of dissipation, according to a new study carried out by researchers in Hungary and the US. The team makes the surprising claim that characteristic features reminiscent of chaos, such as the butterfly effect or fractals, would be seen in a system without any energy input. The new work suggests that a wider range of processes – such as chemical reactions evolving towards their equilibrium or coalescing binary stars that lose energy to gravitational waves – might be chaotic. This would make such processes much less predictable than previously thought.

Sensitive systems

A chaotic system is a dynamic system that is highly sensitive to its initial conditions. The smallest difference in these initial conditions could lead to widely diverging outcomes, making any long-term predictions of its outcome impossible to predict. Normally, such chaotic behaviour is seen in transient systems that are constantly evolving – the system’s dynamics are either non-dissipative or the system is constantly subject to some external force. Such systems undergo transient chaos – a phenomenon in which most trajectories are influenced by a small subset of trajectories that remain chaotic forever.

But what happens with systems in which motion eventually stops due to dissipation? Called “doubly transient”, these systems are the ordinary processes that we encounter in daily life – for example, a spinning coin that wobbles erratically but ultimately stops due to friction. New research led by Adilson Motter of Northwestern University in the US shows that even such systems can exhibit the hallmarks of chaos.

Doubly transient

Motter uses another example of such a system, describing a pendulum that is displaced sideways and then given additional pushes periodically. Thanks to the periodic pushes, the pendulum would never stop as a result of the energy input and this is the scenario of (ordinary) transient chaos. However, if you leave the pendulum alone it will stop thanks to the air friction, regardless of the initial condition. Such a system undergoes “doubly transient chaos” – a phenomenon in which trajectories behave erratically for a transient period of time but then they stop due to dissipation. Systems exhibiting doubly transient chaos do not have any trajectory that remains chaotic forever.

The research was based on both mathematical calculations and numerical experiments. In the experiment, the team’s model system comprised a magnetic pendulum made up of three identical magnets (i.e. three possible final stable states) placed at the corners of a triangle, with the pendulum suspended above the centre of the triangle. The pendulum is subjected to the effects of gravity, attractive magnetic forces, and drag due to air friction. The team found that its pendulum showed signatures of doubly transient chaos.

Surprisingly chaotic

“Perhaps the most surprising result is that, although it looks at first sight similar to the case of transient chaos, doubly transient chaos has fundamentally different dynamical and geometrical properties,” explains Motter. The team found that classical parameters, such as the rate at which the trajectories approach their final states (i.e. toward an attractor) “increases exponentially fast as a function of time, rather than being constant as is the case of transient chaos”, according to Motter. The researchers also found that the fractals separating different basins of attractions have integer dimensions, rather than fractional ones, meaning that their complexity reduces upon magnification.

According to the researchers, their significant and surprising results could have implications for varied and diverse processes from the evolution of chemical reactions towards equilibrium to merging vortices in dissipate flows, to games such as dice throwing, and to large-scale celestial events such as the coalescence of spinning binary-star sources of gravitational waves. “It follows from our study that in all such processes, the outcomes are all far less predictable than anticipated,” says Motter. Now, it is important to study doubly transient chaos in a variety of systems from different domains to identify the universal properties of this phenomenon as well as exploring its applications, according to Motter.

The research is published in Physical Review Letters.