Julien C Sprott - Collected Quotes

"A system of equations is deemed most elegant if it contains no un- necessary terms or parameters and if the parameters that remain have a minimum of digits. [...] Just as one can find the most elegant set of parameters for a given system, it is possible to find the most elegant set of initial conditions within the basin of attraction or chaotic sea. However, it is usually more useful to have initial conditions that are close to the attractor to reduce the transients that would otherwise occur."  (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"Another property of bounded systems is that, unless the trajectory attracts to an equilibrium point where it stalls and remains forever, the points must continue moving forever with the flow. However, if we consider two initial conditions separated by a small distance along the direction of the flow, they will maintain their average separation forever since they are subject to the exact same flow but only delayed slightly in time. This fact implies that one of the Lyapunov exponents for a bounded continuous flow must be zero unless the flow attracts to a stable equilibrium." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"In a chaotic system, there must be stretching to cause the exponential separation of initial conditions but also folding to keep the trajectories from moving off to infinity. The folding requires that the equations of motion contain at least one nonlinearity, leading to the important principle that chaos is a property unique to nonlinear dynamical systems. If a system of equations has only linear terms, it cannot exhibit chaos no matter how complicated or high-dimensional it may be." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"In fact, contrary to intuition, some of the most complicated dynamics arise from the simplest equations, while complicated equations often produce very simple and uninteresting dynamics. It is nearly impossible to look at a nonlinear equation and predict whether the solution will be chaotic or otherwise complicated. Small variations of a parameter can change a chaotic system into a periodic one, and vice versa." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"In fact, there are as many Lyapunov exponents as there are state space variables, and what was calculated is only the largest (or least negative) of them. Fortunately, this is the only one that is required to identify chaos, since if it is positive, the system exhibits sensitive dependence on initial conditions independent of the values of the others, and if it is zero or negative, none of the others can be positive either." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"Systems with dimension greater than four begin to lose their elegance unless they possess some kind of symmetry that reduces the number of parameters. One such symmetry has the variables arranged in a ring of many identical elements, each connected to its neighbors in an identical fashion. The symmetry of the equations is often broken in the solutions, giving rise to spatiotemporal chaotic patterns that are elegant in their own right." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"The main defining feature of chaos is the sensitive dependence on initial conditions. Two nearby initial conditions on the attractor or in the chaotic sea separate by a distance that grows exponentially in time when averaged along the trajectory, leading to long-term unpredictability. The Lyapunov exponent is the average rate of growth of this distance, with a positive value signifying sensitive dependence (chaos), a zero value signifying periodicity (or quasiperiodicity), and a negative value signifying a stable equilibrium." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"The possible existence of multiple attractors means that it is necessary to search different initial conditions as well as different parameters when determining whether a given dynamical system is capable of exhibiting chaos." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)

"Transient chaos can be viewed as a situation in which an attractor touches its basin of attraction but only at places that are rarely visited by the trajectory. The trajectory is initially drawn to the attractor and wanders around on it for a long time before eventually coming to a place outside the basin of attraction, whereupon it escapes. Think of a fly buzzing around in a box for a long time before discovering a small hole in the wall that leads to the outside world. Of course the hole could also be a small patch of flypaper that would bring the °y to a permanent halt, just as a stable equilibrium might for a transiently chaotic trajectory." (Julien C Sprott, "Elegant Chaos: Algebraically Simple Chaotic Flows", 2010)