"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)
"Most systems in nature are inherently nonlinear and can only be described by nonlinear equations, which are difficult to solve in a closed form. Non-linear systems give rise to interesting phenomena such as chaos, complexity, emergence and self-organization. One of the characteristics of non-linear systems is that a small change in the initial conditions can give rise to complex and significant changes throughout the system. This property of a non-linear system such as the weather is known as the butterfly effect where it is purported that a butterfly flapping its wings in Japan can give rise to a tornado in Kansas. This unpredictable behaviour of nonlinear dynamical systems, i.e. its extreme sensitivity to initial conditions, seems to be random and is therefore referred to as chaos. This chaotic and seemingly random behaviour occurs for non-linear deterministic system in which effects can be linked to causes but cannot be predicted ahead of time." (Robert K Logan, "The Poetry of Physics and The Physics of Poetry", 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)
"Complexity carries with it a lack of predictability different to that of chaotic systems, i.e. sensitivity to initial conditions. In the case of complexity, the lack of predictability is due to relevant interactions and novel information created by them." (Carlos Gershenson, "Understanding Complex Systems", 2011)
"The key characteristic of 'chaotic solutions' is their sensitivity to initial conditions: two sets of initial conditions close together can generate very different solution trajectories, which after a long time has elapsed will bear very little relation to each other. Twins growing up in the same household will have a similar life for the childhood years but their lives may diverge completely in the fullness of time. Another image used in conjunction with chaos is the so-called 'butterfly effect' – the metaphor that the difference between a butterfly flapping its wings in the southern hemisphere (or not) is the difference between fine weather and hurricanes in Europe." (Tony Crilly, "Fractals Meet Chaos" [in "Mathematics of Complexity and Dynamical Systems"], 2012)
"The most basic tenet of chaos theory is that a small change in initial conditions - a butterfly flapping its wings in Brazil - can produce a large and unexpected divergence in outcomes - a tornado in Texas. This does not mean that the behavior of the system is random, as the term 'chaos' might seem to imply. Nor is chaos theory some modern recitation of Murphy’s Law ('whatever can go wrong will go wrong'). It just means that certain types of systems are very hard to predict." (Nate Silver, "The Signal and the Noise: Why So Many Predictions Fail-but Some Don't", 2012)
"One of the remarkable features of these complex systems created by replicator dynamics is that infinitesimal differences in starting positions create vastly different patterns. This sensitive dependence on initial conditions is often called the butterfly-effect aspect of complex systems - small changes in the replicator dynamics or in the starting point can lead to enormous differences in outcome, and they change one’s view of how robust the current reality is. If it is complex, one small change could have led to a reality that is quite different." (David Colander & Roland Kupers, "Complexity and the art of public policy : solving society’s problems from the bottom up", 2014)
"The sensitivity of chaotic systems to initial conditions is particularly well known under the moniker of the 'butterfly effect', which is a metaphorical illustration of the chaotic nature of the weather system in which 'a flap of a butterfly’s wings in Brazil could set off a tornado in Texas'. The meaning of this expression is that, in a chaotic system, a small perturbation could eventually cause very large-scale difference in the long run." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)
"A system governed by a deterministic theory can only evolve along a single trajectory - namely, that dictated by its laws and initial conditions; all other trajectories are excluded. Symmetry principles, on the other hand, fit the freedom-inducing model. Rather than distinguishing what is excluded from what is bound to happen, these principles distinguish what is excluded from what is possible. In other words, although they place restrictions on what is possible, they do not usually determine a single trajectory." (Yemima Ben-Menahem, "Causation in Science", 2018)
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