"A fairly general procedure for mathematical study of a physical system with explication of the space of states of that system. Now this space of states could reasonably be one of a number of mathematical objects. However, in my mind, a principal candidate for the state space should be a differentiable manifold; and in case the has a finite number of degrees of freedom, then this will be a finite dimensional manifold. Usually associated with physical is the notion of how a state progresses in time. The corresponding object is a dynamical system or a first order ordinary differential equation on the manifold of states." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)
"The structure of a system is the arrangement of its subsystems and components in three-dimensional space at a given moment of time. This always changes over time. It may remain relatively fixed for a long period or it may change from moment to moment, depending upon the characteristics of the process in the system. This process halted at any given moment, as when motion is frozen by a high-speed photograph, reveals the three-dimensional spatial arrangement of the system's components as of that instant." (James G Miller, "Living systems", 1978)
"Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable." (Stephen Wolfram, "Nonlinear Phenomena, Universality and complexity in cellular automata", Physica 10D, 1984)
"Linking topology and dynamical systems is the possibility of using a shape to help visualize the whole range of behaviors of a system. For a simple system, the shape might be some kind of curved surface; for a complicated system, a manifold of many dimensions. A single point on such a surface represents the state of a system at an instant frozen in time. As a system progresses through time, the point moves, tracing an orbit across this surface. Bending the shape a little corresponds to changing the system's parameters, making a fluid more visous or driving a pendulum a little harder. Shapes that look roughly the same give roughly the same kinds of behavior. If you can visualize the shape, you can understand the system." (James Gleick, "Chaos: Making a New Science", 1987)
"[…] physicists have come to appreciate a fourth kind of temporal behavior: deterministic chaos, which is aperiodic, just like random noise, but distinct from the latter because it is the result of deterministic equations. In dynamic systems such chaos is often characterized by small fractal dimensions because a chaotic process in phase space typically fills only a small part of the entire, energetically available space." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)
"The term chaos is used in a specific sense where it is an inherently random pattern of behaviour generated by fixed inputs into deterministic (that is fixed) rules (relationships). The rules take the form of non-linear feedback loops. Although the specific path followed by the behaviour so generated is random and hence unpredictable in the long-term, it always has an underlying pattern to it, a 'hidden' pattern, a global pattern or rhythm. That pattern is self-similarity, that is a constant degree of variation, consistent variability, regular irregularity, or more precisely, a constant fractal dimension. Chaos is therefore order (a pattern) within disorder" (random behaviour)." (Ralph D Stacey, "The Chaos Frontier: Creative Strategic Control for Business", 1991)
"What is an attractor? It is the set on which the point P, representing the system of interest, is moving at large times (i.e., after so-called transients have died out). For this definition to make sense it is important that the external forces acting on the system be time independent" (otherwise we could get the point P to move in any way we like). It is also important that we consider dissipative systems (viscous fluids dissipate energy by self-friction). Dissipation is the reason why transients die out. Dissipation is the reason why, in the infinite-dimensional space representing the system, only a small set" (the attractor) is really interesting." (David Ruelle, "Chance and Chaos", 1991)
"An attractor that consists of an infinite number of curves, surfaces, or higher-dimensional manifolds - generalizations of surfaces to multidimensional space - often occurring in parallel sets, with a gap between any two members of the set, is called a strange attractor." (Edward N Lorenz, "The Essence of Chaos", 1993)
"As with subtle bifurcations, catastrophes also involve a control parameter. When the value of that parameter is below a bifurcation point, the system is dominated by one attractor. When the value of that parameter is above the bifurcation point, another attractor dominates. Thus the fundamental characteristic of a catastrophe is the sudden disappearance of one attractor and its basin, combined with the dominant emergence of another attractor. Any type of attractor static, periodic, or chaotic can be involved in this. Elementary catastrophe theory involves static attractors, such as points. Because multidimensional surfaces can also attract" (together with attracting points on these surfaces), we refer to them more generally as attracting hypersurfaces, limit sets, or simply attractors." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"The dimensionality and nonlinearity requirements of chaos do not guarantee its appearance. At best, these conditions allow it to occur, and even then under limited conditions relating to particular parameter values. But this does not imply that chaos is rare in the real world. Indeed, discoveries are being made constantly of either the clearly identifiable or arguably persuasive appearance of chaos. Most of these discoveries are being made with regard to physical systems, but the lack of similar discoveries involving human behavior is almost certainly due to the still developing nature of nonlinear analyses in the social sciences rather than the absence of chaos in the human setting. " (Courtney Brown, "Chaos and Catastrophe Theories", 1995)
"A system may be called complex here if its dimension (order) is too high and its model (if available) is nonlinear, interconnected, and information on the system is uncertain such that classical techniques can not easily handle the problem." (M Jamshidi, "Autonomous Control on Complex Systems: Robotic Applications", Current Advances in Mechanical Design and Production VII, 2000)
"Science reveals complexity unfolding in all dimensions and novel features emerging at all scales and organizational levels of the universe. The more we know the more we become aware of how much we do not know. […] Complexity itself is understood as a particular dynamic or 'movement' in time that is simultaneously stable and unstable, predictable and unpredictable, known and unknown, certain and uncertain." (Terry Cooke-Davies et al, "Exploring the Complexity of Projects", 2009)
"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)
"Strange attractors, unlike regular ones, are geometrically very complicated, as revealed by the evolution of a small phase-space volume. For instance, if the attractor is a limit cycle, a small two-dimensional volume does not change too much its shape: in a direction it maintains its size, while in the other it shrinks till becoming a 'very thin strand' with an almost constant length. In chaotic systems, instead, the dynamics continuously stretches and folds an initial small volume transforming it into a thinner and thinner 'ribbon' with an exponentially increasing length." (Massimo Cencini et al, "Chaos: From Simple Models to Complex Systems", 2010)
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