Attractor
An attractor in physics refers to a set of physical properties or states towards which a system tends to evolve over time. It represents the stable equilibrium points or patterns that a system naturally gravitates towards, even in the presence of external disturbances. Attractors are fundamental to understanding the behavior of complex dynamical systems.
6 Key excerpts on "Attractor"
eBook - ePubFractal Sustainability
A systems approach to organizational change
- Isabel Canto de Loura, Robin Dickinson(Authors)
- 2016(Publication Date)
- Routledge(Publisher)
...Lorenz, one of the most eminent researchers in the field of dynamical systems, was among the first to fully elaborate on chaos, and in 1980 he published ‘Attractor Sets and Quasi-geostrophic Equilibrium’ (Lorenz, 1980). This may sound a bit off-putting, and we will not explore these highly complex concepts, apart from attempting to express them in slightly more mundane terms that will then serve to sustain the analogies we want to draw in view of the proposed framework. In the mathematical field of dynamical systems, an Attractor is a set of numerical values towards which a system tends to evolve, for a wide variety of starting conditions of the system. In physical systems, those values may refer to positional coordinates, for example; and in economic systems, those values may be separate variables such as the inflation rate and the unemployment rate. An Attractor is defined as the smallest unit which cannot itself be decomposed into two or more Attractors with distinct areas (or basins) of attraction. This restriction is necessary since a dynamical system may have multiple Attractors, each with its own basin of attraction. Therefore, an Attractor can be a point, a finite set of points, a curve or even a complicated set with a fractal structure known as a strange Attractor. The trajectory (orbit) of the dynamical system in the Attractor does not have to satisfy any special constraints, except for remaining on the Attractor, forward in time. The trajectory may be periodic or random. However, if a set of points flow in the neighbourhood away from the set, then the set is not an Attractor: instead, it is called a repeller (or repellor). The rather curious term strange Attractor was coined by David Ruelle and Floris Takens to describe the Attractor resulting from a series of bifurcations (see §5.6) of a system describing fluid flow...
eBook - ePub- John Vandermeer, Ivette Perfecto(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
...The weather system is chaotic, just like many ecological systems. Furthermore, the initial ideas of transitions among states (all the arrows in Figure 4.4) with probabilities fixed to each of them, would likely have represented a far better sense of understanding. This would have been a stochastic approach to the process. Indeed, the interweaving of chaos and stochasticity is an active area of ecological research, something we deal with more fully in Chapter 6. Chaotic Attractors, transients and Cantor sets Recalling the material in Chapter 2 concerning the idea of stability, it is generally thought that a system is either stable or unstable (e.g., see Figures 2.13 and 2.15). Sometimes it may be an oscillatory system, but it is usually thought that some sort of periodic behavior will characterize a system such that in the limit it will approach a forever repeating cycle, a limit cycle. When a system is NOT at a limit point or in a limit cycle it is said to be “transient.” Trajectories that are on their way toward either a limit point or a limit cycle are said to be transients. A general terminology has evolved concerning these categories of dynamical systems. Limit points can be either stable or unstable (see Chapter 2), and limit cycles similarly can be stable or unstable. Recent literature refers to stable points/cycles as “Attractors” and unstable ones as “repellors,” for obvious reasons. And this leads to possible confusion. A chaotic Attractor is, as might be guessed, an object that is eventually approached, and trajectories that are not on the chaotic Attractor are transients to it. The idea of an “Attractor” that is “chaotic” may seem strange when first encountered. Indeed, that is what everyone thought when we first discovered the phenomenon, which is why another word for a chaotic Attractor is a “strange Attractor.” Consider the example of an artificial population living on a pair of islands...
eBook - ePubLeadership and the New Science
Discovering Order in a Chaotic World
- Margaret J. Wheatley(Author)
- 2006(Publication Date)
- Berrett-Koehler Publishers(Publisher)
...Briggs and Peat paint a similarly compelling picture of the drama and beauty of strange Attractors forming: “Wandering into certain bands, a system is … dragged toward disintegration, transformation, and chaos. Inside other bands, systems cycle dynamically, maintaining their shapes for long periods of time. But eventually all orderly systems will feel the wild, seductive pull of the strange chaotic Attractor” (1989, 76–77). Strange Attractor. 1. Traditional plots of one variable show a system in chaos. 2. If the system is plotted in multiple dimensions in phase space, the shape of chaos, the strange Attractor, gradually becomes visible. 3. As the system’s chaotic wanderings are plotted over time (the system never repeats its behavior exactly), the Attractor reveals itself. This butterfly or owl-shaped strange Attractor reveals the order inherent in a chaotic system. Order always is displayed as a shape or pattern. From Gleick, 1987. Used with permission. Chaos has always partnered with order—a concept that contradicts our common definition of chaos—but until we could see it with computers, we saw only turbulence, energy without predictable form. Chaos is the last state before a system plunges into random behavior where no order exists. Not all systems move into chaos, but if a system becomes unstable, it will move first into a period of oscillation, swinging back and forth between two different states. After this oscillating stage, the next state is chaos, and it is then that the wild gyrations begin. However, in the realm of chaos, where everything should fall apart, the strange Attractor emerges, and we observe order, not chaos. A strange Attractor becomes visible on a computer screen because scientists have developed new ways of observing the system’s wild and rich behavior. Its behavior is displayed in an abstract mathematical space called phase space. In phase space, scientists can track a system’s movement in many more dimensions than was previously possible...
eBook - ePub- Marcio Eisencraft, Romis Attux, Ricardo Suyama(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...The subset A is invariant in the sense that trajectories or orbits starting in A will remain confined in A forever. Attractors can exist only in dissipative systems. In the next three sections, some kinds of Attractors are presented. 3.2 Steady state and Lyapunov stability This section is primarily based on [ 4, 17, 25, 28 ]. A steady-state solution x (t) = x *, represented by a point in the phase space, is called equilibrium point in continuous-time systems or fixed point in discrete-time systems. 12 The steady states of ẋ (t) = f (x (t)) are obtained from f (x *) = 0 (because this implies ẋ (t) = 0); the steady states of x (t +1) = f (x (t)) are calculated from f (x *) = x * (because, in this case, x (t +1) = x (t) = x *). According to Lyapunov, x * is a locally neutrally stable point if, in the phase space, given ϵ > 0, there is an open ball (an. open n -dimensional sphere) of radius δ (ϵ) > 0 centered in x * such that starting from any point x 0 pertaining to this ball, then x (t) remains within a ball of radius ϵ also centered in x * for t > 0. Thus, trajectories or orbits beginning within a distance δ of x * remain within a distance ϵ of x * for t > 0. If there is no value of δ satisfying this condition, then x * is an unstable point. If there is a ball of radius δ > 0 centered in x * such that, beginning in any point pertaining to this ball then x (t) → x * for t → ∞, the point x * is considered locally asymptotically stable. In this case, any trajectories or orbits starting from any initial condition x 0 within a distance δ of x * eventually converges to x *; that is, for ∑ j = 1 n (x 0 j − x j ∗) 2 < δ, then ∑ j = 1 n (x j (t) − x j ∗) 2 → 0 for t → ∞. Roughly speaking, neutral stability of x * means bounded movements around this point; asymptotical stability implies convergence to this point; instability leads to divergence. A way of trying to determine the local stability of x * involves the use of Lyapunov indirect method...
eBook - ePubDevelopmental Psychology
How Nature and Nurture Interact
- Keith Richardson(Author)
- 2005(Publication Date)
- Psychology Press(Publisher)
...There are four sorts of Attractors. The first is called a fixed-point Attractor, and refers to the existence of a single, stable equilibrium point, like the end-point of a reach or a grasp, where action consists in a trajectory (the reach) terminating at a point (location of object) and an accurate form is roughly the same for everyone, there not being alternative stable forms. At first there would be errors in reaching too far, and not far enough, and both of these would reduce over time till the error was minimal. Look at Figure 10. Amplitude refers to amplitude of error, which gradually reduces, for both positive (overreach) and negative (under reach) instances over time, till the attraction of the fixed point prevails. Figure 11 shows the same information in a different way. The meeting of the axes corresponds to zero error in any direction, so the circular trajectory, gradually homing in on zero, corresponds to the decreasing error in reaches, over time, anywhere about the point to be reached. Figure 11 also exactly describes the motion of a real pendulum swinging past the vertical, but gradually reducing in amplitude, because the dynamics are identical. Similarly, the dynamics of the control of the various muscles and joints is exactly the same for an arm as for a spring, involving flexion of the various muscles at various times, in relation to each other, and variation in the angles between limbs at joints. In both, account has to be taken of inertia, a property of all dynamic systems, and the laws of force, mass and acceleration have to be obeyed, because the arm is matter as well, and thus subject to its laws (Zernicke and Schneider, 1993, in Goldfield, 1995, pp. 98-9). Figure 10 Amplitude Source: Thelen and Smith (1994), p. 57. The second sort of Attractor is a limit cycle or periodic Attractor, where a continuous oscillation with a fixed amplitude and frequency occurs...
- Arturo Buscarino, Luigi Fortuna, Mattia Frasca(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...This chapter discusses methods and numerical experiments on continuous-time systems with chaotic Attractors. The electronic implementation and applications of chaotic elementary circuits will be the focus of Chapter 8. 5.1 Features of chaos in continuous-time systems All the fingerprints of chaos in discrete-time systems are also retrieved in continuous-time systems. Therefore aperiodic oscillations of the state variables, high sensitivity to initial conditions, sensitivity to parameter changes, period doubling cascades, if any, ruled by the Feigenbaum constant, long-term unpredictability, signals with a wide spectrum, similar to that of white noise, are all features of continuous-time chaotic systems as well. Moreover, since in deterministic systems state space trajectories have no common points, aperiodicity in autonomous continuous-time systems is only possible if the number of state variables is at least three. In the case of continuous-time systems, Attractors may also be strange. The term refers to the fact that the orbit is bounded, but not periodic or convergent; on the contrary, it has a complex fractal structure. Chaotic systems display strange Attractors, that is, the trajectories are confined in a limit set, in which an infinite number of trajectories approach each other without intersecting one another. Given any point of a trajectory, at some time the trajectory will return arbitrarily close to that point, but will never pass again through the same point, thus forming a dense set of points in a specific geometric structure. Despite the high sensitivity to initial conditions that makes each trajectory unique, the unfolding of the trajectory in the phase space is a geometric structure that is qualitatively the same for each initial condition. Example 5.1 Let us consider the following nonlinear system that was introduced by the American mathematician and meteorologist Edward Lorenz in 1963 [ 52 ], and for this reason it is named the Lorenz system...
