Chapter i overview of chaos theory
Introduction
Confusion is the determination of the initial state of the long-term conduct of the power system, or the fact that the system parameters are extremely sensitive, but they are not dispersed and cannot be accurately repeated, and it is a complex dynamic behaviour that is prevalent in non-linear systems. The confusion variable, which appears to be a tumultuous process, actually contains inherent patterns. Optimized searches are made using the randomity, ubiquitousness and regularity of the mixed variables, the basic idea being to map the linearity of the mixed variables to the value range of the optimized variables, and then to search with the mixed variables. However, the algorithm, in the case of an optimised search for large space, multiple variables, has the problem of long computing time and not finding the best solution. As a result, one type of confusion self-mapping that folds in an unlimited number of areas can be used to generate mixed variables and to select search spaces to optimize them, and to continuously improve search accuracy. Confusion is an important branch of non-linear science, and it's a strange and steady evolution of non-linear power systems that characterizes the nature of a complex phenomenon that prevails in nature and in human society. Thus, a large number of confused scholars, such as the chaos science advocate shlesinger and the famous physicist ford, believe that chaos is the third largest revolution in physics in the 20th century, the first two of which are quantum mechanics and relativism, and that confusion optimization is an important direction of study in the field of engineering applications for the chaos discipline. Its application is characterized by using the characteristics of the chaos movement to overcome the deficiencies of the traditional optimisation method, so that the optimisation results are better。
1. The characteristics of chaos, which, by their nature, appear to be random processes, are in fact fundamentally different from random processes. Confusion movements are caused by the intrinsic properties of the physical law of certainty and are derived from the external expression of the intrinsic properties, and are therefore referred to as certainty confusion, while random processes are caused by the noise of the external characteristics. The chaos has the following characteristics:
(1) internal randomity
Confusion is not a three-state definition of movement under the usual concept: static, cyclical and quasi-cyclical, but rather a complex movement that is always limited to a limited number of regions and is never repeated. First, confusion is inherent and the complexity of the system is demonstrated by its own, internal factors, not caused by outside interference, but by its inherent randomity. Secondly, the randomity of confusion is certain. The certainty of confusion is divided into two aspects, the first being that it is a defined system; second, chaos appears to be random, not real random, and every moment of the system's state is influenced by the previous state of affairs, which is certain, not random, system
The state can be fully recreated, unlike the random system. Thirdly, the performance of chaos systems is complex. Disturbing systems perform seemingly randomly and are not cyclical or quasi-cyclical, but rather characterized by good self-relevance and low-frequency broadband。
(2) long-term unpredictability
Since the initial conditions are limited to a limited precision and the minor differences in the initial conditions may have a significant impact on later time evolution, the dynamic characteristics beyond a certain point in the future cannot be predicted in the long term. The long-term evolution of a confused system is unpredictable. Here, for example, the classic logistic map:
X(n+1) = μx(n) (1-x(n))
= 0, 1, 2, 3 0 x 0 < 1
0<μ4(1-1)
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For the opening value 0. 6, at the end of the spectrometer μ evaporated from 2. 6 to 3e-4 to 3e-4 and 200-time results were modelled as shown in figure 1-1, and it was found that with the increase of the spectrometer, the iterative sequence went through a process of 2 cycles, 4 cycles, 8 cycles, ... Infinity cycles, which validated the chronic unpredictability of the system state from the actual results。
Figure 1-1
Attaching matlab emulator:
Mu=2. 6: 3e-4:4; k=length(mu); x=linspace(0. 6,0,k); for n=1:k
X(n+1)=mu(n)*x(n)*(1-x(n)); plot(mu,x(1:),'k.'); xlabel (`mu'); yabel('x(n)'); end
(3) sensitive dependence on opening value
Over time, the initial conditions of arbitrary proximity will show the evolution of the respective periods of independence, i. E. The sensitivity to the initial conditions. There are small deviations from timely initial data, which can be significant after several overlaps。
(4) universality
When a system becomes confused, the characteristics it displays are general and its systems are not altered by the differences in the specific system and the differences in the system's kinetic equations, even if the confusion is similar from the outside。
(5) fractal
The term fractal is a new term used in the creation of fractal geometry by b. B. Mandelbrot in the 1970s. Fractal means a geometrical nature of a set of points in n-dimensional space, which have an unlimitedly fine structure, are similar in part and in whole at any scale and have a number of integer dimensions smaller than the dimension of n in the space, which is called fractional. The fractional dimension is the basic characteristic of the description of the fraction in terms of the non-integer-dimensional dimension。
(6) spectrum
It is also called mixed. Because chaos is a movement that has always been limited to a limited area and has never been repeated and complex in its course. Thus, over time, the trajectory of the chaos movement will never remain in a state but will pass through every point in the space of the region, i. E., the confusion will not repeat every point as long as it lasts。
(7) boundaries
Its motor trajectory has always been limited to a defined area, known as the confusion attraction。
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Thus, the confusion system is generally stable。
(8) dimensions
The operational state of the mixed system is multileaf and multilayered, and the layer of leaves is more and more nuanced and manifests itself in a self-similar structure of unlimited levels。
(9) statistical characteristics
For a mixed system, the positive lyapunov index shows that the track is unstable in each part and that the adjacent orbit is indexed. However, because of the breadth of the attraction, the orbit can only fold over and over again within a limited area, but never intersect, creating a special structure for the attraction of confusion。
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