Chaos is a kind of pseudo-random behavior generated in deterministic nonlinear systems, showing initial sensitivity. With the increasing application of chaos theory in engineering fields such as communications and radar, the construction of actual chaotic systems and the generation of chaotic signals have become a hot spot in recent research. The realization of chaotic system has laid an important foundation for the application of chaos theory and promoted the further development of chaos theory research.
This thesis first introduced the research background and significance of this topic in general, and then explained some basic concepts of chaos in detail. The establishment of mathematical models, theoretical analysis, circuit realization methods and simulation results of various chaotic system families are described, especially the Lorenz chaotic system modeling and circuit simulation realization based on Matlab, and the related theories of Lorenz chaotic system are discussed. Finally, because this paper introduces a Lorenz chaotic signal generator based on FPGA technology in detail, and simulates the system.
Chaos is a frontier topic in the world today. It reveals the universal complexity in nature and human society: the unification of order and disorder, the unification of determinism and probability, which greatly expands people’s horizons and deepens the objective world. Awareness. The discovery of chaos is the third great revolution in physics after the twentieth century relativity and quantum mechanics. This revolution is impacting and changing almost all fields of science and technology, and presents us with huge challenges.
Chaos science first poses a fundamental question to us: Do people live in a deterministic universe or in a probabilistic universe? This is a question that scientists and philosophers have been discussing for a long time. The world has a fundamentally important issue, but it has not been fully resolved. Chaos theory is narrowing the gap between these two opposing description systems: chaos is the inherent randomness of a deterministic system, and a world of dialectical unity of determinism and probability. Chaotic motion is one of the basic motion forms in nature.
What is the application prospect of chaos? This is a major issue facing the world today. Due to the strange characteristics of chaos (strange attractors), especially the high sensitivity and instability to extremely small changes in initial conditions, the so-called "the smallest difference is a thousand miles away", people have always believed that Chaos is unreliable and uncontrollable, so it is a monster that cannot be applied. In the field of application and engineering, it has always adopted an attitude of avoidance and resistance. For example, people always try to avoid the emergence of chaos in the analysis and design of electronic systems, and strive for the stability and reliability of the system. This one-sided understanding and thinking mode of chaos has restricted the development and application of chaos theory for a long time. From this, people think that the process experienced by the first two revolutions of physics is very similar to this. Since the 1990s, chaotic control and synchronization theories have made breakthroughs in the world, and the theoretical and applied research inspired by this has developed rapidly, giving opportunities for the application and development of chaos. Through a deeper study of chaos, people find that chaos can be controlled and synchronized. It is particularly worth mentioning that the chaos synchronization and chaos control method, which was proposed by the US Naval Laboratory researchers Pecora and Carroll et al. in 1990, made the application of chaos in the field of communication possible. This historic discovery will It is possible to apply chaos to the field of communication, leading to a new communication method in the 21st century—chaotic synchronization communication, which is hailed as a new milestone in chaos theory and application, and opened a new chapter in chaos synchronization communication. . The historical development process from the emergence and formation of chaos theory to the control and application of chaos fully shows that chaos science is changing people's previous thinking mode, enriching and developing people's natural view and epistemology.
1.2 The purpose and significance of the research
The study of chaos has had a huge impact on the development of modern science. From its early explorations to major breakthroughs, the study of chaos has formed a worldwide research boom after the 1970s. Its fields include mathematics, physics, chemistry, biology, meteorology, economics and many other disciplines. The results penetrated and influenced almost the entire disciplinary system of modern science. The study of chaos has opened a new chapter in the development of modern science. The impact of chaos research on modern science is not limited to natural sciences, but also involves economics, sociology, philosophy, and many humanities, which can be said to cover almost all disciplines. Chaos is found in most research fields involving dynamic processes, and the research results of chaotic dynamics need to be applied. In the field of traditional classical science, if you re-examine it according to the chaotic viewpoint, you will discover new phenomena, raise new questions,
New principles: In some non-classical scientific fields, the use of chaos theory can explain phenomena that could not be explained in the past, can process data that has not been processed before, and even form a batch of new disciplines. The more profound influence of chaos on modern science mainly lies in overturning some basic assumptions of classical theories in a broad scientific field, changing the research methods in those fields, and may nurture a scientific revolution. The study of chaos has also revolutionized the classical view of science. Before the discovery of chaos, people long believed that deterministic systems rejected randomness, which was only an attribute of some complex systems. However, chaos studies have shown that some completely deterministic systems, without any additional random factors, have certain initial conditions, but the system itself will inherently produce random behavior, and even a very simple deterministic system has inherent randomness. The scientific fact that the randomness revealed by chaos exists in certainty is the most powerful proof that objective entities can have both certainty and randomness; from Newton to Einstein all believe that the world is essentially orderly and orderly It is equal to regularity, disorder is irregularity, and systematic order and disorder are diametrically opposed. This view has been supported by people for centuries, but the discovery and research of chaos and fractal have shown that chaos contains both order and disorder. Chaos is neither an ordered state with periodicity and other obvious symmetry, nor is it Absolute disorder, which can be regarded as a complex order that must be characterized by strange attractors, is an order implied in disorder.
Chaos research has also made significant contributions to the transformation of traditional methodology, the most prominent of which is the transformation from reductionism to system theory. Classical reductionism believes that overall or high-level properties can also be reduced to partial or low-level properties. However, with the development of modern science, including the exploration of chaotic phenomena, reductionism has hit a wall everywhere. In the 1950s, systems theory began to take shape, advocating to treat the research object as a system. Chaos is an overall behavior of the system, and the research results of chaos have become a powerful proof of system theory. The holistic view and system theory are being expanded to various modern disciplines along with chaos, preparing methodology for the revolutionary change of modern science. Chaos, as a frontier subject and academic hotspot that attracts worldwide attention, not only greatly expands people's horizons and deepens the understanding of the objective world, but also because of the strange characteristics of chaos, it also prompts people to study methods of controlling and using chaos in real life. In recent years, international research on chaos synchronization and chaos control has made some breakthroughs, and the prospects are very attractive. We have every reason to believe that chaotic progress not only breeds a profound scientific revolution, but also promotes the vigorous development of social productive forces and has a huge and fundamental impact on people's lives and production.
When Lorenz was studying atmospheric convection in 1963, Lorenz discovered the first three-dimensional autonomous chaotic system with a simple structure. This is the famous "butterfly effect" model. Its mathematical model is:
When r≤1, it holds for all x, y, and z. The equation is only true when r<1, x=y=z=0 and r=1, x=y, z=0.
When r>1, the system begins to be unstable, and when r increases to rc, there is a subcritical Hopf bifurcation.
When r>rc, the adjacent orbitals near the attractor are separated exponentially on average, so the two orbitals that start to be very close together quickly lose all correlation and appear chaotic.
5.2 MATLAB simulation of Lorenz chaotic system
Run the above program code in the MATLAB environment, enter the program starting point (x0, y0, z0), program running time (t0, tf), parameters (a, b, r), you can get the three-dimensional phase plan simulation results, as shown in the figure Shown in 5-1.
Figure 5-1 xy plane, yz plane, yz plane
Figure 5-2 Lorenz phase plan
The coefficients are quantized, assuming 1 is 1111_1111_1111_1111_1111_1111 after quantization, which is FFFFFF, then we can get
0.99 FD70A2 ；
0.01 ： 028F5C；
0.028 ： 072B02；
0.001 ： 004189；
0.999 ： FFBE75
0.9973333： FF513B；
The calculation results here are all 24 bits, so when writing the hardware description language in the FPGA, we also need to use the 24-bit calculation method. When the multiplier multiplies to obtain a 48-bit result, the 24-bit result is obtained through truncation. Then participate in the next level of calculation.
6.2.1 Multiplier
The multiplier, we use the IP core to complete.
Figure 6-2 Step 1 of multiplier IP core generation
Next step:
Figure 6-3 Step 2 of multiplier IP core generation
Next step:
Figure 6-4 Multiplier IP core generation step 3
Next step:
Figure 6-5 Step 4 of multiplier IP core generation
Next step:
Figure 6-6 Step 5 of multiplier IP core generation
Complete, simulate the core, the simulation results are as follows:
Figure 6-7 Multiplier IP core simulation
6.2.2 add / subtractor
This module is relatively simple and can be implemented with simple code. We will not make a specific introduction here. The simulation results are as follows:
Figure 6-8 Addition simulation results
Figure 6-9 Subtraction simulation results