System [14] etc.display chaotic behavior. A hyper chaos system is considered as a chaotic attractor having more than one positive Lyapunov exponents which gives the randomness and higher unpredictability of the corresponding system so the hyper chaos may be more useful in some fields such as communication, encryption etc. On the other hand the area which attracted much attention is chaos synchronization since the seminal work of Pecora and Carroll [12] recently synchronization of fractional-order chaotic systems starts to attract increasing attention due to its potential applications in secure communication and control processing. There are many types of synchronization for the fractional-order chaotic systems which are investigated, such as Cs [17], Gs [15], PHs [16], As [18], Ps [19, 20, 22] etc. Amongst all projective synchronization, which was first reported by Mainieri and Rehacek [19], is one of the most noticeable one because it can obtain faster communication with its proportional feature [21, 23]. In PS, the responses of the two systems synchronized up to a constant scaling factor. Recently, on account of linear separation Wang and He [26] introduced projective synchronization of the fractional-order chaotic systems Then GPS of the fractional-order chaotic systems was studied in [20, 24]. However, in the above studies, all the states of the drive and response systems synchronize up to an identical constant scaling factor. In [25], Chen et al. proposed a new hyper chaotic system through adding a nonlinear controller of the three- dimensional autonomous chaotic system. More recently, by stability theory of fractional- order systems and tracking control technique the function projective synchronization between fractional-or...
... middle of paper ...
...cal techniques which are numerically stable and can be used to both linear and nonlinear fractional differential equation [28]have been developed in the literature. For finding the numerical approximation more accurate and for reducing the computational cost here we choose the Caputo version and predictor-corrector algorithm for fractional differential equation [29],.According to the fractional predictor-corrector algorithm, we find that hyper chaos exist in new four-dimensional fractional order system. In the numerical results the system parameters are chosen as a = 10, b = 28, c = 8/3 and d = 1. Numeric results shows that when 0.9 1, the fractional- order system (2) always exhibits hyper chaotic behaviors. The two Lyapunov exponents τ_1=0.9,τ_2=0.08 are obtained when = 0.9 is chosen. If < 0.9 then in system (2) there is no hyper chaos exist. As shown
if he were on the ground like he claimed he was, the photos would have
The Scientific Context of the Word Chaos In a scientific context, the word chaos has a slightly different
“Chaos theory proves that unpredictability is built into our daily lives.”(Crichton 313). Ian Malcolm’s words resolve the book, Jurassic Park, in a very absolute way. Throughout the book, Malcolm, spoke about chaos theory and his self proclaimed “Malcolm Effect” to explain his reasoning in his predictions. Ian Malcolm had predicted the demise of Jurassic Park even before its opening, as well as its multiple problems and difficulties. Malcolm’s theory is evidenced countless times throughout the story of Jurassic Park; dinosaurs are breeding, dinosaurs are escaping, and systems fail.
Indirectly through out his novel, Camus compares people who rely too much on their logic and rationality, versus those who accept that our world is confusing and unpredictable. Similar to his thinking, in “Crickets, Bats, Cats and Chaos” Lewis Thomas suggests that chaos stimulates the brain and actually suggests that even crickets or cats have thoughts during chaotic or unpredictable situations. Even though I have always seen chaos as a total lack of order, a desperate situation in which an individual loses control, Thomas gave me a new concept for chaos. He says that it emerges when a system is altered by a small change or small uncertainty in its interior; chaos is then the
According to Edward Lorenz (founder of the Chaos Theory) chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. Chaos Theory deals with nonlinear things that are effectively impossible to predict or control, like turbulence, weather, the stock market, our brain states, and so on. (Lorenz 1995: 187)
I believe that the world is orderly and I base my assumption off of what I know from bible, astronomy, biology, and physics. I do not think the universe is chaotic, I think the world functions uniformly starting from an atomic level.
Henry Adams states “chaos was the law of nature; order was the dream of man”. This statement is clearly used in the novel Crossing to Safety by Larry Morgan, and how this was his belief. However, the novel of Crossing to Safety is filled with chaotic times that happen in Larry Morgan and his wife’s life. This same chaos also affects his close friends who had their own struggles in their lives. This paper will look at how order would not have better for these couples, and how this chaos caused the couples to lose their dream of order.
Dr. Malcom uses the chaos theory to predict the collapse of the park. In the beginning of the book Malcom is introduced to the story. One of the first things Dr. Malcom talks about is the chaos theory. In the first conversation with him in the book Malcom said "Theory tells me that the island will quickly proceed to behave in unpredictable fashion"(Crichton 76). What Malcom means by this is that without even visiting the park he can say that it will fail. Dr. Malcom is a problem in the story because convinces other people that Jurassic Park is a bad idea.
In Heart of Darkness, by Joseph Conrad, the strongest conflict is an internal conflict that is most prominently shown in Marlow and Kurtz. This conflict is the struggle between their image of themselves as civilized human beings and the ease of abandoning their morality once they leave society. This inability has a close resemblance to the chaos theory. This is shown through the contrast of Kurtz as told by others and the actuality of him and through the progression of Marlow's character throughout Heart of Darkness.
Looking at the world where we live everything in someway is connected. Our world is not simple and in fact consists of multiple complex systems. Some everyday examples of complex systems are the brain, immune system, insect colonies, and even social networks such as Facebook and twitter. So what exactly do all these have in common in order to be a complex system? First is the fact that each one has a large amount of simple components that work together by communication through signals without being under leadership. But not all systems are exactly the same so we can break it down further into chaotic systems, complex adaptive systems, and nonlinear systems. Chaotic systems differ in that they are non-linear and are sensitive to initial conditions. Therefore any uncertainty in the system will not produce an outcome that can be predicted later on. A good example of a chaotic system would be the stock market because the prediction of its outcome is unknown due to its sensitivity to initial conditions. Complex adaptive systems are just like they sound. They are capable of adapting to the environment such as the immune system. It’s white blood cells work together to recognize foreign bodies and create antibodies for future encounters.
The dynamic systems view was developed by Arnold Gesell in 1934 and explores how humans develop their motor skills. From Mr. Gesell’s observations, he was able to conclude that children develop their motor skills in a specific order and time frame. He concluded that children roll, walk, sit, and stand as a result of several factors – the ability to move, the environmental support to move and the motivation/goal to move. Once the child has the motivation, ability, and support, they accept the new challenge. After several failed and successful attempts, they begin to fine-tune and master the movement with continued support and motivation. The dynamic systems theory is not a random process that children experience, the skills are calculated and develop over a period of time.
Systems approach is based on the fundamental principle that all aspects of a human problem should be treated together in a rational manner (Healy, 2005). I have divided this essay into relevant sections that cover an overview of systems ideas, general systems theory and ecological systems theory. This assignment will also include Germain and Gittermans life model, and it will be related back to the case study that has been provided. Limitations of systems theory will also be discussed.
Abstract: In this paper, I will attempt to explain the nature of Fractals. Both natural and computer generated fractals will be explained. At the end, I hope the reader has a rudimentary sense of fractals in terms of both art and geometry.
But sometimes, such a linearization may lead to real errors not only of a quantitative but also of a qualitative nature. And when the linearization is not possible, the original nonlinear equation itself must be treated. With the discovery of numerous phenomena of self-excitation of circuits containing nonlinear capacitor of electricity, like electron tubes, gaseous discharge etc. and in many cases of nonlinear mechanical vibrations of special types, the method of small oscillations becomes inadequate for their analytical treatment[*]. To this end, a number of methods are developed, such as straight forward expansion method, Lindtsteadt-Poincare (LP) method, Modified LP method, Struble technique, Harmonic balance method, the Krylov-Bogoliubov-Mitropolskii (KBM) perturbation method
Chaos theory has numerous application including helping explain phenomena or helping to predict the future. Chaos theory is applicable in various fields ranging from weather, business to medicine. Chaos theory explains the reason why it is practically improbable to predict the weather with the current technology as well as providing a way for people to find patterns in the chaotic system of stock exchange. It also helps with the running of organisation by showing what sort of condition is needed for a profitable business as well as helping doctors predict when heart failure may occur. Fractals which is a concept of chaos theory also is portrayed in the natural world in examples such as lightning and neurons in the brains. Chaos theory has