Very small changes in the starting position of a chaotic system make a big difference after a while. Thus, the system can have an effective noninteger dimension. Finally, the important problems of classi cation and normalform of threedimensional quadratic autonomous chaotic systems are formulated and discussed. Chuas system has only one nonlinearity with one variable, whereas the lorenz. Chaotic system the chaotic system employed in this work is the lorenz system one of the earliest indications of chaotic behaviour was developed by edward n.
All your les and all work which you do will be in this folder. Bibliography chaotic flow in the lorenz model what is chaos in mathematical terms, a chaotic system is. Adaptive differential evolutionbased lorenz chaotic. Currently, we can mention some new chaotic systems reported in the literature 2433. The lorenz chaotic attractor was discovered by edward lorenz in 1963 when he was investigating a simplified model of atmospheric convection. It can crop up in unexpected places and in remarkably simple systems, and a great deal of work has been done to describe the behavior of chaotic systems. In particular, the lorenz attractor is a set of chaotic solutions of the lorenz system. The simulation demonstrates chaotic behavior of the numerical solution of the lorenz system of nonlinear ordinary differential equations. When the system is sufficiently simple, the process of formulating the equations may be fairly straightforward. Furthermore, we investigate the global exponential attractive set of this system via the lyapunov function method. Pdf on the dynamics of new 4d lorenztype chaos systems. Bounds for a new chaotic system are very important for the study of the qualitative behavior of a new chaotic system and chaos control. Sprott1, university of wisconsin, madison abstract. The chaotic systems usually are implemented by using electronic circuits in continuous cv and discretized versions dv.
Hence in nity is a xed point of the system, and ingredient 1. Chaos with matlab carlo f barenghi february 24, 2012 contents i matlab 3. Pdf the unified chaotic system describing the lorenz and. An introduction to chaos theory with the lorenz attractor. The lorenz system le temperature delle due superfici sono fissate assenza di flusso attraverso le 2 superfici d. Lorenz system is extended to a new class of generalized lorenz like systems in a canonical form. Using the above numerical solution for fractionalorder lorenz chaotic system 2. It is possible to control and predict the time series outputs of chaotic systems with known equations. Montoya and shujun li abstractthis paper describes how to determine the parameter values of the chaotic lorenz system used in a twochannel cryptosystem.
Bifurcations of the fractionalorder diffusionless lorenz system 3. In this paper, it is based on the application of control by sliding planes, to eliminate the chaotic behavior of the lorenz system and to take the trajectories of the phase space to a particular point, which was taken as the point of equilibrium. Analysis of a lorenzlike chaotic system by lyapunov functions. Additional strange attractors, corresponding to other equation sets that give rise to chaotic systems, have since been discovered. After the discovery of lorenz chaotic system 1, chaos theory, chaosbased applications, and new chaotic systems have been studied intensively 2. Lorenz system parameter determination and application to. This animation, created using matlab, illustrates two chaotic solutions to the lorenz system of odes.
Based on adaptive technique, the lag synchronization of lorenz chaotic system is achieved by designing a novel nonlinear controller. To start up matlab, go to a university computer cluster and login to your account. An attractor describes a state to which a dynamical system evolves after a long enough time. The lorenz chaotic system stimulated the interest of researchers to study chaotic systems and chaotic phenomena. Artificial neural networks model design of lorenz chaotic. In this paper, we investigate the ultimate bound set and positively invariant set of a 3d lorenzlike chaotic system, which is different from the wellknown lorenz system, rossler system, chen system, lu system, and even lorenz system family. The lorenz system 1 formulation 1 formulation the lorenz system was initially derived from a oberbeckboussinesq approximation. Strogatz abstruct a circuit implementation of the chaotic lorenz system is described. The bounds of the famous lorenz system have been studied by leonov et al. Dec 02, 2011 an introduction to chaos theory with the lorenz attractor. Chaotic attractors of the fractionalorder lorenz chaotic system 2. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems.
A new chaotic system and its analysis most researchers developed a new chaotic system depending on one chaotic system like lorenz or rossler systems the proposed scheme in this paper based on merging two chaotic systems lorenz chaotic system and rossler chaotic system. We will wrap up this series of examples with a look at the fascinating lorenz attractor. If, quite unluckily, i happened to know the laws of these phenomena, i could achieve this goal only at the price of inextricable computations, and should renounce to answer you. This paper presents a new threedimensional continuous autonomous chaotic system with ten terms and three quadratic nonlinearities. The unified chaotic system describing the lorenz and chua systems. Lorenz and rossler chaotic system for speech signal encryption. It is a nonlinear system of three differential equations. The basic dynamical properties of the new system are analyzed by means of equilibrium points, eigenvalue. The lorenz attractor is likely the most commonly used example of chaos theory.
Dynamics of a hyperchaotic lorenz system international. Jan 30, 2012 this animation, created using matlab, illustrates two chaotic solutions to the lorenz system of odes. The lorenz system is a system of ordinary differential equations the lorenz equations, note it is not lorentz first studied by the professor of mit edward norton lorenz 19172008 in 1963. Adaptive lag synchronization of lorenz chaotic system with. The lorenz attractor arises in a simplified system of equations describing.
The trajectories are shown to the left, and the x solutions are shown to the upper right as. After an exhaustive research on a new 4d lorenztype hyperchaotic system and a coupled dynamo chaotic system, we obtain the bounds of the new 4d lorenz. Chaos theory is a branch of mathematics focusing on the study of chaosstates of dynamical systems whose apparentlyrandom states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions. It is notable for having chaotic solutions for certain parameter values and initial conditions. We also have confirmed these by calculating the largest lyapunov exponent in some cases using the wolf algorithm 30. Mos realization of the modified lorenz chaotic system. Consequently, cryptosystem based on lorenz system has stronger. Complex dynamical behaviors of lorenzstenflo equations mdpi.
Fractionalorder diffusionless lorenz chaotic system in our simulations, we have visually inspected the bifurcation diagrams to identify chaos. Lorenz system parameter determination and application to break the security of twochannel chaotic cryptosystems a. In this paper, we investigate the lag synchronization of lorenz chaotic system with uncertain parameters. The main challenge for lorenz chaotic system based image encryption techniques is parameter sensitivity and resistance against attacks. The original problem was a 2d problem considering the thermal convection between two parallel horizontal plates.
Specifically, the fluid is heated uniformly from below and cooled. The functionality of the rungekutta method is also considered. Design and fpga implementation of lorenz chaotic system. Image encryption method based on discrete lorenz chaotic. This paper presents a new threedimensional autonomous chaotic system. New chaotic regimes in the lorenz and chen systems j.
A new chaotic behavior from lorenz and rossler systems. Chaos theory simple english wikipedia, the free encyclopedia. Therefore, in order to study the dynamic control of the brain neural networks, an ann architecture is designed and optimized for implementing lorenz attractor to simulate the chaotic states of eeg signals. In this paper, we investigate the dynamics of the lorenz system, linearly extended into one additional dimension. Homoclinic orbits, shilnikov theorem and melnikov function 5. Aes modification using slantlet transform and lorenz. Lorenz, is an example of a nonlinear dynamic system corresponding to the longterm behavior of the lorenz system. The lorenz equations x is proportional to the intensity of convection motion. Various threedimensional 3d chaotic systems have been found and reported in the literature 7. The rst is the logistic map, a rstorder discrete dynamical system, and the second is the lorenz system, a threedimensional system of di erential. Trajectories are repelled to in nity, and never return.
However, we should not consider this system to be chaotic. The parameters of the lorenz attractor were systematically altered using a fortran program to ascertain their effect on the behaviour of the chaotic system and the possible physical consequences of these changes was discussed. Investigate the behavior of a system that exhibits chaotic behavior. Special thanks to paul ohmann, marty johnston, and doug dokken for their support on this project. The system is hyperchaotic in a wide range of parameters. We choose the lorenz chaotic system, which is a continuous system, if you want to use it to encrypt the image, the system should be discrete. A new chaotic behavior from lorenz and rossler systems and. When the present determines the future, but the approximate present does not approximately determine the future. A new simple chaotic lorenztype system and its digital. Pdf a new chaotic behavior from lorenz and rossler. Lorenz equations system the lorenz system, named for edward n. To resolve these issues, a modified image encryption technique based on secure hash algorithm sha3 and adaptive differential evolution ade is proposed. Lyapunov exponents the lyapunov exponent of a dynamical system is one measure of how chaotic a system is.
If, for example, the system is an ordinary pendulum, whose motion, incidentally, is not chaotic, two independent investigators may well arrive at the same set of equations, particularly if they both disregard friction. Article pdf available in international journal of bifurcation and chaos 145. A very small change may make the system behave completely differently. The lorenz attractor chaotic butterflyeffect problem statement. Therefore will be added two chaotic systems in 1 and 2, a new system.
Systems that never reach this equilibrium, such as lorenz s butterfly wings, are known as strange attractors. While there are more conditions for a system to be considered chaotic, one of the primary indicators of achaoticsystemisextreme sensitivity to initial conditions. A rigorous proof of existence and uniqueness of solutions for the fractionalorder lorenz chaotic system has been provided, as well as its stability in contrast with the integerorder lorenz system. The example of chaotic system taken in this paper is the lorenz system lorenz chaotic attractor. Since then, many chaotic systems like lorenz and their chaotic behaviorhave been reported in the literature, for example, 1723. Java code for an animation of the waterwheel model of the lorenz system is included. In this paper, it is based on the application of control by sliding planes, to eliminate the chaotic behavior of the lorenz system and to take the trajectories of the phase space to a particular point, which was taken as the point of. Compared with one dimension chaotic map such as logistic map, tent map, and sine map, the lorenz system has more complicated dynamical property, and number of state variables. The lorenz attractor, a paradigm for chaos 5 you are asking me to predict future phenomena. Index termschaos, cryptography, cryptanalysis, nonlinear systems, security of data, lorenz system. This paper investigates the problem of designing feedback controllers for regulating the output of the simplified lorenz system, which is a novel chaotic system recently discovered by k. Dynamics of the lorenz equations vibrant clean energy. The lorenz systems describes the motion of a fluid between two layers at different temperature. Demonstrate methods for visualizing the evolution of a system.
Attractor and strange attractor, chaos, analysis of lorenz. Fur thermore, the parameters identification is realized simultaneously. This approximation is a coupling of the navierstokes equations with thermal convection. The new system contains five variational parameters and exhibits lorenz and rossler like attractors in numerical simulations. Pdf this article introduces a new chaotic system of threedimensional quadratic autonomous ordinary differential equations, which can display i two. Motivated by the above discussion, a novel control method for the fractionalorder lorenz chaotic system is investigated in this paper. Lorenz like chaotic system on a chip sachin jambovane, hoon suk rho, and jong wook hong materials research and education center, department of mechanical engineering, auburn university, auburn, al, usa abstract the paper reports the development of lorenz like chaotic system in an allfluidic, nanoliter scale, singlephase microfluidic platform. It describes a system very similar to clausewitzs trinity imagery, which has three attractors, but i find the lorenz system to be especially relevant to clausewitzs way of describing the variations in. Kinsy, member, ieee xdepartment of electronic engineering, heilongjiang university, harbin, china yadaptive and secure computing systems laboratory, boston university, boston, usa abstract chaotic systems such as lorenz functions have been proposed as. Analytical and numerical studies of the dynamic properties to generate chaos, for continuous version cv and discretized version dv, for the new chaotic system ncs were conducted. Pdf mos realization of the modified lorenz chaotic. Lorenz chaotic system chaos theory is used to study the behavior of dynamic system that is highly sensitive to initial conditions, which is also known as the butter. In popular media the butterfly effect stems from the realworld implications of the lorenz attractor, i. Thus, this new generalized lorenz canonical form contains as special cases the original lorenz system, the generalized lorenz system, and the chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible.
In this work, by combining the merits of model reference adaptive control, sliding mode, and fuzzy control, a novel control approach is proposed to address the stabilization and tracking problem of lorenz chaotic system without the chattering phenomenon in the face of structured and unstructured uncertainties. Pdf fuzzy sliding mode control of lorenz chaotic system. Adaptive and dynamic device authentication using lorenz chaotic systems lake bu y, hai chengx, michel a. The lorenz system is a 3dimensional dynamical system that. Bifurcation analysis of a new lorenzlike chaotic system. Bifurcations of fractionalorder diffusionless lorenz. It is well known that most of the 3d chaotic systems, such as the lorenz system 1, rossler system 16, chen system 17, or sprott systems cases b to s 18, have a countable number of equilibrium points. Synchronization of lorenz b ased chaotic circuits with applications to communications kevin m. In this letter we introduce a generalization of the lorenz dynamical system using fractional derivatives.
Small differences in initial conditions yield wildly diverging outcomes in a lorenz chaotic system. The lorenz system is discussed as a simple basic model of chaotic behavior. Lassoued and boubaker have done a literature survey on new chaotic and hyperchaotic systems 15. Lorenz and rossler chaotic system for speech signal. Control and synchronization of the fractionalorder lorenz. Adaptive and dynamic device authentication using lorenz. This paper formulates a new threedimensional chaotic system that originates from the lorenz system, which is different from the known lorenz system, rossler system, chen system, and includes lu. Strogatz abstruct a circuit implementation of the chaotic lorenz. Here we choose euler method to discrete the system, then we quantify the obtained sequences to get the two value sequence. Stabilizing the unstable equilibrium points of the fractionalorder. The lorenz system is a system of ordinary differential equations first studied by edward lorenz. The lorenz attractor was once thought to be the mathematically simplest autonomous dissipative chaotic flow, but it is now known that it is only one member of a very large family of such systems, many of which are even simpler.
The dynamics for a class of fractionalorder lorenz chaotic systems has been extensively investigated in this paper. The proposed system generates a chaotic attractor with the variation of two parameters. Therefore, lorenz chaotic system is desirable for encryption which. Dynamic analysis of a fractionalorder lorenz chaotic system. How to get the bounds of chaotic systems is one of very central problems in the theory of dynamical systems. Design and fpga implementation of lorenz chaotic system for. Lorenz chaotic system with three uncertain parameters. A novel hyperchaotic system with fractionalorder terms has been proposed in 14. Bifurcations of fractionalorder diffusionless lorenz system. This paper serves as an introduction to the analysis of chaotic systems, with techniques being developed by working through two famous examples.