Chaotic Differential Equations Research Articles

Transforming chaotic and stiff systems to improve numerical accuracy

AbstractSystems of differential equations can exhibit chaotic or stiff behavior under specific conditions, posing challenges for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution for specified accuracy. In order to improve efficiency, the question arises whether transformation to asymptotically stable solutions can be performed, for which neighbouring solutions converge towards each other at a controlled rate. Employing the concept of local Lyapunov exponents, it is shown that chaotic differential equations can indeed be transformed to obtain high accuracy, whereas stiff equations cannot. For instance, the accuracy of explicit fourth order Runge–Kutta solution of the Lorenz chaotic equations can be increased by several orders of magnitude. Alternatively, the time step can be significantly extended with retained accuracy. The transform method applies broadly to chaotic systems, including weather prediction and turbulence.

3D Chaotic Mixing Application for Polymer Production

Abstract Industrial mixing processes have attracted research over the last decade because rapid and effective mixing of the materials in a mixture is crucial for the production of novel materials. The method of mixing directly affects the physical and chemical properties of the resulting mixture. Different mixing processes have been developed to create a range of polymers with specific properties for
applications such as tissue engineering, artificial muscles, and soft robots.
The mixing process of RTV 2710 silicone rubber and Catalyst CX catalyst was carried out using the Halvorsen, Newton Leipnik, Hadley, and Sprott A. systems. Numerical values were obtained by solving chaotic differential equations. The speed of the DC motor that actuated the mixing propeller changed chaotically. To save energy, the energy values were measured and the Hadley-2 signal was found to be the least energy-consuming chaotic signal. SCARA tracked trajectories for all chaotic systems, samples were obtained, and tensile tests were carried out. The highest average values of force and average tensile strength were obtained in the Sprott A. chaotic system. Finally, samples with constant position and Hadley 2 velocity, and Sprott A. position and Hadley 2 velocity, were produced and subjected to a tensile test. As a result, the highest average force and average tensile strength results were obtained from the Sprott A. position and the Hadley 2 velocity.

Chaos for Differential Equations with Multivalued Impulses

The deterministic chaos in the sense of a positive topological entropy is investigated for differential equations with multivalued impulses. Two definitions of topological entropy are examined for three classes of multivalued maps: [Formula: see text]-valued maps, [Formula: see text]-maps and admissible maps in the sense of Górniewicz. The principal tool for its lower estimates and, in particular, its positivity are the Ivanov-type inequalities in terms of the asymptotic Nielsen numbers. The obtained results are then applied to impulsive differential equations via the associated Poincaré translation operators along their trajectories. The main theorems for chaotic differential equations with multivalued impulses are formulated separately on compact subsets of Euclidean spaces and on tori. Several illustrative examples are supplied.

Simple estimation method for the second-largest Lyapunov exponent of chaotic differential equations

Abstract The largest Lyapunov exponent (LLE) is an important tool used to identify systems. However, it cannot easily distinguish between chaos and hyperchaos. Therefore, the second-largest Lyapunov exponent (SLLE) must be calculated. Some methods to calculate this index have already been proposed, but these require massive computation times and data points. Thus, to reduce the calculating complexity, we propose a simple novel method based on three nearby orbits to directly calculate the finite-time local SLLE from continuous chaotic systems with improved accuracy. First, we obtain a solution orbit of chaotic equations. Then, two selected points from the solution are used as the initial condition to solve the same equation and obtain two solution orbits. Next, we calculate the evolution area of three solution orbits and the distance between the two solution trajectories. The LLE and area exponent are then obtained from the logarithmic curve of the track distance and the area. Finally, the SLLE is obtained from the difference computed between the two indices. Some chaotic differential equations and non-chaotic systems are presented to demonstrate the efficiency of the proposed method. Calculating the finite-time local exponents does not require the calculation of all the evolution lengths. Thus, the proposed method is simple, very fast and robust without the need to reconstruct the phase space. For now, the findings of this research provide new ideas related to the SLLE calculation theory.

On the automatic parameter selection for permutation entropy.

Permutation Entropy (PE) is a cost effective tool for summarizing the complexity of a time series. It has been used in many applications including damage detection, disease forecasting, detection of dynamical changes, and financial volatility analysis. However, to successfully use PE, an accurate selection of two parameters is needed: the permutation dimension n and embedding delay τ. These parameters are often suggested by experts based on a heuristic or by a trial and error approach. Both of these methods can be time-consuming and lead to inaccurate results. In this work, we investigate multiple schemes for automatically selecting these parameters with only the corresponding time series as the input. Specifically, we develop a frequency-domain approach based on the least median of squares and the Fourier spectrum, as well as extend two existing methods: Permutation Auto-Mutual Information Function and Multi-scale Permutation Entropy (MPE) for determining τ. We then compare our methods as well as current methods in the literature for obtaining both τ and n against expert-suggested values in published works. We show that the success of any method in automatically generating the correct PE parameters depends on the category of the studied system. Specifically, for the delay parameter τ, we show that our frequency approach provides accurate suggestions for periodic systems, nonlinear difference equations, and electrocardiogram/electroencephalogram data, while the mutual information function computed using adaptive partitions provides the most accurate results for chaotic differential equations. For the permutation dimension n, both False Nearest Neighbors and MPE provide accurate values for n for most of the systems with a value of n=5 being suitable in most cases.

Provide a New Encryption Algorithm for Medical Images and Evaluate the Proposed Algorithm

Introduction: In this paper, an encryption algorithm for the security of medical images is presented, which has extraordinary security. Given that the confidentiality of patient data is one of the priorities of medical informatics, the algorithm can be used to store and send medical image.Material and Methods: In this paper, the solutions of chaotic differential equations are used to generate encryption keys. This method is more than other methods used in encoding medical images, resistant to statistics attacks, low encryption and decryption time and very high key space. In the proposed algorithm, unlike other methods that use random key generation, this method uses the production of solutions of the chaotic differential equations in a given time period for generating a key. All simulations and coding are done in MATLAB software.Results: Chaotic Differential Equations have two very important features that make it possible to encode medical images. One is the unpredictability of the system's behavior and the other is a severe sensitivity to the initial condition.Conclusion: These two features make the method resistant to possible attacks to decode the concept of synchronization chaotic systems. Using the results of the method, medical information can be made safer than existing ones.

Provide a New Encryption Algorithm for Medical Images and Evaluate the Proposed Algorithm

Introduction : In this paper, an encryption algorithm for the security of medical images is presented, which has extraordinary security. Given that the confidentiality of patient data is one of the priorities of medical informatics, the algorithm can be used to store and send medical image. Material and Methods : In this paper, the solutions of chaotic differential equations are used to generate encryption keys. This method is more than other methods used in encoding medical images, resistant to statistics attacks, low encryption and decryption time and very high key space. In the proposed algorithm, unlike other methods that use random key generation, this method uses the production of solutions of the chaotic differential equations in a given time period for generating a key. All simulations and coding are done in MATLAB software. Results : Chaotic Differential Equations have two very important features that make it possible to encode medical images. One is the unpredictability of the system's behavior and the other is a severe sensitivity to the initial condition. Conclusion : These two features make the method resistant to possible attacks to decode the concept of synchronization chaotic systems. Using the results of the method, medical information can be made safer than existing ones.

Riemann-Liouville Fractional Derivative and Application to Model Chaotic Differential Equations

Riemann-Liouville Fractional Derivative and Application to Model Chaotic Differential Equations

Computed chaos or numerical errors

Discrete numerical methods with finite time-steps represent a practical technique to solve initial-value problems involving nonlinear differential equations. These methods seem particularly useful to the study of chaos since no analytical chaotic solution is currently available. Using the well-known Lorenz equations as an example, it is demonstrated that numerically computed results and their associated statistical properties are time-step dependent. There are two reasons for this behavior. First, chaotic differential equations are unstable so that any small error is amplified exponentially near an unstable manifold. The more serious and lesser-known reason is that stable and unstable manifolds of singular points associated with differential equations can form virtual separatrices. The existence of a virtual separatrix presents the possibility of a computed trajectory actually “jumping” through it due to the finite time-steps of discrete numerical methods. Such behavior violates the uniqueness theory of differential equations and amplifies the numerical errors explosively. These reasons imply that, even if computed results are bounded, their independence on time-step should be established before accepting them as useful numerical approximations to the true solution of the differential equations. However, due to these exponential and explosive amplifications of numerical errors, no computed chaotic solutions of differential equations independent of integration-time step have been found. Thus, reports of computed non-periodic solutions of chaotic differential equations are simply consequences of unstably amplified truncation errors, and are not approximate solutions of the associated differential equations.

Comment on “Computational periodicity as observed in a simple system,” by Edward N. Lorenz (2006a)

Systems of ordinary differential equations that exhibit chaotic responses have yet to be correctly integrated. So far no ‘convergent’ computational results have been determined for chaotic differential equations. Various computed numbers are not solutions of the continuous differential equations; all chaotic responses are simply numerical noise and have nothing to do with the solutions of differential equations. It would be an exciting contribution if a convergent computed chaotic solution for a Lorenz model could be obtained.

Minimum description length neural networks for time series prediction.

Artificial neural networks (ANN) are typically composed of a large number of nonlinear functions (neurons) each with several linear and nonlinear parameters that are fitted to data through a computationally intensive training process. Longer training results in a closer fit to the data, but excessive training will lead to overfitting. We propose an alternative scheme that has previously been described for radial basis functions (RBF). We show that fundamental differences between ANN and RBF make application of this scheme to ANN nontrivial. Under this scheme, the training process is replaced by an optimal fitting routine, and overfitting is avoided by controlling the number of neurons in the network. We show that for time series modeling and prediction, this procedure leads to small models (few neurons) that mimic the underlying dynamics of the system well and do not overfit the data. We apply this algorithm to several computational and real systems including chaotic differential equations, the annual sunspot count, and experimental data obtained from a chaotic laser. Our experiments indicate that the structural differences between ANN and RBF make ANN particularly well suited to modeling chaotic time series data.

Estimation of chaotic ordinary differential equations by coevolutional genetic programming

AbstractThis paper describes a method of estimation of ordinary differential dynamics by Genetic Programming (GP) and discusses the accuracy of estimation. The dynamics is expressed by a tree structure and we obtain a numerical solution by carrying out digital integration (Runge‐Kutta‐Gill method) of the function system. The square error of the numerical solution and the original data is calculated and the inverse of the error is used as an evaluation of the individuals (functions) of the GP. Genetic operations such as crossover and mutation are carried out on pairs of solution candidates with high evaluation values, and the solutions (functions) are optimized. In addition, a coevolutional model that can efficiently perform simultaneous estimation of multiple functions is proposed.In the study of the estimation error, two systems, the Lorenz and Rössler systems, which are representatives of the chaotic ordinary differential model, are estimated and it is shown that a function of the same shape as the original one can be estimated in one‐function estimation, and that in multiple‐function (three‐function) estimation, the coevolutional model is effective. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 86(2): 1–12, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10057

Defect-controlled numerical methods and shadowing for chaotic differential equations

It is well-known that chaotic ODE's are, in the classical sense, unstable or ill-conditioned. It is not clear that variable step-size codes based on local error control can solve such problems in a useful way. In particular, it is usually difficult to show that the apparent chaos in the resulting solution is not a numerical artifact. I show here that a defect-controlled method gives useful solutions for chaotic problems. A pragmatically modified definition of what it means for a dynamical system to be chaotic is also presented.

Rigorous verification of trajectories for the computer simulation of dynamical systems

The authors present a new technique for constructing a computer-assisted proof of the reliability of a long computer-generated trajectory of a dynamical system. Auxiliary calculations made along the noise-corrupted computer trajectory determine whether there exists a true trajectory which follows the computed trajectory closely for long times. A major application is to verify trajectories of chaotic differential equations and discrete systems. They apply the main results to computer simulations of the Henon map and the forced damped pendulum.

Chaotic Dynamical Systems as Automata

We discuss the replacement of discrete maps by automata, algorithms for the transformation of finite length digit strings into other finite length digit strings, and then discuss what it required in order to replace chaotic phase flows that are generated by ordinary differential equations by automata without introducing unknown and uncontrollable errors. That question arises naturally in the discretization of chaotic differential equations for the purpose of computation. We discuss as examples an autonomous and a periodically driven system, and a possible connection with cellular automata is also discussed. Qualitatively, our considerations are equivalent to asking when can the solution of a chaotic set of equations be regarded as a machine, or a model of a machine.