Multimedia Image Encryption Analysis Based on High-Dimensional Chaos Algorithm

With the development of network and multimedia technology, multimedia communication has attracted the attention of researchers. Image encryption has become an urgent need for secure multimedia communication. Compared with the traditional encryption system, encryption algorithms based on chaos are easier to implement, which makes them more suitable for large-scale data encryption. The calculation method of image encryption proposed in this paper is a combination of high-dimensional chaotic systems. This algorithm is mainly used for graph mapping and used the Lorenz system to expand and replace them one by one. Studies have shown that this calculation method causes mixed pixel values, good diffusion performance, and strong key performance with strong resistance. The pixel of the encrypted picture is distributed relatively random, and the characteristics of similar loudness are not relevant. It is proved through experiments that the above calculation methods have strong safety performance.

Dynamic Mathematical Models’ System and Synchronization

We created the equilibrium, which includes sickness outcomes, health and risk behaviors, environmental factors, and health-related assets and delivery systems, and it should be incorporated in system Dyc (dynamic) modelling of chronic disease prevention. System Dyc has the ability to model a variety of interconnected illnesses and dangers, as well as the interaction between delivery systems and afflicted people, as well as state and national policies. This paper proposes a unique idea. Hybrid synchronization utilizes four positive LYP (Lyapunov) exponents based on state feedback management with two identical systems of the Lorenz system 6D HYCH system.

Stochastic Chaos and Markov Blankets

In this treatment of random dynamical systems, we consider the existence—and identification—of conditional independencies at nonequilibrium steady-state. These independencies underwrite a particular partition of states, in which internal states are statistically secluded from external states by blanket states. The existence of such partitions has interesting implications for the information geometry of internal states. In brief, this geometry can be read as a physics of sentience, where internal states look as if they are inferring external states. However, the existence of such partitions—and the functional form of the underlying densities—have yet to be established. Here, using the Lorenz system as the basis of stochastic chaos, we leverage the Helmholtz decomposition—and polynomial expansions—to parameterise the steady-state density in terms of surprisal or self-information. We then show how Markov blankets can be identified—using the accompanying Hessian—to characterise the coupling between internal and external states in terms of a generalised synchrony or synchronisation of chaos. We conclude by suggesting that this kind of synchronisation may provide a mathematical basis for an elemental form of (autonomous or active) sentience in biology.

Using the (3N)-dimensional generalized Lorenz systems as a testbed for data assimilation: The ensemble Kalman filter

The feasibility of using a (3N)-dimensional generalization of the Lorenz system in testing a traditional implementation of the ensemble Kalman filter is explored through numerical experiments. The generalization extends the Lorenz system, known as the Lorenz ’63 model, into a (3N)-dimensional nonlinear system for any positive integer N. Because the extension involves inclusion of additional wavenumber modes, raising the dimension allows the system to resolve smaller-scale motions, a unique characteristic of the present generalization that can be relevant to real modeling scenarios. Model imperfections are simulated by assuming a high-dimensional generalized Lorenz system as the true system and a generalized system of dimension less than or equal to the dimension of the true system as the model system. Different scenarios relevant to data assimilation practices are simulated by varying the dimensional-differences between the model and true systems, ensemble size, and observation frequency and accuracy. It is suggested that the present generalization of the Lorenz system is an interesting and flexible tool for evaluating the effectiveness of data assimilation methods and a meaningful addition to the portfolio of testbed systems that includes the Lorenz ’63 and ’96 models, especially considering its relationship with the Lorenz ’63 model. The results presented in this study can serve as useful benchmarks for testing other data assimilation methods besides the ensemble Kalman filter.

A New Nonlinear Ninth-Order Root-Finding Method with Error Analysis and Basins of Attraction

Nonlinear phenomena occur in various fields of science, business, and engineering. Research in the area of computational science is constantly growing, with the development of new numerical schemes or with the modification of existing ones. However, such numerical schemes, objectively need to be computationally inexpensive with a higher order of convergence. Taking into account these demanding features, this article attempted to develop a new three-step numerical scheme to solve nonlinear scalar and vector equations. The scheme was shown to have ninth order convergence and requires six function evaluations per iteration. The efficiency index is approximately 1.4422, which is higher than the Newton’s scheme and several other known optimal schemes. Its dependence on the initial estimates was studied by using real multidimensional dynamical schemes, showing its stable behavior when tested upon some nonlinear models. Based on absolute errors, the number of iterations, the number of function evaluations, preassigned tolerance, convergence speed, and CPU time (sec), comparisons with well-known optimal schemes available in the literature showed a better performance of the proposed scheme. Practical models under consideration include open-channel flow in civil engineering, Planck’s radiation law in physics, the van der Waals equation in chemistry, and the steady-state of the Lorenz system in meteorology.

Parameter Identification of Lorenz System with Incomplete Information: Case of One Unknown Function

Inverse problem of the Lorenz system parametric identification is considered in the case of incomplete information about solutions of the system. In the present paper, it is assumed that only two solutions of the system from three are known in different combinations. The problem of the parameter identification of the system is solved by means of elimination of unknown functions from the original system. The obtained system of equations has the same order as the original one, but contains the unknown original parameters in new combinations. Sometimes, the number of new unknown parameters is higher than number of the original unknowns. In this case, the method of the constrained least squares minimization is used in the special formulation, developed by the authors. This novel formulation exploits linearity of the system with respect to the new unknown parameters, by means of which the number of nonlinear equations becomes equal to the number of the constraints between the new parameters. Two methods of the constraint minimization are considered: the classical method of Lagrange’s multipliers and a novel method of the auxiliary parameters. Numerical simulations demonstrate effectiveness of the algorithms.

Chaos in the Periodically Parametrically Excited Lorenz System

We demonstrate in this paper a new chaotic behavior in the Lorenz system with periodically excited parameters. We focus on the parameters with which the Lorenz system has only two asymptotically stable equilibrium points, a saddle and no chaotic dynamics. A new mechanism of generating chaos in the periodically excited Lorenz system is demonstrated by showing that some trajectories can visit different attractor basins due to the periodic variations of the attractor basins of the time-varying stable equilibrium points when a parameter of the Lorenz system is varying periodically.

Weighted recurrence network for characterizing continuous dynamical systems

Recurrence network (RN) is a powerful tool for the analysis of complex dynamical systems. It integrates complex network theory with the idea of recurrence of a trajectory, i.e. whether two state vectors are close neighbors in a phase space. However, the differences in proximity between connected state vectors are not considered in the RN construction. Here, we propose a weighted state vector recurrence network method which assigns weights to network links based on the proximity of the two connected state vectors. On the basis, we further propose a weighted data segment recurrence network that takes continuous data segments as nodes for the analysis of noisy time series. The feasibility of the proposed methods is illustrated based on the Lorenz system. Finally, an application to five types of EEG recordings is conducted to demonstrate the potentials of the proposed methods in the study of real-world data.