New universal aspects of diffusion in strongly chaotic systems

A minimum principle for chaotic dynamical systems

The classical fourth-order Runge-Katla integrator and the third-order force gradient symplectic integrator are used to solve the two-dimensional H'enon-Heiles system respectively. Numerical results, including the relative energy error, Poincare section, the largest Lyapunov exponent and Fast Lyapunov Indicator, are compared in detail. It is found that the Runge-Katla algorithm does not conserve the energy of the system, but the symplectic one does. On the other hand, the former method gives some spurious descriptions of the dynamics, while the latter one does not.

Nonhyperbolicity, as characterized by the coexistence of Kolmogorov-Arnold-Moser (KAM) tori and chaos in the phase space, is generic in classical Hamiltonian systems. An open but fundamental question in physics concerns the relativistic quantum manifestations of nonhyperbolic dynamics. We choose the mushroom billiard that has been mathematically proven to be nonhyperbolic, and study the resonant tunneling dynamics of a massless Dirac fermion. We find that the tunneling rate as a function of the energy exhibits a striking ``clustering'' phenomenon, where the majority of the values of the rate concentrate on a narrow region, as a result of the chaos component in the classical phase space. Relatively few values of the tunneling rate, however, spread outside the clustering region due to the integrable component. Resonant tunneling of electrons in nonhyperbolic chaotic graphene systems exhibits a similar behavior. To understand these numerical results, we develop a theoretical framework by combining analytic solutions of the Dirac equation in certain integrable domains and physical intuitions gained from current understanding of the quantum manifestations of chaos. In particular, we employ a theoretical formalism based on the concept of self-energies to calculate the tunneling rate and analytically solve the Dirac equation in one dimension as well as in two dimensions for a circular-ring-type of tunneling systems exhibiting integrable dynamics in the classical limit. Because relatively few and distinct classical periodic orbits are present in the integrable component, the corresponding relativistic quantum states can have drastically different behaviors, leading to a wide spread in the values of the tunneling rate in the energy-rate plane. In contrast, the chaotic component has embedded within itself an infinite number of unstable periodic orbits, which provide far more quantum states for tunneling. Due to the nature of chaos, these states are characteristically similar, leading to clustering of the values of the tunneling rate in a narrow band. The appealing characteristic of our work is a demonstration and physical understanding of the ``mixed'' role played by chaos and regular dynamics in shaping relativistic quantum tunneling dynamics.

Designing and analyzing new chaotic systems with desired properties has gotten much interest recently. Through applying modification in a recent rare chaotic flow which has adjustable Kaplan–Yorke dimension, a flexible chaotic system is introduced in this work. In the mentioned modification, two parameters are added which can control amplitude of the attractor and largest Lyapunov exponent. Statistical and dynamical properties of such system could be discovered by applying stability analysis, estimating Lyapunov exponents, local Kaplan–Yorke dimension, and plots of state space. Interestingly, this system is in group of systems which have coexisting attractors, so it is a multistable chaotic system. By considering a practical application of this rare system, meaningful relation between the value of local Kaplan–Yorke dimension and the ability of the system as a random number generator can be seen in statistical tests.

We investigate the optimization performance of multi-objective chaotic evolution (MOCE) algorithm with implementations using different chaotic systems. A comparison experiment of MOCE algorithms with four chaotic systems are employed in MOCE to analyse whether chaotic systems will affect the optimization performance of the MOCE algorithms. We analyze and discuss the performance of the MOCE algorithms implemented using different chaotic systems. Four chaotic systems are introduced in this work, i.e., the logistic map, the Hénon map, the tent map, and the Gauss map, respectively. The number of Pareto solution and the diversity of Pareto solution are two evaluation metrics to evaluate the performance of the multi-objective optimization algorithm. We apply the statistical tests to analyse and investigate the number of Pareto solution and their diversity. The evaluation results indicate that the MOCE with the logistic map has the best optimization performance in both the number of Pareto solution and their diversity. The statistical significance demonstrates that chaotic systems have a great influence on the optimization performance of MOCE algorithms.

In this research article, we have proposed a novel image encryption scheme for the confidentiality of digital information. The modern block ciphers based on confusion and diffusion characteristic, as proposed by Claude Shannon in 1949. Firstly, we have designed a nonlinear confusion component of a block cipher and apply the action of symmetry group S8 to generate a pool of 40,320 substitution boxes with the same cryptographic strength. These nonlinear components are responsible for adding confusion in the encryption algorithm. Secondly, we have utilized a nonlinear chaotic dynamical system to add diffusion capability in our proposed encryption scheme. The suggested scheme is further examined under security performance evaluations, which shows the appropriateness of our offered scheme for digital contents.

Periodicity in trajectories of chaotic systems in phase space

This article is devoted to forecasting the dynamics of processes occurring in the stock market, the methods of chaotic dynamics. Chaotic dynamics is one of the areas of synergetics. Although there are many effective methods for modeling securities price forecasts, emerging stock markets are characterized by long-term stability and short-term instability. In this situation, to study the processes occurring in the markets, chaos theory can be used, which includes methods of theory of dynamical systems and fractal analysis. An example of practical implementation of this approach using neural networks for the share price of the company "Pharmacy 36 and 6" at the close of trading for 2014–2019. The company "Pharmacy 36 and 6" is a retail network of pharmacies "Lekpharm", which is actively working and developing in the pharmaceutical market of Ukraine since 2001. The initial data were obtained on the website of the holding "FINAM". Neural networks have the ability to learn. At the same time, in the process of learning there is a phenomenon similar to self-organization. This property is characteristic of chaotic systems. The most important characteristics of chaos-dynamic analysis are calculated: Hausdorff dimension and Kolmogorov entropy. Their values indicate that the studied statistical series contains a chaotic component. The algorithm of technical application of neural networks is offered in the article. The model is built using neural network technology. Some of the input parameters of the network are determined based on the analysis performed. Comparison of the results obtained on the model deviate slightly from the actual data. It could be considered. that the model is adequate and can be used in practice. Forecasting the dynamics of securities can give a certain economic effect, a specific financial benefit to the owners of securities. Synergistic methods do not always lead to a solution to a problem or task. But they lead to a deeper and more accurate understanding of reality, which allows one to give an explanation for incomprehensible phenomena, such as pre-construction stability and short-term instability. Therefore, the synergistic economic paradigm has an interest on the part of researchers and policy makers.

Duality analysis of time series and complex networks has been a frontier topic during the last several decades. According to some recent approaches in this direction, the intrinsic dynamics of typical nonlinear systems can be better characterized by considering the related nonlinear time series from the perspective of networks science. In this paper, the associated network family of the unified piecewise-linear (PWL) chaotic family, which can bridge the gap of the PWL chaotic Lorenz system and the PWL chaotic Chen system, was firstly constructed and analyzed. We constructed the associated network family via the original and the modified frequency-degree mapping strategy, as well as the classical visibility graph and horizontal visibility graph strategy, after removing the transient states. Typical related network characteristics, including the network fractal dimension, of the associated network family, are computed with changes of single key parameter α. These characteristic vectors of the network are also compared with the largest Lyapunov exponent (LLE) vector of the related original dynamical system. It can be found that, some network characteristics are highly correlated with LLE vector of the original nonlinear system, i.e., there is an internal consistency between the largest Lyapunov exponents, some typical associated network characteristics, and the related network fractal dimension index. Numerical results show that the modified frequency-degree mapping strategy can demonstrate highest correlation, which means it can behave better to capture the intrinsic characteristics of the unified PWL chaotic family.

Correlation dimension (CD) and the largest Lyapunov exponent (LLE), which are two most important nonlinear invariant measures of nonlinear system, are adopted to characterize the complexity and stability of human brain during all sleep stages. In order to find an optimal embedding to reconstruct phase space, the C-C method and Cao's algorithm were thoroughly analyzed and implemented in efficient ways. Additionally, the G-P algorithm and Kantz's algorithm are also implemented for calculation of CD and LLE respectively. Eight normal sleep EEG records are acquired from MIT-BIH sleep- EDF database for this study. The values of CD and LLE for 30s epochs are calculated, and their mean and standard deviation values are evaluated as well. The results indicate that CD and LLE decrease from the waking stage to stages I, II and III-IV, and they have large values during rapid eye movement (REM) sleep stage. Then, a qualitative conclusion may be drawn that during waking stage and REM sleep stage, brain activity has higher complexity and lower stability than during other stages.

Motor disorders in Parkinson's Disease (PD) show high inter-individual variability which challenges the current observational-based strategies in the clinical setting to determine the actual disease evolution and monitoring the therapy response. In spite the recent development of the motion capture technology, it is still hardly transferable to the routine exam and the non-linear disease patterns are poorly explored. Because gait pattern could be approached as deterministic chaotic system, this work aimed to non-linearly represent lower limb kinematic standing out the differences among PD stages. For doing so, a widely used deep learning framework was implemented for obtaining the body landmarks and their temporal series and thereafter, construing the phase space based on the first order derivatives. Largest Lyapunov exponent, correlation dimension and approximate entropy were then computed resulting in statistically significant differences (Wilcoxon rank test, p < 0.05), particularly between healthy controls and stages 3, the most advanced stage, and comparing stage 1 face to stage 3. These finding providing insights how the complex patterns may be related with the disease progression in PD and easily implemented using affordable video devices.

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos

In this work, a secure second-order discrete time delta sigma (ΔΣM) modulator is used to securely modulate two physiological signals for e-health applications. The chaotic system used is a new 3D chaotic system that is used to generate random keys for encryption. Quintic nonlinearity is integrated into the existing sprott chaotic system and then studied via nonlinear dynamics tools, which reveal complex dynamics, including the coexistence of three distinct chaotic attractors and the phenomenon of intermittency. A microcontroller implementation of the system is performed to demonstrate its real-life applicability. The bits generated from the chaotic regime of the proposed oscillator are tested successfully via the NIST-800.22 test. The pseudo random bit streams generated from the chaoticity of the oscillator using the system parameters and initial conditions as secret keys are used in a 2-stage confusion process to encrypt two physiological signals. At the receiving end, the physiological signals are successfully recovered, and some standard metrics, such as key space analysis, key sensitivity, and cross-correlation, are applied to verify the strength of the encryption scheme. A comparison with recent and good standing contributions in the literature is also provide at the end of this work, which shows that the proposed scheme is robust against common attacks. The results obtained in this study open the possibility of applying chaos in digital communication systems for the conception of secure e-health services.p. 2Highlights• A new random chaotic system applicable in information protection is conceived and experimented.• Random encryption data are successfully generated from the chaotic system in its confused state and tested statistically. • Patient's ECG signal is securely transferred via a secure and simple communication scheme.

At present, most of the manufacturers are increasingly laying emphasis on dual-channel selling. The level of consumer’s preference is diverse with different channels. This study discusses a supply chain where one common manufacturer uses two different channels to sell product. The bounded rationality expectation has been applied to explore the decision-making mechanism in finalizing the wholesale prices and sales commissions for the manufacturer and retailers. In addition, the dynamic features of the system are simulated by 2D bifurcation diagram, the largest Lyapunov exponent, attractor variation, and time series. The simulation results suggest that if the adjustment speeds of the wholesale prices and sales commissions change drastically, the system would fall into a chaotic state. If the consumers prefer the online channel, the wholesale prices and sales commissions may be raised and vice versa. However, too strong preference of each channel will make the system appear to have a considerably periodic fluctuation, even chaos. In the meantime, the parameter adjustment method can help make the periodic or chaotic system go back to stability. Therefore, the significance of this study is with a practical meaning to make better pricing strategy for players in dual-channel selling supply chain system.

This paper discusses the design process toward new lumped chaotic systems that originates in higher-order ordinary differential equations commonly used as description of ideal oscillators. In investigated third-order case, two chaotic oscillators were constructed. These systems are dual in the sense of vector field geometry local to fixed points. The existence of robust chaos was proved by both standard routines of numerical analysis and practical measurement. For the fourth-order oscillatory equation, the concept based on interaction between superinductor and supercapacitor was examined in detail. Since both “superelements” are active, the nonlinearity essential to the evolution of chaos is fully passive. It is demonstrated that complex motion is robust and does not represent long transient behavior or numerical artefact. The existence of chaos was verified using standard quantifiers of the flow, such as the largest Lyapunov exponents, recurrence plots, approximate entropy and sensitivity calculation. A good final agreement between theoretical assumptions and practical results will be concluded, on a visual comparison basis.