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(a) Bifurcation diagram and (b) largest Lyapunov exponent of Case IE 10 with respect to changing parameter b. Initial conditions are set with backward continuation, and the first initial conditions are (17.82, −61.74, −82.95).
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In this paper, some new three-dimensional chaotic systems are proposed. The special property of these autonomous systems is their identical eigenvalues. The systems are designed based on the general form of quadratic jerk systems with 10 terms, and some systems with stable equilibria. Using a systematic computer search, 12 simple chaotic systems wi...
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... are shown in Fig. 1. It can be seen that the topologies of attractors are very different. In the interest of brevity, we focus on System IE 10 . Case 10 is one of the interesting systems with three equal nonzero eigenvalues and its bifurcation diagram is interesting. A bifurcation diagram and plot of the largest Lyapunov exponent for IE 10 in Fig. 2 Table 1. Lyapunov exponents of the system are calculated using Wolf's method [Wolf et al., 1985] with the runtime 20 000. The figure shows that the dynamic of the system is very sensitive to variations of parameter b in its smaller values. Kaplan-Yorke dimension (D KY ) is a complex degree and conjectures the dimension of strange ...
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Autonomous driving has seen rapid growth with advancements in Deep Learning. Various types of research have been done that focused on distinct machine-learning approaches for autonomous driving. In this paper, we are going to discuss two paradigms used in autonomous driving known as Modular systems and End-to-end systems. We are going to define the...
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... Additionally, there has been a continuous effort towards developing new chaotic models and circuit realizations of various chaotic generators [23][24][25][26][27]. In this context, there is a substantial amount of work presented in the literature on the implementation of chaotic circuits suitable for random bit generation. ...
... The fractional derivative provides a physical memory. 22,23 Based on these applications, fractional derivatives are also used to study the chaotic systems, which can be seen in Refs. 24-28. ...
The use of fractal-fractional derivatives has attracted considerable interest in the analysis of chaotic and nonlinear systems, as they provide a unique capability to represent complex dynamics that cannot be fully described by integer-order derivatives. The Fractal-Fractional derivative with a power law kernel is used in this paper as an analytical tool to analyze the dynamics of a chaotic integrated circuit. Using coupled ordinary differential equations of classical order, the complexity of an integrated circuit is modeled. The classical order model is generalized via fractal-fractional derivatives of power law kernel. Moreover, the paper is concerned with investigating the Ulam Stability of the model and conducting theoretical studies in order to analyze equilibrium points, identify unique solutions, and verify the existence of such solutions. By examining the complex dynamics that result in chaotic behavior, these investigations shed light on the fundamental properties of integrated circuits. For the purpose of exploring the non-linear fractal-fractional order system, a numerical algorithm has been developed to facilitate our analysis. MATLAB software has been used to implement this algorithm, making it possible to carry out detailed simulations. Simulating solutions are accomplished using 2D and 3D portraits, which provide visual and graphical representations of the results. Throughout the simulation phase, particular attention is given to the impact of fractional order parameter and fractal dimension. As a result of this study, we have gained a comprehensive understanding of the behavior of the system and its response to variations in values.
... Lassoued and Boubaker [9] provides a general survey about newly designed chaotic and hyperchaotic systems, this involves an analysis of the dynamical behaviors of these complex systems Salih et al. [10] designed a new approach by using a 3D Logistic map as a proposed key through the replacement of the predetermined XOR operation in advanced encryption standard (AES) algorithms. Faghani et al. [11] designed a new 3D chaotic system depending on the basic formula of quadratic Jerk systems having terms, and other systems having a stable node equilibrium point. Patra and Banerjee [12] prove that 3D systems have attractors such as higher [13] try to improve the Rivest-Shamir-Adleman (RSA) algorithms by applying a 3D chaotic system with an experiment characteristic designed for this purpose. ...
In this paper, a description, and analysis of a novel 3-D dimension hyperchaotic system is implemented. The proposed system oscillation is two-order autonomous and consisted of a nine-term and symmetric oscillation w.r.t x-axis. It is proved analysis by Kaplan-York dimension, waveform analysis, phase portrait, and Lyapunov exponent. This work-study stability and equilibrium point and Routh stability criteria produced that the new system has one unstable point from the type saddle-focus point. One of the characteristics of the proposed system is hyperchaotic since this system has two Lyapunov large than zero. This system is applied to generate a chaotic matrix 16 * 16 (S-box) based in advanced encryption standard (AES) algorithm for text encryption and gives a high level of security. In addition to the description, and analysis S-box. Therefore. the proposed algorithm is satisfied the high randomness of entropy value and passes the National Institute of Standards and Technology (NIST) parameters and another test. Mathematica and MATLAB programs simulated some results. Keywords: 3-D hyperchaotic system AES algorithm Fractional-Kaplien dimension Lyapunov exponent SDIC Waveform analysis This is an open access article under the CC BY-SA license.
... While analog circuit realizations of several chaotic systems with nonhyperbolic equilibria have been presented in Panahi et al., 2018;Kamdem Kuate et al., 2019;Kengne et al., 2020], digital microcontroller-based implementations [Kamdem Kuate et al., 2019] and DSPbased implementations [Yang et al., 2020] have also been reported. It is worth noting that a class of chaotic systems with nonhyperbolic equilibrium and identical eigenvalues was recently reported in [Faghani et al., 2020]. Digital realizations of chaotic systems can be performed in either floating-point or fixed-point representations, where the difference is shown in Fig. 1 for 32 bits. ...
We introduce a new chaotic system with nonhyperbolic equilibrium and study its sensitivity to different numerical integration techniques prior to implementing it on an FPGA. We show that the discretization method used in numerically integrating the set of differential equations in MATLAB and Mathematica does not yield chaotic behavior except when a low accuracy Euler method is used. More accurate higher-order numerical algorithms (such as midpoint and fourth-order Runge–Kutta) result in divergence in both MATLAB and Mathematica (but not Python), which agrees with the divergence observed in an analog circuit implementation of the system. However, a fixed-point digital FPGA implementation confirms the chaotic behavior of the system using Euler and fourth-order Runge–Kutta realizations. Therefore, the increased sensitivity of chaotic systems with nonhyperbolic equilibrium should be carefully considered for reproducibility.
... A three-dimensional continuous autonomous chaotic system is proposed in literature [19], and its state equation is ...
... After calculation and analysis, the three Lyapunov exponents of the system when g = 20, h = 5, c = 10, and d = 2 are LE 1 = 2:051, LE 2 = −0:017, and LE 3 = −17:406. Compared with the chaotic system proposed in literature [19], it is obvious that the three Lyapunov exponents of the chaotic system in this study are much larger than those in reference [19]. Therefore, the chaotic system in this study has more complex dynamic characteristics than the chaotic system in literature [19] and is more suitable for the research of big data encryption than the chaotic system in literature [19]. ...
... After calculation and analysis, the three Lyapunov exponents of the system when g = 20, h = 5, c = 10, and d = 2 are LE 1 = 2:051, LE 2 = −0:017, and LE 3 = −17:406. Compared with the chaotic system proposed in literature [19], it is obvious that the three Lyapunov exponents of the chaotic system in this study are much larger than those in reference [19]. Therefore, the chaotic system in this study has more complex dynamic characteristics than the chaotic system in literature [19] and is more suitable for the research of big data encryption than the chaotic system in literature [19]. ...
Powered by the rapid development of the information age, big data technology has had a great impact on China’s socialist economic development. Big data technology is integrated into all fields of nowadays society, especially in the accounting industry of enterprises. Accounting computerization quickly replaces the traditional manual bookkeeping, realizes the timeliness and accuracy of accounting information data processing, and improves the work efficiency and quality of accounting staff to a great extent, but at the same time, the security of financial accounting information is also very prominent. To tackle this problem, this paper proposes a hybrid encryption algorithm based on double chaotic system and improved AES encryption algorithm. The improved AES algorithm uses affine transformation pairs (A7 and 6F) to generate new S-boxes. The double four-dimensional hyperchaotic system is transformed from two three-dimensional chaotic systems, and then, the transformed hyperchaotic system is used to generate chaotic sequences, and a block encryption scheme is designed. On Hadoop big data platform, double hyperchaotic encryption scheme and improved AES algorithm are combined. Simulation test results show that this method can safely transmit enterprise financial accounting information data packets. With the increase of computing cluster nodes, its encryption transmission efficiency continues to improve. This scheme not only solves the problem of enterprise financial accounting data security encryption but also realizes the parallel transmission of encrypted data in the information age, forming a double guarantee of data transmission security and efficiency.
... e stability of this equilibrium point cannot be determined under the Shilnikov criteria. So, the stability of this case is investigated numerically by tracking the final state of the system with initial values around the equilibrium point (Faghani et al. [53]). ...
In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z-axis. Numerical analysis of the system reveals many strong dynamics. The new system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system.
... This paper deals with certain new fractional-order three-dimensional chaotic systems. These autonomous systems are the fractional version of dynamical systems introduced recently by Faghani et al. [6]. The feature property of these systems is the presence of fractional order derivatives as well as equality of their eigenvalues. ...
... Recently, Faghani et al. [6] defined a new category of chaotic systems with identical eigenvalues, proposed three general structures with special conditions and described their chaotic attractors. ...
... In this paper, we propose the fractional version of systems studied by Faghani et al. [6]. These systems have the features of the presence of fractional derivatives and the equality of eigenvalues. ...
... which can exhibit many features of regular and chaotic motions. The study of chaos (either self-excited or hidden) in jerk systems has attracted significant attention in [8,17,23,42,70,73,74]. By considering many thousands of combinations of the coefficients, Molaie, Jafari and Sprott [42] identified 23 chaotic flows with a stable equilibrium, in which the cases SE 1 -SE 6 are some elegant quadratic jerk systems. ...
This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.
... which can exhibit many features of the regular and chaotic motions. The study of 75 chaos (either self-excited or hidden) in jerk systems has attracted significant attention in 76 [13,16,37,41,50,58,75]. 77 By considering many thousands of combinations of the coefficients, Molaie, Jafari and Sprott [37] identified 23 chaotic flows with a stable equilibrium, in which the cases SE 1 -SE 6 are some elegant quadratic jerk systems. ...
... An application of the formula (12) to the tranformed system (8) yields the first 105 order focus quantity W 1 shown in (13), whose sign determines the criticality of Hopf 106 bifurcation. In view of a 2 > 0, a 3 > 0 and the transversality condition (6), the bifurcation 107 is supercritcal if I < 0, and subcritical if I > 0. In the supercritical case, the bifurcation 108 generates a family of stable limit cycles for a 1 > a 2 a 3 , while in the subcritical case, it 109 generates a family of unstable limit cycles for a 1 < a 2 a 3 . ...
Citation: Liu, M.; Sang, B.; Wang, N.; Ahmad, I. Title. Axioms 2021, 1, 0. open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecom-mons.org/licenses/by/ 4.0/). Abstract: This paper is about the dynamical evolution of a family of chaotic jerk systems, which 1 have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, 2 bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden 3 attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf 4 bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a 5 subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-6 doubling cascades to chaos. A circuit implement is presented for the hidden chaotic attractor. The 7 methods used in this paper will help understand and predict the chaotic dynamics of quadratic 8 jerk systems. 9
... which can exhibit many features of the regular and chaotic motions. The study of 43 chaos (either self-excited or hidden) in jerk systems has attracted significant attention in 44 [7,10,20,22,26,32,47]. 45 By considering many thousands of combinations of the coefficients, Molaie, Jafari and Sprott [20] identified 23 chaotic flows with a stable equilibrium, in which the cases SE 1 -SE 6 are some elegant quadratic jerk systems. ...
This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period doubling cascades to chaos. A circuit implement is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.
