Simin Yu

Experimental verification of multidirectional multiscroll chaotic attractors

A systematic methodology for circuit design is proposed for experimental verification of multidirectional multiscroll chaotic attractors, including one-directional (1-D) n-scroll, 2-D n/spl times/m-grid scroll, and 3-D n/spl times/m/spl times/l-grid scroll chaotic attractors. Two typical cases are investigated in detail: the hysteresis and saturated multiscroll chaotic attractors. A simple blocking circuit diagram is designed for experimentally verifying 1-D 5/spl sim/11-scroll, 2-D 3/spl times/5/spl sim/11-grid scroll, and 3-D 3/spl times/3/spl times/5/spl sim/11-grid scroll hysteresis chaotic attractors by manipulating the switchers. Moreover, a block circuitry is also designed for physically realizing 1-D 10, 12, 14-scroll, 2-D 10, 12, 14/spl times/10-grid scroll, and 3-D 10/spl times/10/spl times/10-grid scroll saturated chaotic attractors via switching. In addition, one can easily realize chaotic attractors with a desired odd number of scrolls by slightly modifying the corresponding voltage saturated function series of the circuit, to produce for instance a 1-D 13-scroll saturated chaotic attractor. This is the first time in the literature to report an experimental verification of a 1-D 14-scroll, a 2-D 14/spl times/10-grid scroll and a 3-D 10/spl times/10/spl times/10-grid (totally 1000) scroll chaotic attractors. Only the 3-D case is reported in detail for simplicity of presentation. It is well known that hardware implementation of 1-D n-scroll with n/spl ges/10, 2-D n/spl times/m-grid scroll with n,m/spl ges/10, and 3-D n/spl times/m/spl times/l-grid scroll with n,m,l/spl ges/10 chaotic attractors is very difficult technically, signifying the novelty and significance of the achievements reported in this paper. Finally, this circuit design approach provides some principles and guidelines for hardware implementation of chaotic attractors with a multidirectional orientation and with a large number of scrolls, useful for future circuitry design and engineering applications.

Design and implementation of n-scroll chaotic attractors from a general jerk circuit

This paper proposes a novel nonlinear modulating function approach for generating n-scroll chaotic attractors based on a general jerk circuit. The systematic nonlinear modulating function methodology developed here can arbitrarily design the swings, widths, slopes, breakpoints, equilibrium points, shapes, and even the general phase portraits of the n-scroll chaotic attractors by using the adjustable sawtooth wave, triangular wave, and transconductor wave functions. The dynamic mechanism and chaos generation condition of the general jerk circuit are further investigated by analyzing the system stability. A simple block circuit diagram, including integrator, sawtooth wave and triangular wave generators, buffer, switch linkages, and voltage-current conversion resistors, is designed for the hardware implementations of various 3-12-scroll chaotic attractors via switchings of the switch linkages. This is the first time to experimentally verify a 12-scroll chaotic attractor generated by an analog circuit. In particular, the recursive formulas of system parameters and real physical circuit parameters are rigorously derived for the hardware implementations of the n-scroll chaotic attractors. Moreover, the adjustability of the nonlinear modulating function and the rigorous recursive formulas together provide a theoretical principle for the hardware implementations of various chaotic attractors with a large number of scrolls.

Theoretical Design and FPGA-Based Implementation of Higher-Dimensional Digital Chaotic Systems

Traditionally, chaotic systems are built on the domain of infinite precision in mathematics. However, the quantization is inevitable for any digital devices, which causes dynamical degradation. To cope with this problem, many methods were proposed, such as perturbing chaotic states and cascading multiple chaotic systems. This paper aims at developing a novel methodology to design the higher-dimensional digital chaotic systems (HDDCS) in the domain of finite precision. The proposed system is based on the chaos generation strategy controlled by random sequences. It is proven to satisfy the Devaneys definition of chaos. Also, we calculate the Lyapunov exponents for HDDCS. The application of HDDCS in image encryption is demonstrated via FPGA platform. As each operation of HDDCS is executed in the same fixed precision, no quantization loss occurs. Therefore, it provides a perfect solution to the dynamical degradation of digital chaos.

On the cryptanalysis of Fridrich's chaotic image encryption scheme

Utilizing complex dynamics of chaotic maps and systems in encryption was studied comprehensively in the past two and a half decades. In 1989, Fridrichs chaotic image encryption scheme was designed by iterating chaotic position permutation and value substitution some rounds, which received intensive attention in the field of chaos-based cryptography. In 2010, Solak et al. proposed a chosen-ciphertext attack on the Fridrichs scheme utilizing influence network between cipher-pixels and the corresponding plain-pixels. Based on their creative work, this paper scrutinized some properties of Fridrichs scheme with concise mathematical language. Then, some minor defects of the real performance of Solaks attack method were given. The work provides some bases for further optimizing attack on the Fridrichs scheme and its variants. HighlightsSome properties of Fridrichs chaotic image encryption scheme are represented with concise mathematical language.Real performance of Solaks chosen-plaintext attack on Fridrichs chaotic image encryption scheme is tested with detailed experiments.Extension of the attack idea to Chens scheme and its variants is briefly evaluated.

Design and Implementation of Grid Multiwing Hyperchaotic Lorenz System Family via Switching Control and Constructing Super-Heteroclinic Loops

This paper initiates a systematic methodology for generating various grid multiwing hyperchaotic attractors by switching control and constructing super-heteroclinic loops from the piecewise linear hyperchaotic Lorenz system family. By linearizing the three-dimensional generalized Lorenz system family at their two symmetric equilibria and then introducing the state feedback, two fundamental four-dimensional linear systems are obtained. Moreover, a super-heteroclinic loop is constructed to connect all equilibria of the above two fundamental four-dimensional linear systems via switching control. Under some suitable conditions, various grid multiwing hyperchaotic attractors from the real world applications can be generated. Furthermore, a module-based circuit design approach is developed for realizing the designed piecewise linear grid multiwing hyperchaotic Lorenz and Chen attractors. The experimental observations validate the proposed systematic methodology for grid multiwing hyperchaotic attractors generation. Our theoretical analysis, numerical simulations and circuit implementation together show the effectiveness and universality of the proposed systematic methodology.

A Systematic Methodology for Constructing Hyperchaotic Systems With Multiple Positive Lyapunov Exponents and Circuit Implementation

This paper aims at developing a systematic methodology for constructing continuous-time autonomous hyperchaotic systems with multiple positive Lyapunov exponents. To overcome the essential difficulty in balancing between local instability and global convergence, this paper initiates a new methodology for designing a dissipative hyperchaotic system with a desired number of positive Lyapunov exponents. A general design principle and the corresponding implementation steps are then developed. Three representative examples are shown to validate the proposed principle and implementation scheme. Moreover, a hyperchaotic circuit is constructed to verify a 6-dimensional hyperchaotic system with four positive Lyapunov exponents. Comparing with the traditional trial-and-error approach, the proposed method can design various hyperchaotic systems with any desired number of positive Lyapunov exponents in a systematic way.

A general multiscroll Lorenz system family and its realization via digital signal processors

This paper proposes a general multiscroll Lorenz system family by introducing a novel parameterized nth-order polynomial transformation. Some basic dynamical behaviors of this general multiscroll Lorenz system family are then investigated, including bifurcations, maximum Lyapunov exponents, and parameters regions. Furthermore, the general multiscroll Lorenz attractors are physically verified by using digital signal processors.

Generation of

This paper explores the generation of n- and n times m-wing Lorenz-type attractors from a modified Shimizu-Morioka system. The basic idea is to increase the number of index-2 equilibrium points by introducing a multisegment quadratic function and a stair function in the 2-D state-space of the system. The design is verified by both simulation and experiment, where multiwing attractors over a grid can be clearly observed.

n\times m

This paper introduces a new and unified approach for designing desirable dissipative hyperchaotic systems. Based on the anti-control principle of continuous-time systems, a nominal system of n (n ≥ 5) independent first-order linear differential equations are coupled through all state variables, making the controlled system be in a closed-loop cascade-coupling form, where each equation contains only two state variables therefore the system is quite simple. Based on this setting, a simple model for dissipative hyperchaotic systems is constructed, with an adjustable parameter which can ensure the dissipation of the system. In the closed-loop cascade-coupling form, it is shown that all the eigenvalues are symmetrically distributed in a circumferential manner. Consequently, a universal law is derived on the relationship of the number of positive Lyapunov exponents and the number of positive real parts of its Jacobian eigenvalues. For the above-mentioned simple model, the number of positive Lyapunov exponents for any n-dimensional dissipative hyperchaotic system is given by N = round((n-1)/2), n ≥ 5. Therefore, in theory, the system can generate any desired number of positive Lyapunov exponents as long as the dimension of the system is sufficiently high. Thus, the proposed method provides a new approach for purposefully constructing desirable dissipative hyperchaotic systems. Finally, two examples are given to demonstrate the feasibility of the proposed design method.

-Wing Lorenz-Like Attractors From a Modified Shimizu–Morioka Model

In this paper, the generation of n × m-scroll attractors under a Chua-circuit framework is presented. By using a sawtooth function, f1(x), and a staircase function, f2(y), n × m-scroll attractors can be generated and observed from a third-order circuit. Its dynamical behaviors are investigated by means of theoretical analysis as well as numerical simulation. Moreover, two electronic circuits are designed for its realization, and experimental observations of n × m-scroll attractors based on Chuas circuit are reported, for the first time in the literature.