Robust Chaos Research Articles

Observation of robust chaos in 3D electronic system

Robust Chaos occurring in piecewise smooth dynamical systems is very important in practical applications. It is defined by the absence of periodic windows and coexisting attractors in some neighbourhood of the parameter space. In earlier works, the occurrence of robust chaos was reported in the context of piecewise linear 1D and 2D maps, and regions of occurrences have been investigated in 1D and 2D switching circuits. Here, it has been reported the first experimental observation of this phenomenon in a 3D electronic switching system and obtain the region of parameter space by constructing a discrete map of the system.

Robust Chaos of Cubic Polynomial Discrete Maps with Application to Pseudorandom Number Generators

Based on the robust chaos theorem of S-unimodal maps, this paper studies a kind of cubic polynomial discrete maps (CPDMs) and sets up a novel theorem. This theorem gives general conditions for the occurrence of robust chaos in the CPDMs. By using the theorem, we construct a CPDM. The parameter regions of chaotic robustness of the CPDM are larger than these of Logistic map. By using a fixed point arithmetic, we investigate the cycle lengths of the CPDM and a Logistic map. The results show that the maximum cycle lengths of 1000 chaotic sequences with length 3×107 generated by different initial value conditions exponentially increase with the resolutions. When the resolutions reach 10-7~10-13, the maximum cycle lengths of the cubic polynomial chaotic sequences are significantly greater than these of the Logistic map. When the resolution reaches 10-14, there is the situation without cycle for 1000 cubic polynomial chaotic sequences with length 3×107. By using the CPDM and Logistic map, we design four chaos-based pseudorandom number generators (CPRNGs): CPRNGI, CPRNGII, CPRNGIII, and CPRNGIV. The randomness of two 1000 key streams consisting of 20000 bits is tested, respectively, generated by the four CPRNGs. The result suggests that CPRNGIII based on the cubic polynomial chaotic generalized synchronic system has better performance.

Prediction of robust chaos in micro and nanoresonators under two-frequency excitation.

Robust chaos in a dynamical system is characterized by the persistence of the chaotic attractor with changes in the system parameters and is generally required in practical applications based upon physical sources of chaos. However, for applications that rely upon continuous time chaotic signals, there are now very few alternatives of dynamical systems with robust chaos that could be used. In this context, it is important to find a new dynamical system and, particularly, new physical systems that present robust chaos. In this work, we show through simulations that a relevant physical system, suspended beam micro and nanoelectromechanical resonators, can present robust chaos when excited by two distinct frequencies. To demonstrate the existence of robust chaos in the system, we perform an extensive numerical analysis, showing that the attractor is unique and changes smoothly in a large region of the relevant physical parameter space. We find that the robustness of the chaotic dynamics depends crucially on the dissipation, which must be sufficiently small. When the dissipation is small, we find a large range of frequencies, frequency ratios, and applied voltages where robust chaos is observed. These findings turn these systems into viable and strong candidates for practical applications since the chaotic dynamics becomes quite insensitive to fabrication tolerances, changes in the physical parameters induced by the environment, and aging.

Design new chaotic maps based on dimension expansion**Project supported by the National Natural Science Foundation of China (Grant Nos. 61161006 and 61573383).

Based on the high-dimensional (HD) chaotic maps and the sine function, a new methodology of designing new chaotic maps using dimension expansion is proposed. This method accepts N dimensions of any existing HD chaotic map as inputs to generate new dimensions based on the combined results of those inputs. The main principle of the proposed method is to combine the results of the input dimensions, and then performs a sine-transformation on them to generate new dimensions. The characteristics of the generated dimensions are totally different compared to the input dimensions. Thus, both of the generated dimensions and the input dimensions are used to create a new HD chaotic map. An example is illustrated using one of the existing HD chaotic maps. Results show that the generated dimensions have better chaotic performance and higher complexity compared to the input dimensions. Results also show that, in the most cases, the generated dimensions can obtain robust chaos which makes them attractive to usage in a different practical application.

Dynamics and Optimization Control of a Robust Chaotic Map

Robust chaos in the discrete system is suggested to have practical as well as theoretical importance since it can obtain reliable operation in the chaotic mode. However, it receives only moderate attention and only focuses on a finite chaotic parameter space and small Lyapunov exponents. This paper introduces a two-dimensional smooth map and studies its robustness of chaos in the infinite parameter space. Then, a compound operation-based optimization control method is introduced to increase the map complexity in the measure of Lyapunov exponent. The introduced method is simple and provides a new pathway for exploring the robustness and complexity of discrete chaotic system. Finally, we design a chaos-based pseudo-random number generator (CPNG) based on the optimized robust chaotic map, and the careful analysis shows that the proposed CPNG has high quality of randomness and has passed the rigorous National Institute of Standards and Technology (NIST) test.

A Handwriting Document Encryption Scheme Based on Segmentation and Chaotic Logarithmic Map

A one-dimensional chaotic logarithmic map (CLM) is proposed. Characteristics of the CLM are investigated and used to develop a symmetric handwriting document encryption scheme that consists of three phases. The segmentation phase divides a handwriting image into graphemes. The permutation phase shuffles pixel locations of each grapheme. Finally, the substitution phase modifies pixel intensity values of the corresponding permutated grapheme utilising the chaotic logarithmic map. Experimental results indicate that the proposed CLM poses interesting characteristics such as wide range chaotic behaviour, robust chaos, s-unimodality, and high sensitivity to a small change in initial condition. The results also indicate that the randomly generated keystreams by the CLM pass the NIST statistical tests. Furthermore, the proposed segmentation and encryption scheme achieves a key space of 2180 and provides a high encryption performance for handwriting documents. In addition, statistical results indicate the superior permutation and substitution properties of the proposed encryption scheme over other image encryption schemes of the same structure.

Robust chaos in 3-D piecewise linear maps.

A chaotic attractor is called robust if there is no periodic window or any coexisting attractor in some open subset of the parameter space. Such a chaotic attractor cannot be destroyed by a small change in parameter values since a small change in the parameter value can only make small changes in the Lyapunov exponents. Earlier investigations have calculated the existence and the stability conditions of robust chaos in 1D and 2D piecewise linear maps. In this work, we demonstrate the occurrence of robust chaos in 3D piecewise linear maps and derive the conditions of its occurrence by analyzing the interplay between the stable and unstable manifolds.

Transition to high-dimensional chaos in nonsmooth dynamical systems

We uncover a route from low-dimensional to high-dimensional chaos in nonsmooth dynamical systems as a bifurcation parameter is continuously varied. The striking feature is the existence of a finite parameter interval of periodic attractors in between the regimes of low- and high-dimensional chaos. That is, the emergence of high-dimensional chaos is preceded by the system's settling into a totally nonchaotic regime. This is characteristically distinct from the situation in smooth dynamical systems where high-dimensional chaos emerges directly and smoothly from low-dimensional chaos. We carry out an analysis to elucidate the underlying mechanism for the abrupt emergence and disappearance of the periodic attractors and provide strong numerical support for the typicality of the transition route in the pertinent two-dimensional parameter space. The finding has implications to applications where high-dimensional and robust chaos is desired.

Parameter estimation for 1D PWL chaotic maps using noisy dynamics

Many physical situations involve chaotic systems implemented in hardware. Among them one-dimensional piecewise linear maps are popular candidates for such applications because of their property of generating robust chaos. In physical implementations, the control parameter of these maps may deviate from its ideal value due to hardware imprecision. Since the dynamics of a chaotic map is completely defined by its control parameter, one needs to know the value of the parameter in a hardware realisation. In this paper, we show that it is possible to determine the parameter, through the realisation of the unstable fixed point of the map, by utilising noise that is always present in the system. We present this in the form of an algorithm and demonstrate its efficacy through simulated results. We also determine the bounds on the signal-to-noise ratio required for successful parameter estimation. The proposed approach is expected to be beneficial to the existing noise reduction techniques and time series recovery algorithms that require a reasonably accurate knowledge of the map.

Necessary and sufficient conditions of full chaos for expanding Baker-like maps and their use in non-expanding Lorenz maps

In this work we give necessary and sufficient conditions for a discontinuous expanding map f of an interval into itself, made up of N pieces, to be chaotic in the whole interval. For N=2 we consider the class of expanding Lorenz maps, for N ≥ 3 a class of maps whose internal branches are onto, called Baker-like. We give the necessary and sufficient conditions for a discontinuous expanding map to be chaotic in the whole interval and persistent under parameter perturbations (robust full chaos in short). These classes of maps represent a suitable first return in non-expanding Lorenz maps. Thus the obtained conditions can be used to prove robust full chaos in non-expanding Lorenz maps. An example from the engineering application is illustrated.

A family of fully integrated CMOS chaos generators with strictly 1-D linear-piecewise chaos maps

A family of three complementary metal-oxide semiconductor (CMOS) chaos generators is presented. The proposed scheme starts by obtaining strictly 1-D piecewise-linear chaos maps using a stroboscopic sampling method. The chaotic behavior is then analyzed with the help of numerical simulations using Mathematica. As required for practical applications, the robustness of the derived chaos maps was also confirmed by analyzing bifurcation and Lyapunov exponent diagrams. For verification, a tent-shape chaos generator was implemented using the 0.18- $$\upmu $$ m CMOS process. The simulation as well as measurement results confirmed the proposed scheme. The key advantages of the proposed chaos generators with varied fabrication parameters, including compact size, low power consumption, simple achievement of chaos, and robust chaos generation, recommend them as a solution in design of fully integrated chaotic circuits for applications where noise-like signal sources are required.

Improving particle beam acceleration in plasmas

The dynamics of wave-particle interactions in magnetized plasmas restricts the wave amplitude to moderate values for particle beam acceleration from rest energy. We analyze how a perturbing invariant robust barrier modifies the phase space of the system and enlarges the wave amplitude interval for particle acceleration. For low values of the wave amplitude, the acceleration becomes effective for particles with initial energy close to the rest energy. For higher values of the wave amplitude, the robust barrier controls chaos in the system and restores the acceleration process. We also determine the best position for the perturbing barrier in phase space in order to increase the final energy of the particles.

Sine-Transform-Based Chaotic System With FPGA Implementation

As chaotic dynamics is widely used in nonlinear control, synchronization communication, and many other applications, designing chaotic maps with complex chaotic behaviors is attractive. This paper proposes a sine-transform-based chaotic system (STBCS) of generating one-dimensional (1-D) chaotic maps. It performs a sine transform to the combination of the outputs of two existing chaotic maps (seed maps). Users have the flexibility to choose any existing 1-D chaotic maps as seed maps in STBCS to generate a large number of new chaotic maps. The complex chaotic behavior of STBCS is verified using the principle of Lypunov exponent. To show the usability and effectiveness of STBCS, we provide three new chaotic maps as examples. Theoretical analysis shows that these chaotic maps have complex dynamics properties and robust chaos. Performance evaluations demonstrate that they have much larger chaotic ranges, better complexity, and unpredictability, compared with chaotic maps generated by other methods and the corresponding seed maps. Moreover, to show the simplicity of STBCS in hardware implementation, we simulate the three new chaotic maps using the field-programmable gate array (FPGA).

Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation.

The search for generation approaches to robust chaos has received considerable attention due to potential applications in cryptography or secure communications. This paper is of interest regarding a 1-D sigmoidal chaotic map, which has never been distinctly investigated. This paper introduces a generic form of the sigmoidal chaotic map with three terms, i.e., xn+1 = ∓AfNL(Bxn) ± Cxn ± D, where A, B, C, and D are real constants. The unification of modified sigmoid and hyperbolic tangent (tanh) functions reveals the existence of a “unified sigmoidal chaotic map” generically fulfilling the three terms, with robust chaos partially appearing in some parameter ranges. A simplified generic form, i.e., xn+1 = ∓fNL(Bxn) ± Cxn, through various S-shaped functions, has recently led to the possibility of linearization using (i) hardtanh and (ii) signum functions. This study finds a linearized sigmoidal chaotic map that potentially offers robust chaos over an entire range of parameters. Chaos dynamics are described in terms of chaotic waveforms, histogram, cobweb plots, fixed point, Jacobian, and a bifurcation structure diagram based on Lyapunov exponents. As a practical example, a true random bit generator using the linearized sigmoidal chaotic map is demonstrated. The resulting output is evaluated using the NIST SP800-22 test suite and TestU01.

Hybrid robust discrete sliding mode control for generalized continuous chaotic systems subject to external disturbances

This paper presents a discrete sliding mode control (DSMC) for the robust chaos suppression of the generalized continuous-time chaotic systems subject to matched/mismatched disturbances. The proposed DSMC first ensures the existence of the sliding manifold. Then, the effect of external disturbances including matched and mismatched cases are discussed after the controlled system is driven into the sliding manifold. The proposed results show that the chaotic behaviour of controlled systems with matched disturbances can be fully suppressed to zero or robustly driven into an estimated bound in the state–space form, which was not explicitly addressed in the literature. Finally, two different chaotic systems are utilized to study the hybrid chaos suppression. The corresponding numerical results show the effectiveness and robustness of the proposed DSMC.

Quasiperiodicity and suppression of multistability in nonlinear dynamical systems

It has been known that noise can suppress multistability by dynamically connecting coexisting attractors in the system which are otherwise in separate basins of attraction. The purpose of this mini-review is to argue that quasiperiodic driving can play a similar role in suppressing multistability. A concrete physical example is provided where quasiperiodic driving was demonstrated to eliminate multistability completely to generate robust chaos in a semiconductor superlattice system.

Secure communication in wireless sensor networks based on chaos synchronization using adaptive sliding mode control

Due to resource constraints in wireless sensor networks and the presence of unwanted conditions in communication systems and transmission channels, the suggestion of a robust method which provides battery lifetime increment and relative security is of vital importance. This paper considers the secure communication in wireless sensor networks based on new robust adaptive finite time chaos synchronization approach in the presence of noise and uncertainty. For this purpose, the modified Chua oscillators are added to the base station and sensor nodes to generate the chaotic signals. Chaotic signals are impregnated with the noise and uncertainty. At first, we apply the modified independent component analysis to separate the noise from the chaotic signals. Then, using the adaptive finite-time sliding mode controller, a control law and an adaptive parameter-tuning method is proposed to achieve the finite-time chaos synchronization under the noisy conditions and parametric uncertainties. Synchronization between the base station and each of the sensor nodes is realized by multiplying a selection matrix by the specified chaotic signal which is broadcasted by the base station to the sensor nodes. Simulation results are presented to show the effectiveness and applicability of the proposed technique.

From Geodesic Flow on a Surface of Negative Curvature to Electronic Generator of Robust Chaos

Departing from the geodesic flow on a surface of negative curvature as a classic example of the hyperbolic chaotic dynamics, we propose an electronic circuit operating as a generator of rough chaos. Circuit simulation in NI Multisim software package and numerical integration of the model equations are provided. Results of computations (phase trajectories, time dependencies of variables, Lyapunov exponents and Fourier spectra) show good correspondence between the chaotic dynamics on the attractor of the proposed system and of the Anosov dynamics for the original geodesic flow.

Robust finite-time chaos synchronization of time-delay chaotic systems and its application in secure communication

This paper presents finite-time chaos synchronization of time-delay chaotic systems with uncertain parameters. According to the proposed method, a lot of coupled items can be treated as zero items. Thus, the whole system can be simplified greatly. Based on robust chaotic synchronization, secure communication can be realized with a wide range of parameter disturbance and time-delay. Numerical simulations are provided to illustrate the effectiveness of the proposed method.

ОТ ДИНАМИКИ АНОСОВА НА ПОВЕРХНОСТИ ОТРИЦАТЕЛЬНОЙ КРИВИЗНЫ К ЭЛЕКТРОННОМУ ГЕНЕРАТОРУ ГРУБОГО ХАОСА

ОТ ДИНАМИКИ АНОСОВА НА ПОВЕРХНОСТИ ОТРИЦАТЕЛЬНОЙ КРИВИЗНЫ К ЭЛЕКТРОННОМУ ГЕНЕРАТОРУ ГРУБОГО ХАОСА