Output tracking and synchronization of chaotic Chua’s circuit with disturbances via model predictive regulator

In this paper, we consider the output tracking and synchronization problem for a piecewise linear chaotic Chua’s circuit, which is expected to follow the reference signal generated by another autonomous chaotic Chua’s circuit. A novel model predictive regulator is proposed to guarantee the chaos synchronization in the presence of external disturbance. Moreover with this regulator, the optimal transient tracking performance for synchronization is ensured even when the piecewise autonomous and controlled Chua’s circuits switch asynchronously. The proposed approach is highly efficient in the sense that the regulation conditions can be satisfied by solving a set of linear matrix inequalities and several Sylvester equations instead of many partial differential regulator equations in the well-known nonlinear output regulation theory.     

An improvement of chaos-based hash function in cryptanalysis approach: An experience with chaotic neural networks and semi-collision attack

In this paper, the chaos-based hash function is analyzed, then an improved version of chaos-based hash function is presented and discussed using chaotic neural networks. It is based on the piecewise linear chaotic map that is used as a transfer function in the input and output of the neural network layer. The security of the improved hash function is also discussed and a novel type of collision resistant hash function called semi-collision attack is proposed, which is based on the collision percentage between the two hash values. In the proposed attack particle swarm optimization algorithm is used to define the fitness function parameters. Finally, numerical and simulation results provides strong collision resistance and high performance efficiency.     

Constructing piecewise linear chaotic system based on the heteroclinic Shil’nikov theorem

In this paper, a kind of piecewise linear chaotic system is constructed based on the Shil’nikov theorem. These systems have the same Jacobian in each equilibrium, and the piecewise linear functions in them are discontinuous, piecewise constants. The condition for the existence of the heteroclinic orbits in this kind of system is discussed. According to the separating plane and the position of the equilibriums, four different chaotic systems are given. Computer simulations confirm that the proposed method can be used to construct arbitrary chaotic attractors with multi-scrolls.     

Comparison of different chaotic maps in particle swarm optimization algorithm for long-term cascaded hydroelectric system scheduling

The goal of this paper is to present a novel chaotic particle swarm optimization (CPSO) algorithm and compares the efficiency of three one-dimensional chaotic maps within symmetrical region for long-term cascaded hydroelectric system scheduling. The introduced chaotic maps improve the global optimal capability of CPSO algorithm. Moreover, a piecewise linear interpolation function is employed to transform all constraints into restrict upriver water level for implementing the maximum of objective function. Numerical results and comparisons demonstrate the effect and speed of different algorithms on a practical hydro-system.     

Analytical solution for a class of learning by doing models with multiplicative uncertainty

We find the closed form optimal solution for a class of learning by doing models, where multiplicative uncertainty is introduced in a piecewise linear cost reduction function. Previous literature does not find the closed form optimal solution for these models. We consider a monopolist, facing a linear demand function. The optimal policy for the resulting problem is shown to be piecewise linear and continuous. The optimal output increases with unit cost for certain values of the latter. Numerical examples are provided.     

An improved particle swarm optimization algorithm combined with piecewise linear chaotic map

Particle swarm optimization (PSO) has gained increasing attention in tackling complex optimization problems. Its further superiority when hybridized with other search techniques is also shown. Chaos, with the properties of ergodicity and stochasticity, is definitely a good candidate, but currently only the well-known logistic map is prevalently used. In this paper, the performance and deficiencies of schemes coupling chaotic search into PSO are analyzed. Then, the piecewise linear chaotic map (PWLCM) is introduced to perform the chaotic search. An improved PSO algorithm combined with PWLCM (PWLCPSO) is proposed subsequently, and experimental results verify its great superiority.     

Snap-back repellers in non-smooth functions

In this work we consider the homoclinic bifurcations of expanding periodic points. After Marotto, when homoclinic orbits to expanding periodic points exist, the points are called snap-back-repellers. Several proofs of the existence of chaotic sets associated with such homoclinic orbits have been given in the last three decades. Here we propose a more general formulation of Marotto’s theorem, relaxing the assumption of smoothness, considering a generic piecewise smooth function, continuous or discontinuous. An example with a two-dimensional smooth map is given and one with a two-dimensional piecewise linear discontinuous map.     

Controlling chaotic Chua''s circuit based on piecewise quadratic Lyapunov functions method

In this paper, we present a linear feedback controller design method for controlling chaotic Chua's circuit based on piecewise quadratic Lyapunov functions. Firstly, we get the piecewise linear differential inclusions (pwLDIs) model of tracking error dynamics, then we design a linear state feedback controller to stabilize it based on the piecewise quadratic Laypunov functions. Finally, we give some numerical simulations to demonstrate the effectiveness of our theoretical results.     

Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.     

Reservoir flood control operation based on chaotic particle swarm optimization algorithm

Reservoir flood control operation is a complex engineering optimization problem with a large number of constraints. In order to solve this problem, a chaotic particle swarm optimization (CPSO) algorithm based on the improved logistic map is presented, which uses the discharge flow process as the decision variables combined with the death penalty function. According to the principle of maximum eliminating flood peak, a novel flood control operation model has been established with the goal of minimum standard deviation of the discharge flow process. At the same time, a piecewise linear interpolation function (PLIF) is applied to deal with the constraints for solving objective function. The performance of the proposed model and method is evaluated on two typical floods of Three Gorges reservoir. In comparison with existing models and other algorithms, the proposed model and algorithm can generate better solutions with the minimal flood peak discharge and the maximal peak-clipping rate for reservoir flood control operation.     

Isotopic Implicit Surface Meshing

This paper addresses the problem of piecewise linear approximation of implicit surfaces. We first give a criterion ensuring that the zero-set of a smooth function and the one of a piecewise linear approximation of it are isotopic. Then, we deduce from this criterion an implicit surface meshing algorithm certifying that the output mesh is isotopic to the actual implicit surface. This is the first algorithm achieving this goal in a provably correct way.     

Quick estimation of escape rate with the help of fractal dimension

The non-linearity caused by the application of digital control may lead to transient chaotic behaviour. In the present paper, we analyse a simple model of a digitally controlled mechanical system with dry friction, which may perform transient chaotic vibrations. As a consequence of the digital effects, the behaviour of this system can be described by a 1D piecewise linear map. The fractal dimension of the so-called chaotic repeller set is calculated and the results are used for the quick estimation of the mean lifetime of chaotic transients.     

Life Expectancy of Transient Microchaotic Behaviour

The application of digital control may lead to so-called transient chaotic behaviour. In the present paper, we analyse a simple model of a digitally controlled mechanical system, which may create such vibrations. As a consequence of the digital effects, i.e., the sampling and the round-off error, the behaviour of this system can be described by a one-dimensional piecewise linear map. The lifetime of chaotic transients is usually characterized by the so-called escape rate. In the literature, the reciprocal of the escape rate is considered to be the expected duration of the transient chaotic phenomenon. We claim that this approach is not always fruitful, and present a different way of calculating the mean lifetime in the case of one-dimensional piecewise linear maps. Our method might also be used to solve diffusion problems in one-dimensional models of periodic arrays.     

Approximate solutions and chaotic motions of a piecewise nonlinear–linear oscillator

A global symmetric period-1 approximate solution is analytically constructed for the non-resonant periodic response of a periodically excited piecewise nonlinear–linear oscillator. The approximate solutions are found to be in good agreement with the exact solutions that are obtained from the numerical integration of the original equations. In addition, the dynamic behaviour of the oscillator is numerically investigated with the help of bifurcation diagrams, Lyapunov exponents, Poincare maps, phase portraits and basins of attraction. The existence of subharmonic and chaotic motions and the coexistence of four attractors are observed for some combinations of the system parameters.     

Chaotic hash function based on circular shifts with variable parameters

We propose a chaotic hash algorithm based on circular shifts with variable parameters in this paper. We exploit piecewise linear chaotic map and one-way coupled map lattice to produce initial values and variable parameters. Circular shifts are introduced to improve the randomness of hash values. We evaluate the proposed hash algorithm in terms of distribution of the hash value, sensitivity of the hash value to slight modifications of the original message and secret keys, confusion and diffusion properties, robustness against birthday and meet-in-the-middle attacks, collision tests, analysis of speed, randomness tests, flexibility, computational complexity, and the results demonstrate that the proposed algorithm has strong security strength. Compared with the existing chaotic hash algorithms, our algorithm shows moderate statistical performance, better speed, randomness tests, and flexibility.     

Suppressing grazing chaos in impacting system by structural nonlinearity

In this note we investigate the influence of structural nonlinearity of a simple cantilever beam impacting system on its dynamic responses close to grazing incidence by a means of numerical simulation. To obtain a clear picture of this effect we considered two systems exhibiting impacting motion, where the primary stiffness is either linear (piecewise linear system) or nonlinear (piecewise nonlinear system). Two systems were studied by constructing bifurcation diagrams, basins of attractions, Lyapunov exponents and parameter plots. In our analysis we focused on the grazing transitions from no impact to impact motion. We observed that the dynamic responses of these two similar systems are qualitatively different around the grazing transitions. For the piecewise linear system, we identified on the parameter space a considerable region with chaotic behaviour, while for the piecewise nonlinear system we found just periodic attractors. We postulate that the structural nonlinearity of the cantilever impacting beam suppresses chaos near grazing.     

Switched safe tracking control design for unmanned autonomous helicopter with disturbances

In this paper, the switched safe tracking control scheme is investigated for the attitude and altitude system of a medium-scale unmanned autonomous helicopter with output constraints and unknown external disturbances. To keep the attitude angles and altitude within the desired constrained range, an output boundary protection approach is adopted to generate an output constrained trajectory which is piecewise differentiable. The disturbance observer-based control method is employed to handle the unknown external disturbances of the system. Because of the piecewise differentiability of the output constrained trajectory, the closed-loop error system with the safe tracking controller can be seen as a switched system with jump dynamics. The multiple Lyapunov function method is adopted to guarantee the tracking performance with designed average dwell time. Simulation results of an example are provided to illustrate the effectiveness of the proposed control scheme for the unmanned autonomous helicopter system.     

Global asymptotical synchronization of chaotic neural networks by output feedback impulsive control: An LMI approach

In this paper, impulsive control for master–slave synchronization schemes consisting of identical chaotic neural networks is studied. Impulsive control laws are derived based on linear static output feedback. A sufficient condition for global asymptotic synchronization of master–slave chaotic neural networks via output feedback impulsive control is established, in which synchronization is proven in terms of the synchronization errors between the full state vectors. An LMI-based approach for designing linear static output feedback impulsive control laws to globally asymptotically synchronize chaotic neural networks is discussed. With the help of LMI solvers, linear output feedback impulsive controllers can be easily obtained along with the bounds of the impulsive intervals for global asymptotic synchronization. The method is finally illustrated by numerical simulations.     

Border collision bifurcations and chaotic sets in a two-dimensional piecewise linear map

A two-dimensional piecewise linear continuous model is analyzed. It reflects the dynamics occurring in a circuit proposed as chaos generator, in a simplified case. The parameter space is investigated in order to classify completely regions of existence of stable cycles, and regions associated with chaotic behaviors. The border collision bifurcation curves are analytically detected, as well as the degenerate flip bifurcations of k-cycles and the homoclinic bifurcations occurring in cyclic chaotic regions leading to chaos in one-piece.     

Optimality of a Standard Adaptive Finite Element Method

In this paper an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever, for some s > 0, the solution can be approximated within a tolerance ε > 0 in energy norm by a continuous piecewise linear function on some partition with O(ε^-1/s) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that converge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, the results generalize in several respects.