Chaotic single neuron model with periodic coefficients with period two

In this paper, we examine the pattern separation ability in the Chaotic Quaternionic Associative Memory. The Chaotic Quaternionic Associative Memory is composed of chaotic quaternionic neuron model which is based on the chaotic neuron model and the quaternionic neuron model. If the parameters such as damping factor and scaling factor of refractoriness are set appropriately, the chaotic quaternionic neuron model can generate chaotic response. The Chaotic Quaternionic Associative Memory makes use of the dynamic association ability of the chaotic quaternionic neuron model in order to realize pattern separation. However, pattern separation ability is very sensitive to the parameters in chaotic quaternionic neuron model and the connection weight from external input. In this research, we examine the relation between pattern separation ability and the connection weight from external input in the Chaotic Quaternionic Associative Memory. We carried out a series of computer experiments and confirmed that the influence of the connection weight from external input to the pattern separation ability becomes large when the number of neurons increases.

Pulse-type hardware chaotic neuron model and its bifurcation phenomena

The optimization performance of transiently chaotic neural network (TCNN) is affected by various factors such as chaotic characteristic, model parameters, and annealing function, and its capacity of global optimization is limited. It is demonstrated that the non-monotonic activation function can generate richer chaotic characteristic than the monotonic activation function in the TCNN model. Besides, the activation function involving neurobiological mechanism can not only reflect the rich brain activity in brain waves, but also enhance the non-linear dynamic characteristic, which may further improve the global optimization ability. Hence, a novel chaotic neuron model is proposed with the non-monotonic activation function based on the neurobiological mechanisms from the electroencephalogram. The electroencephalogram consists of five brain waves (i.e., , , , , and waves) which are defined by the quality and intensity of brain waves with different frequency bands ranging from 0.5 Hz to 100 Hz. The brain wave with a higher frequency and a lower amplitude represents a more active brain. Researches demonstrate that the five brain waves can be simplified into sinusoidal waves with different frequencies. Hence, a frequency conversion sinusoidal (FCS) function which has the consistent frequency range and features with brain waves is designed based on the above neurobiological mechanisms. Then a novel chaotic neuron model with non-monotonic activation function which is composed of the FCS function and sigmoid function, is proposed for richer chaotic dynamic characteristic. The reversed bifurcation and the Lyapunov exponent of the chaotic neuron are given and the dynamic system is analyzed, indicating that the proposed FCS neuron model owns richer chaotic dynamic characteristic than transiently chaotic neuron model due to its special non-monotonic activation function. Based on the neuron model, a novel transiently-chaotic neural networkfrequency conversion sinusoidal chaotic neural network (FCSCNN) is constructed and the basis of model parameter selection is provided as well. To validate the effectiveness of the proposed model, the FCSCNN is applied to nonlinear function optimization and 10-city, 30-city, 75-city traveling salesman problem. The experimental results show that 1) the FCSCNN has a good performance under the condition of moderate a, smaller cA(0) and 2(0); 2) on the basis of the appropriate model parameters, the FCSCNN has better global optimization ability and optimization accuracy than Hopfield neural network, TCNN, improved-TCNN due to its richer chaotic characteristic in complicated combinational optimization problem, especially in middle and large scale problem.

<p>Interactions between genes and the proteins they synthesize shape genetic regulatory networks (GRN). Several models have been proposed to describe these interactions, been the most commonly used those based on ordinary differential equations (ODEs). Some approximations using piecewise linear differential equations (PLDEs), have been proposed to simplify the model non linearities. However they not allways give good results. In this context, it has been developed a model capable of representing small GRN, combining characteristics from the ODE’s models and fuzzy inference systems (FIS). The FIS is trained through an artificial neural network, which forms an Adaptive Nertwork-based Fuzzy Inference System (ANFIS). This network allows to adapt the membership and output functions from the FIS according to the training data, thus, reducing the previous knowledge needed to model the specific phenomenon.<br /> In addition, Fuzzy Logic allows to express their rules through linguistic labels, which also allows to incorporate expert knowledge in a friendly way. The proposed model has been used to describe the Lac Operon in E. Coli and it has been compared with the models already mentioned. The outcome errors due to the training process of the ANFIS network are comparable with those of the models based on ODEs. Additionally, the fuzzy logic approach provides modeling flexibility and knowledge acquisition advantages.</p>

Due to the voltage-commutation, one has to take an additional subinterval into account during analysis. The difference equation of sample values is set up starting from piecewise linear differential equations. For pulse-width modulated control a dynamic model follows, linearized around the steady state working point. In the case of dc choppers the useful signal is the low-frequency mean value. The difference equation of the mean value behaviour can be obtained via a discrete relation between mean- and sample value. If sufficient smoothing is provided a continuous equivalent differential equation can be derived. One obtains via series expansions an analytical approximation model, which simply can be interpreted as continuous equivalent circuit. Then a further analysis follows with the well known methods of dc networks.

This paper focuses on implementations of two modified Aihara's chaotic neuron models and a simple chaotic neural network constructed with two chaotic neurons in a programmable and reconfigurable manner with an analog programmable device, FPAA (Field Programmable Analog Array). After testing the chaotic behaviors of two chaotic neuron models and a simple chaotic neural network through numerical analyses that consist of time domain responses, phase portrait illustrations and bifurcation diagrams, the experimental setup is constructed with a FPAA device. The parametric adjustments of chaotic neural structures are possible with the proposed flexible design methodology and different chaotic neuron models are constructed on the same reconfigurable device without any hardware changes. Experimental results verify the dynamic behaviors of these chaotic neural structures and demonstrate the efficiency of programmable implementations.

Robust model predictive control of discrete nonlinear systems with time delays and disturbances via T–S fuzzy approach

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Qualitative Analysis of Behavior of Two Dimensional Piecewise Linear Differential Equations : (1) 0verview and Theoretical Analysis of Characteristics of Behavior of Two Dimensional Piecewise Linear Differential Equations

A novel chaotic-neuron model is presented by introducing the non-monotonous activation function which is composed of the Legendre function and the Sigmoid function.The reversed bifurcation of the chaotic neuron model is given and analyzed, meanwhile, how do parameters influence the network convergence speed is discussed.Based on the neuron model, the piecewise simulated annealing SLF chaotic neural network was made by introducing the simulated annealing idea, the model improve the convergence speed, at the same time, the precision of this network have not being influenced.The simulation experiment of function optimization and TSP problem verify the effectiveness of the segmented simulated annealing strategy.

The timing of DNA synthesis, mitosis, and cell division is regulated by a complex network of biochemical reactions that control the activities of a family of cyclin-dependent kinases. The temporal dynamics of this reaction network is typically modeled by nonlinear differential equations describing the rates of the component reactions. This approach provides exquisite details about molecular regulatory processes but is hampered by the need to estimate realistic values for the many kinetic constants that determine the reaction rates. It is difficult to estimate these kinetic constants from available experimental data. To avoid this problem, modelers often resort to ‘qualitative’ modeling strategies, such as Boolean switching networks, but these modelsdescribe only the coarsest features of cell cycle regulation. In this chapter it describes a hybrid approach that combines the best features of continuous differential equations and discrete Boolean networks. Cyclin abundances are tracked by piecewise linear differential equations for cyclin synthesis and degradation. The cyclin synthesis is regulated by transcription factors whose activities are represented by discrete variables (0 or 1) and likewise for the activities of the ubiquitin-ligating enzyme complexes that govern cyclin degradation. The discrete variables change according to a predetermined sequence, with the times between transitions determined in part by cyclin accumulation and degradation and as well by exponentially distributed random variables. The model is evaluated in terms of flow cytometry measurements of cyclin proteins in asynchronous populations of human cell lines. The few kinetic constants in the model are easily estimated from the experimental data. Using this hybrid approach, modelers can quickly create quantitatively accurate, computational models of protein regulatory networks in cells.

A new method to investigate asymptotic properties of linear differential equations with strong threshold and switching effects is presented. The method is applied to systems of equations of the form dx/dt = F(x) - yx, where y = constant and the dependence of F on x is mediated by sigmoid functions. Using a special sigmoid function called a logoid, which rises monotonically from zero to one in a narrow interval surrounding the threshold value, exact analytical expressions for the limiting value of all steady points can be found in the limit when the logoid approaches a step function. The limiting values are independent of the shape of the logoid for a large class of logoids. Relations between steady points and limit cycles of the equations with logoids, their step function limit and the corresponding piecewise linear equations are derived. It is found that the approximation of sigmoids by the step function idealization is not always warranted. The results strongly suggest the use of logoids instead of othe...

Here, first we shall set together various basic statements on the periodicity of the solutions of first order linear difference equations, then we shall define various discrete functions which are in a sense connected with the concept of periodicity. Finally, we will formulate a decomposition theorem for the solutions of first order linear difference equations with periodic coefficients.

A definite theory exists for solving linear systems of ordinary differential equations with constant coefficients. “This theory is essentially a branch of linear algebra, and allows us to solve all autonomous linear equations”, ([4], p. 95). Furthermore, “In view of the fact that the solution of these equations does not, in principle, present any great difficulties, they are often considered to be of no great interest for theory, and in textbooks they are usually relegated to the position of simple exercises appended to the general theory of linear equations”, ([81], p. 41). Modern textbooks [8, 21, 26, 44, 45, 46, 50, 90] give a detailed and rigorous treatment of the standard methods available for linear problems, supplemented with numerous mechanical, electrical, and engineering applications. The voluminous literature accumulated on the general properties of the solutions of linear differential equations must therefore no doubt leave the impression that no further contribution could be made to the standard normal linear dynamic system with real and constant coefficients. This impression, however, will fade rather quickly as we go on. Paradoxically, some fundamental aspects of this simple dynamic system are still unnoticed.

An analysis technique is presented to provide an essentially explicit solution for a system of n simultaneous first-order linear differential equations with periodic coefficients. This representation of a periodic variable-parameter linear system of arbitrary finite order is chosen for its theoretical and practical advantages over the classical nth order linear differential equation. Emphasis is placed on natural mode solutions of a homogeneous set of equations. The characteristic exponents for these solutions are determined from a polynomial equation the coefficients of which are linear combinations of n — 1 convergent infinite-order determinants. Approximate calculation of these determinants is feasible for problems of moderate order.