We show by numerical calculations that a chaotic neuron model driven by a weak sinusoid has resonance. This resonance phenomenon has a peak at a drive frequency similar to that of noise-induced stochastic resonance (SR). This neuron model was proposed from biological studies and shows a chaotic response when a parameter is varied. SR is a noise induced effect in driven nonlinear dynamical systems. The basic SR mechanism can be understood through synchronization and resonance in a bistable system driven by a subthreshold sinusoid plus noise. Therefore, background noise can boost a weak signal using SR. This effect is found in biological sensory neurons and obviously has some useful sensory function. The signal-to-noise ratio (SNR) of the driven chaotic neuron model is improved depending on the drive frequency; especially at low frequencies, the SNR is remarkably promoted. The resonance mechanism in the model is different from the noise-induced SR mechanism. This paper considers the mechanism and proposes possible explanations. Also, the meaning of chaos in biological systems based on the resonance phenomenon is considered.

Enhancement of resonance is shown by coupling and summing in sinusoidally driven chaotic neural networks. This resonance phenomenon has a peak at a drive frequency similar to noise-induced stochastic resonance (SR), however, the mechanism is different from noise-induced SR. We numerically study the properties of resonance in chaotic neural networks in the turbulent phase with summing and homogeneous coupling, with particular consideration of enhancement of the signal-to-noise ratio (SNR) by coupling and summing. Summing networks can enhance the SNR of a mean field based on the law of large numbers. Global coupling can enhance the SNR of a mean field and a neuron in the network. However, enhancement is not guaranteed and depends on the parameters. A combination of coupling and summing enhances the SNR, but summing to provide a mean field is more effective than coupling on a neuron level to promote the SNR. The global coupling network has a negative correlation between the SNR of the mean field and the Kolmogorov-Sinai (KS) entropy, and between the SNR of a neuron in the network and the KS entropy. This negative correlation is similar to the results of the driven single neuron model. The SNR is saturated as an increase in the drive amplitude, and further increases change the state into a nonchaotic one. The SNR is enhanced around a few frequencies and the dependence on frequency is clearer and smoother than the results of the driven single neuron model. Such dependence on the drive amplitude and frequency exhibits similarities to the results of the driven single neuron model. The nearest neighbor coupling network with a periodic or free boundary can also enhance the SNR of a neuron depending on the parameters. The network also has a negative correlation between the SNR of a neuron and the KS entropy whenever the boundary is periodic or free. The network with a free boundary does not have a significant effect on the SNR from both edges of the free boundaries.

A chaotic noise is one of the most important implements for information processing such as neural networks. It has been suggested that chaotic neural networks have high performance ability for information processing. In this paper, we report two designs of a compact chaotic noise generator for large integration circuits using CMOS technology. The chaotic noise is generated using map chaos. We design both of the logistic map type and the tent map type circuits. These chaotic noise generators are compact as compared with the other circuits. The results show that the successful chaotic operations of the circuits because of the positive Lyapunov number. We calculate the Lyapunov exponents to certify the results of the chaotic operations. However, it is hard to estimate its accurate number for noisy data using the conventional method. And hence, we propose the modified calculation of the Lyapunov exponent for noisy data. These two circuits are expected to be utilized for various applications.

Synchronization between two fully stretching piecewise affine Markov maps in the usual master-slave configuration has been proven to be possible in some interesting 2-dimensional and 3-dimensional cases. Aim of this contribution is to make a further step in the study of this phenomenon by showing that, if the two systems synchronize, the probability of having a certain synchronization time is bounded from above by an exponentially vanishing distribution. This result gives some formal ground to the numerical evidence shown in [2].

We here consider an extension of the validity of classical criteria ensuring the robustness of the statistical features of discrete time dynamical systems with respect to implementation inaccuracies and noise. The result is achieved by proving that, whenever a discrete time dynamical system is robust, all the discrete time dynamical systems topologically conjugate with it are also robust. In particular, this result offer an explanation for the stochastic robustness of the logistic map, which is confirmed by the reported experimental measurements.

In this study, a coupled chaotic circuits network is realized by real circuit elements. By using a simple circuit converting generating spatial patterns to digital signal, irregular self-switching phenomenon of the appearing patterns can be observed as real physical phenomenon.

In order to demodulate a Differential Chaos Shift Keying (DCSK) signal, the energy carried by the received chaotic signal must be determined. Since a chaotic signal is not periodic, the energy per bit carried by the chaotic signal can only be estimated, even in the noise-free case. This estimation has a non-zero variance that limits the attainable data rate. In this paper the DCSK technique is combined with frequency modulation in order to overcome the estimation problem and to improve the data rate of DCSK modulation.

In this paper some new results for analog hardware realization of secure communication system using chaos synchronization have been presented. In particular the effect of the use of transmission line as channel has been considered assuming practical implementation. The influence of the loss of transmission line and mismatching on synchronization has been investigated in chaotic systems based on the Pecora-Carroll concept. It has been shown that desynchronization due to loss can be checked by using an amplifier with appropriate gain. Moreover the bit error rate (BER) has been evaluated in a digital communication system based on the principle of chaotic masking.

In the paper a new type of Multilayer Perceptron, developed in Quaternion Algebra, is adopted to realize short-time prediction of chaotic time series. The new introduced neural structure, based on MLP and developed in the hypercomplex quaternion algebra (HMLP) allows accurate results with a decreased network complexity with respect to the real MLP. The short term prediction of various chaotic circuits and systems has been performed, with particular emphasys to the Chua's circuit, the Saito's circuit with hyperchaotic behaviour and the Lorenz system. The accuracy of the prediction is evaluated through a correlation index between the actual predicted terms of the time series. A comparison of the performance obtained with both the real MLP and the hypercomplex one is also reported.

Deterministic nonlinear prediction is applied to both artificial and real time series data in order to investigate orbital-instabilities, short-term predictabilities and long-term unpredictabilities, which are important characteristics of deterministic chaos. As an example of artificial data, bimodal maps of chaotic neuron models are approximated by radial basis function networks, and the approximation abilities are evaluated by applying deterministic nonlinear prediction, estimating Lyapunov exponents and reconstructing bifurcation diagrams of chaotic neuron models. The functional approximation is also applied to squid giant axon response as an example of real data. Two metnods, the standard and smoothing interpolation, are adopted to construct radial basis function networks; while the former is the conventional method that reproduces data points strictly, the latter considers both faithfulness and smoothness of interpolation which is suitable under existence of noise. In order to take a balance between faithfulness and smoothness of interpolation, cross validation is applied to obtain an optimal one. As a result, it is confirmed that by the smoothing interpolation prediction performances are very high and estimated Lyapunov exponents are very similar to actual ones, even though in the case of periodic responses. Moreover, it is confirmed that reconstructed bifurcation diagrams are very similar to the original ones.

Discrete-time chaotic circuits realizing a tent map and a Bernoulli map are synthesized using switched-current (SI) techniques. For these proposed circuits, simulations are performed concerning the return maps and bifurcation trees. The theoretical analysis is carried out to predict the bifurcation tree under the existence of the nonidealities in the return map. This analysis has been done by assuming the return maps to be piecewise linear. The proposed circuits are built with commerciallyavailable IC's. And their return maps and bifurcation trees are measured in the experiment. The design formulas are obtained for the bifurcation trees and they are confirmed by the simulation results. The proposed circuits are integrable by a standard BiCMOS technology.

In this study, multimode chaos observed from two coupled chaotic oscillators with hard nonlinearities is investigated. At first, a simple chaotic oscillator with hard nonlinearities is realized. It is confirmed that in this chaotic oscillator the origin is always asymptotically stable and that the solution, which is excited by giving relatively large initial conditions, undergoes period-doubling bifurcations and bifurcates to chaos. Next, the coexistence of four different modes of oscillations are observed from two coupled chaotic oscillators with hard nonlinearities by both of circuit experiments and computer calculations. One of the modes of oscillation is a nonresonant double-mode oscillation and this oscillation is stably generated even in the case that oscillation is chaotic. Namely, for this oscillation mode, chaotic oscillation and periodic oscillation can be simultaneously excited. This phenomenon has not been reported yet, and we name this phenomenon as double-mode chaos. Finally, the beat frequency of the double-mode chaos is confirmed to be changed by varying the value of the coupling capacitor.

Secure communications via chaotic synchronization is experimentally demonstrated using 3-pieces of commercial integrated circuit phase-locked loops, MC14046. Different from the conventional chaotic synchronization secure communication systems where one channel is used, our system uses two channels to send one signal to be concealed. Namely, one channel is used to send a synchronizing chaotic signal. The other channel is used to send the informational signal superimposed on the chaotic masking signal at transmitter side. The synchronizing chaotic signal is applied as a common input to two identical PLL's located at both transmitter and receiver sides. It has been shown previously by us that the VCO inputs of almost identical two PLL's driven by a common chaotic signal become chaotic, and synchronized with each other. This synchronization is only possible for those who knows exact internal configuration and exact parameter values of the PLL at transmitter side. Therefore, we can use the synchronized VCO input signal as a masking signal which can be used as a key for secure communications. The advantage of this method compared to the previous one channel method is that informational signal frequency range does not affect the quality of recovered signal. Our experiments demonstrate good masking and recovery characteristics for sinusoidal, triangular, and square waves.

The dynamics of an artificial neural network derived from a biological system, and its two applications to engineering problems are examined. The model has a multi-layer structure simulating the primary and secondary components in the olfactory system. The basic element in each layer is an oscillator which simulates the interactions between excitatory and inhibitory local neuron populations. Chaotic dynamics emerges from interactions within and between the layers, which are connected to each other by feedforward and feedback lines with distributed delays. A set of electroencephalogram (EEG) obtained from mammalian olfactory system yields aperiodic oscillation with 1/f characteristics in its FFT power spectrum. The EEG also reveals abrupt state transitions between a basal and an activated state. The activated state with each inhalation consists of a burst of oscillation at a common time-varying instantaneous frequency that is spatially amplitude-modulated (AM). The spatial pattern of the activated state seems to represent the class of the input ot the system, which simulates the input from sensory receptors. The KIII model of the olfactory system yields sustained aperiodic oscillation with "1/f" spectrum by adjustment of its parameters. Input in the form of a spatially distributed step funciton induces a state transition to an activated state. This property gives the model its utility in pattern classification. Four different methods (SD, RMS, PCA and FFT) were applied to extract AM patterns of the common output wave forms of the model. The pattern classification capability of the model was evaluated, and synchronization of the output wave form was shown to be crucial in PCA and FFT methods. This synchronization has also been suggested to have an important role in biological systems related to the information extraction by spatiotemporal integration of the output of a transmitting area of cortex by a receiving area.

In this study, a ring of simple chaotic circuits coupled by inductors is investigated. An extremely simple three-dimensional autonomous circuit is considered as a chaotic subcircuit. By carrying out circuit experiments and computer calculations for two, three or four subcircuits case, various synchronization phenomena of chaos are confirmed to be stably generated. For the three subcircuits case, two different synchronization modes coexist, namely in-phase synchronization mode and three-phase synchronization mode. By investigating Poincar map, we can see that two types of synchronizations bifurcate to quasi-synchronized chaos via different bifurcation route, namely in-phase synchronization undergoes period-doubling route while three-phase synchronization undergoes torus breakdown. Further, we investigate the effect of the values of coupling inductors to bifurcation phenomena of two types of synchronizations.

In this paper, we demonstrate how Yamakawa's chaotic chips and Chua's circuits can be used to implement a secure communication system. Furthermore, their performance for the secure communication is discussed.

In this paper, bifurcation and chaotic behavior which occur in simple looped MOS inverters with high speed operation are described. The most important point in this work is to change a nonlinear transfer characteristic of a MOS inverter to the nonlinearity generating a chaos. Three types of circuits which include four, three and one MOS inverters, respectively, are proposed. A switched capacitor (SC) circuit to operate sampling holding is added in the loop in each of the circuits. The bifurcation and chaotic behavior have been found along with a variation of an external input, and/or a sampling clock frequency. The bifurcation and chaotic behavior of the proposed simple looped MOS inverters are verified by employing SPICE circuit simulator as well as the experiments. For the first type of four looped CMOS inverters, Lyapunov exponent λ which has the positive regions for the chaotic behavior can be calculated by use of the fitting nonlinear function synthesized from two sigmoid functions. For the second type of three looped CMOS inverters and the third type of one looped MOS inverter, the nonlinear charge/discharge characteristics of the hold capacitor in the SC circuit is utilized efficiently for forming the nonlinearity generating the bifurcation and chaotic behavior. Their bifurcation can be generated by the sampling clock frequency parameter which is controlled easily.

We introduce recurrent networks that are able to learn chaotic maps, and investigate whether the neural models also capture the dynamical invariants (Correlation Dimension, largest Lyapunov exponent) of chaotic time series. We show that the dynamical invariants can be learned already by feedforward neural networks, but that recurrent learning improves the dynamical modeling of the time series. We discover a novel type of overtraining which corresponds to the forgetting of the largest Lyapunov exponent during learning and call this phenomenon dynamical overtraining. Furthermore, we introduce a penalty term that involves a dynamical invariant of the network and avoids dynamical overtraining. As examples we use the Hnon map, the logistic map and a real world chaotic series that correspond to the concentration of one of the chemicals as a function of time in experiments on the Belousov–Zhabotinskii reaction in a well–stirred flow reactor.

The analytic structure of the governing equation for a 2nd order Phase–Locked Loops (PLL) is studied in the complex time plane. By a local reduction of the PLL equation to the Ricatti equation, the PLL equation is analytically shown to have singularities which form a fractal structure in the complex time plane. Such a fractal structure of complex time singularities is known to be characteristic for nonintegrable, especially chaotic systems. On the other hand, a direct numerical detection of the complex time singularities is performed to verify the fractal structure. The numerical results show the reality of complex time singularities and the fractal structure of singularities on a curve.

In this paper, we show that the neural network can approximate the chaotic behavior in nonlinear dynamical system by experimental study. Chaotic neural activities have been reported in many respects including neural network field. On the contrary, can the neural network learn the chaotic behavior? There have been explored the neural network architecture for predicting successive elements of a sequence. Also there have been several studies related to learning algorithms for general recurrent neural networks. But they often require complicated procedure in time calculation. We use simple standard backpropagation for a kind of simple recurrent neural network. Two types of chaotic system, differential equation and difference equation, are examined to compare characteristics. In the experiments, Lorenz equation is used as an example of differential equation. One-dimensional logistic equation and Henon equation are used as examples of difference equation. As a result, we show the approximation ability of chaotic dynamics in difference equation, which is logistic equation and Henon equation, by neural network. To indicate the chaotic state, we use Lyapunov exponent which represents chaotic activity.