Attractors demonstrating randomly transitional (or chaotic) phenomena usually exhibit the following three outstanding properties:
·Sensitive dependence on initial conditions.
·Fractal structure of alpha-branches of principal saddles.
·Transitive property, i.e., one trajectory or sequence of images can come arbitrarily close to every point in the attractor.
Among the three properties given above, the third property is always observed in simulation experiments. However, rigorous mathematical demonstration has not been provided yet.
It supports the proposition that the omega branches of the principal saddle in a chaotic attractor must be basin-filling Peano curves.

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Electronic oscillators have to be described by nonlinear differential equations with a limit cycle where analytical solutions are rarely predictable. Therefore, design concepts of oscillators are not available where a designer can start with a complete set of specifications of the desired oscillator. Only linear aspects of the oscillator design can be solved in a reasonable manner. In this paper we present a mathematical concept that is appropriate to construction nonlinear differential equations with a limit cycle. For this purpose we use a concept from theoretical physics, the so-called Nambu mechanics, and add canonical dissipation. It can be shown that a limit cycle arises under certain conditions. Based on the corresponding system of differential equations a SIMULINK model is constructed that can be the basis of a circuit realization.

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The response of some oscillatory circuit to a periodic singular input wave, possessing “pauses” that are not directly seen via the frequency spectrum of the oscillations, but very simply result from the reflection symmetry of the input waveform and that of the forced describing equation, is considered. The symmetry argument is similar for linear and nonlinear (quasi-linear) versions of the oscillatory circuit or equation, and the simplicity of the argument is a noticeable methodological point. The pauses are not ideal; the inevitable power losses in the circuit cause breaking of the reflection symmetry of the response function, and some relatively weak oscillations occur in the pauses. The pause oscillations can be resonantly suppressed with respect to the non-pause oscillations. The discussion is oriented towards scientists and teachers.

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The nanoscale memristor is a serious candidate to become the core element of novel ultra-high density low-power non-volatile memories and innovative pattern recognition systems based upon oscillatory associative and dynamic memories. Furthermore, this peculiar device also has the potential to capture the behavior of a biological synapse more efficiently and accurately than any conventional electronic emulator since it exhibits the unique capability of performing computation and storing data at the same physical location and at same time. In addition, it has a flux-controlled conductance which is analogous to the ionic flow-controlled synaptic weight. This chapter gives some insight into the mechanisms underlying the emergence of synchronization between two oscillatory cells coupled through an ideal memristor. The investigations show that in some cases the nonlinear dynamics of the memristor play a key role in the development of synchronous oscillations in the two oscillators. This work sheds light on some aspects of the nonlinear behavior of the still largely unexplored memristor, which is doomed to make an impact in integrated circuit design in the years to come.

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It is shown that the Muthuswamy Chua memristor circuit is chaotic in the topological sense. A rigorous analysis of the circuit is performed using interval analysis based tools.

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We consider the methods for guaranteed computations of solutions for nonlinear parabolic initial-boundary value problems. First, in order to make the basic principle clear, we briefly introduce the numerical verification methods of solutions for elliptic problems which we have developed up to now. Next, under some fundamental procedures of verification for parabolic problems based on the fixed point theorem with Newton's method, we describe a summary of our methods including additional new technique which could yield some improvements. The main contents of the paper consist of the guaranteed a posteriori estimates for the linearized inverse operators of parabolic type. In order to confirm the effectiveness of our methods, we give some numerical examples for the guaranteed bounds of iverse operators as well as give some prototype results for numerical verification of solutions of nonlinear parabolic problems. Moreover, we will mention an extention of the present technique to the verification of time-periodic solutions.

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An algorithm is presented for computing verified and accurate bounds for the value of the gamma function over the entire real double precision floating-point range. It means that for every double precision floating-point number x except the poles -k for 0 ≤ k ∈ N the true value of Γ(x) is included within an almost maximally accurate interval with floating-point bounds. One motivation to write this note are some erroneous results in Matlab's gamma function. The application to interval arguments x ∈ IF, thus enclosing the range of Γ over x, is discussed as well.

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Static charge pump mismatch can cause fractional-N frequency synthesizers to exhibit both an elevated in-band noise floor and in-band spurious tones (spurs). To first order, the transfer characteristic of the phase-frequency detector/charge pump (PFD/CP) is piecewise-linear with two segments. The mismatch-related nonidealities can be alleviated by offsetting the operating point so that the phase excursions at the input to the PFD/CP lie completely within one segment. By considering a simplified two-segment piecewise-linear model of the PFD/CP, we show how a combination of delay and/or current offset can be used to combat mismatch. We illustrate our analysis using a simplified Matlab model and confirm its predictions by simulation using the CppSim behavioral simulator.

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This paper presents new insights and a review of recent results on the application of phase models to phase noise analysis in nonlinear oscillators. It is shown that an approach based on stochastic processes and the theory of stochastic differential equations offers several advantages. It is also shown that white noise is responsible for phase diffusion and frequency shift in oscillators, that is, the phase noise problem is best described as a convection diffusion process. An approach for the derivation of a reduced phase model and the associated Fokker-Planck equation is presented. These equations allow an easy determination of fundamental properties as the expectation value for the oscillation frequency, the probability distribution of the phase, the correlation function and the power spectral density.

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Neuromorphic systems are designed by mimicking or being inspired by the nervous system, which realizes robust, autonomous, and power-efficient information processing by highly parallel architecture. It is a candidate of the next-generation computing system that is expected to have advanced information processing ability by power-efficient and parallel architecture. A silicon neuronal network is a neuromorphic system with a most detailed level of analogy to the nervous system. It is a network of silicon neurons connected via silicon synapses;they are electronic circuits to reproduce the electrophysiological activity of neuronal cells and synapses, respectively. There is a trade-off between the proximity to the neuronal and synaptic activities and simplicity and power-consumption of the circuit. Power-efficient and simple silicon neurons assume uniform spikes, but biophysical experimental data suggest the possibility that variety of spikes given to a synapse is playing a certain role in the information processing in the brain. In this article, we review our design approach of silicon neuronal networks where uniform spikes are not assumed. Simplicity of the circuits is brought by mathematical techniques of qualitative neuronal modeling. Though it is neither simpler nor low-power consuming than above silicon neurons, it is expected to be more appropriate for silicon neuronal networks applied to brain-morphic computing.

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The great majority of currently used computational models and devices are using discrete computation. Discrete in time, in value , in parameters. In nature, on the contrary, like in the mammalian retina, difficult tasks are solved with simplicity, low power and elegance, in a wavelike manner, without discretization. Spatial-temporal event detection is a prototype problem in many sophisticated problems of recognition, identification, associative memory, machine-vision, multimodal sensory systems, navigation, and control. In this paper we introduce how continuous spatial-temporal dynamics can be used as algorithmic tools for computation and show a few examples about their usage in practice. These techniques introduce a new kind of thinking also in the field on nonlinear dynamical systems, in general.

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