Time-delayed Circuit Research Articles

Snap-Back Repellers and Chaos in Time-delayed Chua's Circuit

The chaotic behavior of one-dimensional, 2-segment and 3-segment piecewise-linear maps is examined by using the concept of snap-back repellers introduced by Marotto and the parameters conditions of existence for snap-back repeller are obtained. Simulation results are presented to show the snap-back repeller, some periodic points and attracting interval cycles with chaotic intervals.

PIV를 이용한 인공심장용 폴리우레탄 인공판막 하류의 유동 측정 : 맥동유동실험

In-vitro flow characteristics downstream of a polyurethane artificial heart valve and a Bj??rk-Shiley Monostrut mechanical valve have been comparatively investigated in pulsatile flow using particle image velocimetry (PIY). With a triggering system and a time-delayed circuit the velocity distributions on the two perpendicular measurement planes downstream of the valves are evaluated at any given instant in conjunction with the opening behaviors of valve leaflets during a cardiac cycle. The regions of stasis and high shear stress can be found simultaneously by examining the entire view of the instantaneous velocity and Reynolds shear stress fields. It is known that high shear stress regions exist at the interface between strong axial jet flows along the wall and vortical flows in the central area distal to the valves. In addition. there are large stagnation or recirculation regions in the vicinity of the valve leaflet. where thrombus formation can be induced by accumulation of blood elements damaged in the high shear stress zones. A correlation between the unsteady flow patterns downstream of the valve and the corresponding opening postures of the polyurethane valve membrane gives useful data necessary for improved design of the frame structure and leaflet geometry of the polyurethane valve.

A discrete-time approach to the steady-state and stability analysis of distributed nonlinear autonomous circuits

We present a direct method for the steady-state and stability analysis of autonomous circuits with transmission lines and generic nonlinear elements. With the discretization of the equations that describe the circuit in the time domain, we obtain a nonlinear algebraic formulation where the unknowns to determine are the samples of the variables directly in the steady state, along with the oscillation period, the main unknown in autonomous circuits. An efficient scheme to build the Jacobian matrix with exact partial derivatives with respect to the oscillation period and with respect to the samples of the unknowns is described. Without any modification in the analysis method, the stability of the solution can be computed a posteriori constructing an implicit map, where the last sample is viewed as a function of the previous samples. The application of this technique to the time-delayed Chua's circuit (TDCC) allows us to investigate the stability of the periodic solutions and to locate the period-doubling bifurcations.

A new technique for chaos prediction in RF circuit design using harmonic-balance commercial simulators

In this paper a new method is presented for the detection of chaotic responses using harmonic balance commercial simulators. The method is applicable to circuits exhibiting homoclinic bifurcation routes. The successive period-doublings, often preceding the formation of homoclinic orbits, are predicted and calculated. This is done by means of a new probe technique that overcomes the harmonic balance difficulties when dealing with subharmonic frequencies. This calculation of the steady-state frequency divided responses has allowed an accurate prediction of the onset of chaos, through a homoclinic orbit, in the time-delayed Chua's circuit (TDCC). The method is specially suitable for circuits containing transmission lines that cannot be analyzed by standard time-domain simulations.

Predicting period-doubling bifurcations and multiple oscillations in nonlinear time-delayed feedback systems

In this work, a graphical approach is developed from an engineering frequency-domain approach enabling prediction of period-doubling bifurcations (PDB's) starting from a small neighborhood of Hopf bifurcation points useful for analysis of multiple oscillations of periodic solutions for time-delayed feedback systems. The proposed algorithm employs higher order harmonic-balance approximations (HBA's) for the predicted periodic solutions of the time-delayed systems. As compared to the same study of feedback systems without time delays, the HBA's used in the new algorithm include only some simple modifications. Two examples are used to verify the graphical algorithm for prediction: one is the well-known time-delayed Chua's circuit (TDCC) and the other is a time-delayed neural-network model.

Qualitative analysis of the dynamics of the time-delayed Chua's circuit

Several variants of the Chua's circuit have been recently proposed in order to enlarge the class of nonlinear phenomena that can be generated by relatively simple circuits. In particular, Sharkovsky et al. proposed the so called time-delayed Chua's circuit (TDCC), where the original lumped LC resonator is substituted by an ideal transmission line, thereby generating an infinite dimensional system, The TDCC has been studied in details in the absence of the capacitor C, the only lumped dynamic element left in the circuit. This paper studies the effects of the presence of C on the dynamics of the circuit. After recasting the circuit equations in a suitable normalized form, their characteristic equation is theoretically investigated and the regions in the parameter space where all the eigenvalues have negative real part are exactly evaluated along with all the possible qualitative eigenvalue distributions. This analysis allows for a qualitative description of the TDCC dynamics in presence of the capacitor C. In particular, it is shown that, for particular sets of circuit parameters, an even small value of C, e.g., a parasitic element, may completely change the behavior of TDCC and that, on the other hand, any TDCC, exhibiting the period-adding phenomenon for C=0, stilt continues to present this phenomenon even if a small capacitor C is added to the circuit.

Predicting chaos through an harmonic balance technique: an application to the time-delayed Chua's circuit

The time-delayed Chua's circuit (TDCC) can be considered as a paradigm for studying chaos in circuits described by difference-differential equations. The dynamics of such circuits cannot be investigated by means of the standard time-domain techniques developed for finite dimensional systems. We show that through a spectral approach the occurrence of periodic limit cycles and of chaotic attractors can be easily predicted, without performing any simulations.

DRY TURBULENCE AND PERIOD-ADDING PHENOMENA FROM A 1-D MAP WITH TIME DELAY

In this tutorial paper, we consider an infinite-dimensional extension of Chua's circuit, as shown in Fig. 1, where the transmission line is lossless. As we shall see, if the capacitance C1 is set to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced, without any approximation, to that of a continuous scalar nonlinear difference equation. This type of equation can lead to space-time chaos which, due to the absence of viscosity in our system, will be termed "dry turbulence". Another interesting property of this system occurs under certain conditions, when the corresponding 1-D map has two segments and is piecewise-linear. The extreme simplicity of this map will allow us to derive, without any approximation, the exact analytical solution of the stability boundaries of stable cycles of every period n. Since the stability region is non-empty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.

Analysis of chaotic behavior in lumped-distributed circuits applied to the time-delayed Chua's circuit

A general method for the analysis of prechaotic and chaotic behaviors in lumped-distributed circuits has been developed. The method is applied to the time-delayed Chua's circuit and the analysis predicts the possibility of the existence of multilevel oscillations and chaotic behavior of the circuit. The proposed procedure predicts the presence of multilevel oscillations which may lead to a chaotic behavior of this circuit. A bifurcation diagram, and phase plane are presented which verify the proposed procedure. >

DRY TURBULENCE FROM A TIME-DELAYED CHUA'S CIRCUIT

In this paper, we consider an infinite-dimensional extension of Chua's circuit (Fig. 1) obtained by replacing the left portion of the circuit composed of the capacitance C2 and the inductance L by a lossless transmission line as shown in Fig. 2. As we shall see, if the remaining capacitance C1 is equal to zero, the dynamics of this so-called time-delayed Chua's circuit can be reduced to that of a scalar nonlinear difference equation. After deriving the corresponding 1-D map, it will be possible to determine without any approximation the analytical equation of the stability boundaries of cycles of every period n. Since the stability region is nonempty for each n, this proves rigorously that the time-delayed Chua's circuit exhibits the "period-adding" phenomenon where every two consecutive cycles are separated by a chaotic region.

SOUND AND MUSIC FROM CHUA'S CIRCUIT

Nonlinear Dynamics have been very inspiring for musicians, but have rarely been considered specifically for sound synthesis. We discuss here the signals produced by Chua's circuit from an acoustical and musical point of view. We have designed a real-time simulation of Chua's circuit on a digital workstation allowing for easy experimentation with the properties and behaviors of the circuit and of the sounds. A surprisingly rich and novel family of musical sounds has been obtained. The audification of the local properties of the parameter space allows for easy determination of very complex structures which could not be computed analytically and would not be simple to determine by other methods. Finally, we have found that the time-delayed Chua's circuit can model the basic behavior of an interesting class of musical instruments.

Chaos from a time-delayed Chua's circuit

By replacing the parallel LC resonator in Chua's circuit by a lossless transmission line that is terminated by a short we obtain a Chua's circuit, whose time evolution is described by a pair of linear partial differential equations with a nonlinear boundary condition. If we neglect the capacitance across the Chua's diode, which is described by a nonsymmetric piecewise-linear V/sub R/-I/sub R/ characteristic, the resulting idealized Chua's circuit is described exactly by a scalar nonlinear difference equation with continuous time, which makes it possible to characterize its associated nonlinear dynamics and spatial chaotic phenomena. From a mathematical view point, circuits described by ordinary differential equations can generate only temporal chaos, whereas the time-delayed Chua's circuit can generate spatial temporal chaos. Except for stepwise periodic oscillations, the typical solutions of the idealized time-delayed Chua's circuit consist of either weak turbulence or strong turbulence, which are examples of ideal (or dry) turbulence. In both cases, we can observe infinite processes of spatial-temporal coherent structure formations. Under weak turbulence, the graphs of the solution tend to sets that are fractals with a Hausdorff dimension between 1 and 2 and is therefore larger than the topological dimension (of sets). Under strong turbulence, the limit oscillations are oscillations whose amplitudes are random functions. This means that the attractor of the idealized time-delayed Chua's circuit already contains random functions, and spatial self-stochasticity phenomenon can be observed. >

Models of musical instruments from Chua's circuit with time delay

We study a new version of Chua's circuit where the linear elements are replaced by a delay line. We show that this circuit is a model of an interesting class of musical instruments, namely, those, like the clarinet, consisting of a massless reed coupled to a passive linear system. The properties and behaviors of this circuit are studied with or without a filter in the feedback loop. In particular, the oscillation conditions are determined, and the respective role of the nonlinearity and of the linear elements are explained. Real-time simulations of the time-delayed Chua's circuit on a digital workstation makes possible easy experimentation with signals and sounds. A surprisingly rich and novel family of periodic and chaotic musical sounds has been obtained. The audification of the local properties of the parameter space permits easy determination of very complex structures that could not be computed analytically and would be difficult to determine by other methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>