Numerical Investigation and Experimental Demonstration of Chaos from Two-Stage Colpitts Oscillator in the Ultrahigh Frequency Range

A hardware prototype of the two-stage Colpitts oscillator employing the microwave BFG520 type transistors with the threshold frequency of 9 GHz and designed to operate in the ultrahigh frequency range (300–1000 MHz) is described. The practical circuit in addition to the intrinsic two-stage oscillator contains an emitter follower acting as a buffer and minimizing the influence of the load. The circuit is investigated both numerically and experimentally. Typical phase portraits, Lyapunov exponents, Lyapunov dimension and broadband continuous power spectra are presented. The main advantage of the two-stage chaotic Colpitts oscillator against its classical single-stage version is in the fact that operating in a chaotic mode it exhibits higher fundamental frequencies and smoother power spectra.     

Experimental demonstration of 1.5 GHz chaos generation using an improved Colpitts oscillator

Experimental study of the ultrahigh-frequency chaotic dynamics generated in an improved Colpitts oscillator is performed. Reliable and reproducible chaos can be generated at the fundamental frequency up to 1.5 GHz using the microwave BFG520 type transistors with the threshold frequency of 9 GHz. By the tuning of the supply voltages, we observe complex nonlinear dynamics like period-one oscillation, period-two oscillation, multiple-period oscillation, and chaotic oscillation. Typical time series, autocorrelation, and broadband continuous power spectrum are presented. Furthermore, compared with the corresponding classical Colpitts oscillator, the main advantage of the improved circuit is in the fact that by operating in a chaotic mode it exhibits higher fundamental frequencies and a lower peak side-lobe level.     

Synchronization of improved chaotic Colpitts oscillators using nonlinear feedback control

The model and the normalized state equations of the novel version of the Colpitts oscillator designed to operate in the ultra-high frequency range are presented. The circuit is investigated numerically and simulations demonstrate chaos in the microwave frequency range. Typical phase portrait, Lyapunov exponent and Lyapunov dimension are calculated using a piece-wise linear approximation of nonlinear I–V characteristic of the bipolar junction transistor. In addition, the feedback controller is applied to achieve chaos synchronization for two identical improved chaotic Colpitts oscillators. In the frame the nonlinear function of the system is used as a nonlinear feedback term for the stability of the error dynamics. Finally, numerical simulations show that this control method is feasible for this oscillator.     

On the analysis of bipolar transistor based chaotic circuits: case of a two-stage colpitts oscillator

We perform a systematic analysis of a system consisting of a two-stage Colpitts oscillator. This well-known chaotic oscillator is a modification of the standard Colpitts oscillator obtained by adding an extra transistor and a capacitor to the basic circuit. The two-stage Colpitts oscillator exhibits better spectral characteristics compared to a classical single-stage Colpitts oscillator. This interesting feature is suitable for chaos-based secure communication applications. We derive a smooth mathematical model (i.e., sets of nonlinear ordinary differential equations) to describe the dynamics of the system. The stability of the equilibrium states is carried out and conditions for the occurrence of Hopf bifurcations are obtained. The numerical exploration reveals various bifurcation scenarios including period-doubling and interior crisis transitions to chaos. The connection between the system parameters and various dynamical regimes is established with particular emphasis on the role of both bias (i.e., power supply) and damping on the dynamics of the oscillator. Such an approach is particularly interesting as the results obtained are very useful for design engineers. The real physical implementation (i.e., use of electronic components) of the oscillator is considered to validate the theoretical analysis through several comparisons between experimental and numerical results.     

Response and stability analysis of piecewise-linear oscillators under multi-forcing frequencies

Steady-state solutions of a piecewise-linear oscillator under multi-forcing frequencies are obtained using the fixed point algorithm (FPA). Stability analysis is also performed using the same technique. For the periodic solutions of a piecewise-linear oscillator with single forcing frequency, the harmonic balance method (HBM) is also used along with the FPA. Although both FPA and HBM generate accurate solutions, it is observed that the HBM failed to converge to solutions in the superharmonic range of the forcing frequency.The fixed point algorithm was also applied to the oscillator under multifrequency excitation. The algorithm proved to be very effective in obtaining torus solutions and in locating corresponding bifurcation thresholds. A piecewise-linear oscillator model of an offshore articulated loading platform (ALP) subjected to two incommensurate wave frequencies is found to exhibit chaotic behavior. A second order Poincaré mapping technique reveals the hidden fractal-like nature of the resulting chaotic response. A parametric study is performed for the response of the ALP.     

Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink

We explore the conservative and dissipative dynamics of a two-degree-of-freedom (2-DoF) system consisting of a linear oscillator and a lightweight nonlinear rotator inertially coupled to it. When the total energy of the system is large enough, the motion of the rotator is, generically, chaotic. Moreover, we show that if the damping of the rotator is sufficiently small and the damping of the linear oscillator is even smaller, then the system passes through a cascade of resonance captures (transient internal resonances) as the total energy gradually decreases. Rather unexpectedly, all these captures have the same principal frequency but correspond to different nonlinear normal modes (NNMs). In each NNM, the rotator is phase-locked into periodic motion with two frequencies. The NNMs differ by the ratio of these frequencies, which is approximately an integer for each NNM. Essentially non-integer ratios lead to incommensurate periods of ??slow?? and ??fast?? motions of the rotator and, thus, to its chaotic behavior between successive resonance captures. Furthermore, we show that these cascades of resonance captures lead to targeted energy transfer (TET) from the linear oscillator to the rotator, with the latter serving, in essence, as a nonlinear energy sink (NES). Since the inertially-coupled NES that we consider has no linearized natural frequency, it is capable of engaging in resonance with the linear oscillator over broad frequency and energy ranges. The results presented herein indicate that the proposed rotational NES appears to be a promising design for broadband shock mitigation and vibration energy harvesting.     

SD振子的设计及非线性特性实验研究初探

具有光滑与不连续转迁特征的SD振子发现和提出以来, 引起了广泛关注. 基于双稳系统大位移特征的测量法困难, SD振子的实验研究还未见报道. 该文提出并设计了具有SD振子系统光滑特征的非线性实验装置, 用实验的方法揭示由几何关系产生的强非线性系统的非线性动力学行为. 设计的非线性实验装置基本振动参数均有良好的可调性和可测量性, 对SD振子在不同频率及幅值的简谐激励作用下的非线性动力学响应进行了实验研究. 为克服大位移测量难题, 研究采用高速摄像机采集振子振动视频信号并进行分析. 结果表明, SD振子系统在一定的参数条件下会产生周期振动、周期5振动及混沌运动等复杂非线性动力学现象, 在相同实验参数条件下进行了数值仿真, 仿真结果与实验结果一致.      

On the Start-Up Behavior and Steady-State Oscillation of Singularly Perturbed Harmonic Oscillators

In this paper we present an analytic methodology for the analysis of a class of electrical harmonic oscillators. We combine geometric methods with the theory of singularly perturbed systems, which we use as tool for reduced order modeling. So we are able to define an easy to use formula in order to reduce an oscillatory system to center manifold. Thus we get a model for the start-up behavior as well as for the steady-state oscillation of sinusoidal oscillators. Furthermore, we demonstrate our technique by means of the Clapp oscillator which is an important member of the Colpitts, Clapp and Pierce oscillator family.     

Experimental study of regular and chaotic transients in a non-smooth system

This paper focuses on thoroughly exploring the finite-time transient behaviors occurring in a periodically driven non-smooth dynamical system. Prior to settling down into a long-term behavior, such as a periodic forced oscillation, or a chaotic attractor, responses may exhibit a variety of transient behaviors involving regular dynamics, co-existing attractors, and super-persistent chaotic transients. A simple and fundamental impacting mechanical system is used to demonstrate generic transient behavior in an experimental setting for a single degree of freedom non-smooth mechanical oscillator. Specifically, we consider a horizontally driven rigid-arm pendulum system that impacts an inclined rigid barrier. The forcing frequency of the horizontal oscillations is used as a bifurcation parameter. An important feature of this study is the systematic generation of generic experimental initial conditions, allowing a more thorough investigation of basins of attraction when multiple attractors are present. This approach also yields a perspective on some sensitive features associated with grazing bifurcations. In particular, super-persistent chaotic transients lasting much longer than the conventional settling time (associated with linear viscous damping) are characterized and distinguished from regular dynamics for the first time in an experimental mechanical system.     

CHAOTIC BEHAVIOUR OF FORCED OSCILLATOR CONTAINING A SQUARE NONLINEAR TERM ON PRINCIPAL RESONANCE CURVES

ＣＨＡＯＴＩＣＢＥＨＡＶＩＯＵＲＯＦＦＯＲＣＥＤＯＳＣＩＬＬＡＴＯＲＣＯＮＴＡＩＮＩＮＧＡＳＱＵＡＲＥＮＯＮＬＩＮＥＡＲＴＥＲＭＯＮＰＲＩＮＣＩＰＡＬＲＥＳＯＮＡＮＣＥＣＵＲＶＥＳＰｅｉＱｉｎ－ｙｕａｎ（裴钦元）（ＣｈａｎｇｓｈａＲａｉｌｗａｙＵｎｉ...     

Strong chaotification and robust chaos in the Duffing oscillator induced by two-frequency excitation

In this work, we demonstrate numerically that two-frequency excitation is an effective method to produce chaotification over very large regions of the parameter space for the Duffing oscillator with single- and double-well potentials. It is also shown that chaos is robust in the last case. Robust chaos is characterized by the existence of a single chaotic attractor which is not altered by changes in the system parameters. It is generally required for practical applications of chaos to prevent the effects of fabrication tolerances, external influences, and aging that can destroy chaos. After showing that very large and continuous regions in the parameter space develop a chaotic dynamics under two-frequency excitation for the double-well Duffing oscillator, we demonstrate that chaos is robust over these regions. The proof is based upon the observation of the monotonic changes in the statistical properties of the chaotic attractor when the system parameters are varied and by its uniqueness, demonstrated by changing the initial conditions. The effects of a second frequency in the single-well Duffing oscillator is also investigated. While a quite significant chaotification is observed, chaos is generally not robust in this case.

     

Chaotic motion and internal resonance of forced surface waves in a water-filled circular basin

The objectives of this paper are to investigate the chaotic motions and internal resonance of nonlinear surface waves generated by a harmonic vibration applied to the side wall of a water-filled circular basin. The harmonic forcing consists of a component with a frequency near twice a fundamental resonance frequency of the basin and a smaller component with a frequency near the fundamental frequency. The amplitude equation for the excited eigenmode corresponding to the fundamental frequency is derived and the existence of chaotic motion of this equation is studied by Melnikov method. At certain critical radii of the basin, twice the fundamental frequency is also a resonance frequency and the so-called internal resonance takes place. Eigenmode corresponding to this resonance frequency can also be excited.     

Conjugate resonances and bifurcations in nonlinear systems under biharmonical excitation

Using a bistable oscillator described by a Duffing equation as an example, resonances caused by a biharmonical external force with two different frequencies (the so-called vibrational resonances) are considered. It is shown that, in the case of a weakly damped oscillator, these resonances are conjugate; they occur as either the low and high frequency is varied. In addition, the resonances occur as the amplitude of the high-frequency excitation is varied. It is also shown that the high-frequency action induces the change in the number of stable steady states; these bifurcations are also conjugate, and are the cause of the seeming resonance in an overdamped oscillator.     

Relaxation oscillations,subharmonic orbits and chaos in the dynamics of a linear lattice with a local essentially nonlinear attachment

We study the dynamic interactions between traveling waves propagating in a linear lattice and a lightweight, essentially nonlinear and damped local attachment. Correct to leading order, we reduce the dynamics to a strongly nonlinear damped oscillator forced by two harmonic terms. One of the excitation frequencies is characteristic of the traveling wave that impedes to the attachment, whereas the other accounts for local lattice dynamics. These two frequencies are energy-independent; a third energy-dependent frequency is present in the problem, characterizing the nonlinear oscillation of the attachment when forced by the traveling wave. We study this three-frequency strongly nonlinear problem through slow-fast partitions of the dynamics and resort to action-angle coordinates and Melnikov analysis. For damping below a critical threshold, we prove the existence of relaxation oscillations of the attachment; these oscillations are associated with enhanced targeted energy transfer from the traveling wave to the attachment. Moreover, in the limit of weak or no damping, we prove the existence of subharmonic oscillations of arbitrarily large periods, and of chaotic motions. The analytical results are supported by numerical simulations of the reduced order model.     

Two fundamental modes of disturbances and their development in a plane plume above a horizontal heat source

Experimental investigations are conducted on development of the disturbances which appear in the transition to turbulence in a natural convection plume above a horizontal line heat source in air. Both the power spectra of velocity and temperature in natural transition show that there seem to be two fundamental modes of disturbances. One is an outstanding peak about 0.8 Hz and the other a small one about 1.1 Hz in the spectra. The disturbances of these fundamental frequencies are observed as anti-symmetric modes around the entrance to the transition region. The disturbance of the first fundamental frequency is a selectively amplified anti-symmetric mode in that area. In contrast, the disturbance of the second fundamental frequency is thought to be originated from a symmetric mode and then transformed into an anti-symmetric mode of the same frequency during its growth.     

A chaotic memcapacitor oscillator with two unstable equilibriums and its fractional form with engineering applications

A novel charge-controlled memcapacitor 3D chaotic oscillator with two unstable equilibriums is proposed. Various dynamic properties of the proposed system are derived and investigated to show the existence of chaotic oscillations. Fractional-order analysis of the chaotic oscillator shows that the maximum value for the largest positive Lyapunov exponent is exhibited in fractional order. Adomian decomposition method is used to discretize the fractional-order system. Field-programmable gate arrays are used to realize the proposed oscillator. In addition, random number generator is designed by employing this novel chaotic system in its fractional-order form.     

Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations

Stable bifurcating solutions may appear in an autonomous time-delayed nonlinear oscillator having quadratic nonlinearity after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. For the corresponding non-autonomous time-delayed nonlinear oscillator, the dynamic interactions between the periodic excitation and the stable bifurcating solutions can induce resonant behaviour in the forced response when the forcing frequency and the frequencies of Hopf bifurcations satisfy certain relationships. Under hard excitations, the forced response of the time-delayed nonlinear oscillator can exhibit three types of secondary resonances, which are super-harmonic resonance at half the lower Hopf bifurcation frequency, sub-harmonic resonance at two times the higher Hopf bifurcation frequency and additive resonance at the sum of two Hopf bifurcation frequencies. With the help of centre manifold theorem and the method of multiple scales, the secondary resonance response of the time-delayed nonlinear oscillator following two-to-one resonant Hopf bifurcations is studied based on a set of four averaged equations for the amplitudes and phases of the free-oscillation terms, which are obtained from the reduced four-dimensional ordinary differential equations for the flow on the centre manifold. The first-order approximate solutions and the nonlinear algebraic equations for the amplitudes and phases of the free-oscillation terms in the steady state solutions are derived for three secondary resonances. Frequency-response curves, time trajectories, phase portraits and Poincare sections are numerically obtained to show the secondary resonance response. Analytical results are found to be in good agreement with those of direct numerical integrations.     

A sensitive and rapid bridge for the study of magnetic susceptibilities at frequencies from 200 to 106 Hz

Summary A new piece of equipment is described for measuring / ₀ and / ₀ as a function of field (0 to 4250 Oe), temperature (1.2°K to room temperature) and frequency (200 Hz to 1 MHz). It is about ten times more sensitive than the Hartshorn bridge used in Leiden²⁾ and it is more convenient to operate as the measuring procedure is automatic after initial adjustments have been made. The main component is a bridge circuit of four inductors, built as closely similar to each other as possible. The output from the bridge goes to two phase sensitive detectors which monitor the two outputs, one inphase, one /2 out of phase, of the bridge. A heterodyne system is used where the input signal to the bridge is obtained by mixing the output from a variable frequency oscillator with that from a 1.5 MHz oscillator and taking the difference frequency, which is phase locked to a master oscillator. The output from the bridge, after preamplification, is mixed with a second output from the variable oscillator and the difference taken again. This gives a 1.5 MHz signal modulated by the magnetic effects in the bridge which is used in the two phase-sensitive detectors. Their output is recorded on an x–y writer. The bridge needs only be balanced to an output of about 50 mV because the sample is moved between two coils and the difference voltage is measured. The use of the same equipment to measure relaxation times longer than 100 ms is also described.Communication No. 349a from The Kamerlingh Onnes Laboratorium, Leiden, The Netherlands     

A Theoretical and Experimental Implementation of a Control Method Based on Saturation

A novel approach for implementing an active nonlinear vibration absorber is presented. The absorber, which is built in electronic circuitry, takes advantage of the saturation phenomenon that occurs when two natural frequencies of a system with quadratic nonlinearities are in the ratio of two-to-one. When the system is excited at a frequency near the higher natural frequency, there is a small ceiling for the system response at the higher frequency and the rest of the input energy is channeled to the low-frequency mode.A working model of using saturation to suppress the vibrations of a rigid beam connected to a DC motor has been built. An electronic oscillator is built, and its frequency is set at one-half the frequency of the beam. The output from a sensor on the beam is multiplied by the output from the electronic oscillator and a suitable gain, and the result is used as the forcing term for the oscillator. At the same time, the output from the oscillator is squared and multiplied by a suitable gain, and that result is used as the input to the motor. The oscillator/actuator and the beam act as the two modes of a two-degree-of-freedom quadratically coupled system with a 2:1 autoparametric resonance. When the beam is excited by a harmonic force, its motion quickly becomes saturated, and most of the energy imparted to the beam by the harmonic force is transferred to the electronic circuit and from there to the actuator. Thus, the harmonic force is made to work against itself. As a result, the motion of the beam always remains small.     

Coupled Chaotic Colpitts Oscillators: Identical and Mismatched Cases

A system consisting of two linearly coupled chaotic Colpitts oscillators is considered. Two different coupling configurations, namely coupled collector nodes (C–C) and coupled emitter nodes (E–E) have been investigated. In addition to identical oscillators the case of mismatched circuits has been studied. Specifically the influence of the transistor parameter mismatch has been analyzed. The relative synchronization error has been estimated for different mismatch levels provided the coupling coefficient is twice larger than the synchronization threshold. Illustrative experimental results, including phase portraits and synchronization error are presented.