George L. Johnston

Amplitude-mediated chimera states

We investigate the possibility of obtaining chimera state solutions of the nonlocal complex Ginzburg-Landau equation (NLCGLE) in the strong coupling limit when it is important to retain amplitude variations. Our numerical studies reveal the existence of a variety of amplitude-mediated chimera states (including stationary and nonstationary two-cluster chimera states) that display intermittent emergence and decay of amplitude dips in their phase incoherent regions. The existence regions of the single-cluster chimera state and both types of two-cluster chimera states are mapped numerically in the parameter space of C(1) and C(2), the linear and nonlinear dispersion coefficients, respectively, of the NLCGLE. They represent a new domain of dynamical behavior in the well-explored rich phase diagram of this system. The amplitude-mediated chimera states may find useful applications in understanding spatiotemporal patterns found in fluid flow experiments and other strongly coupled systems.

Collective dynamics of delay-coupled limit cycle oscillators

Coupled limit cycle oscillators with instantaneous mutual coupling offer a useful but idealized mathematical paradigm for the study of collective behavior in a wide variety of biological, physical and chemical systems. In most real-life systems however the interaction is not instantaneous but is delayed due to finite propagation times of signals, reaction times of chemicals, individual neuron firing periods in neural networks etc. We present a brief overview of the effect of time-delayed coupling on the collective dynamics of such coupled systems. Simple model equations describing two oscillators with a discrete time-delayed coupling as well as those describing linear arrays of a large number of oscillators with time-delayed global or local couplings are studied. Analytic and numerical results pertaining to time delay induced changes in the onset and stability of amplitude death and phase-locked states are discussed. A number of recent experimental and theoretical studies reveal interesting new directions of research in this field and suggest exciting future areas of exploration and applications.

Driven response of time delay coupled limit cycle oscillators

We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. For an appropriate choice of time delay, synchronization can occur with infinitesimal forcing amplitudes even at off-resonant driving. The system is also found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Numerical simulations with a large number of coupled driven oscillators display similar behavior. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications. 2003 Elsevier B.V. All rights reserved.

Cylindrical Brillouin Flow In Relativistic Smooth-Bore Magnetrons

A macroscopic cold-fluid model is used to determine the influence of cylindrical effects on the operating range and properties of the electron flow in relativistic smooth-bore magnetrons. Assuming operation at Brillouin flow, it is found that cylindrical effects (such as the centrifugal force on an electron fluid element) can significantly modify several features of the equilibrium flow and diode operating range relative to the case of planar flow.

Hamiltonian structure of particle motion in an ideal helical wiggler with guide field

Abstract The ideal helical wiggler with guide field is shown to possess an integrable hamiltonian. Explicit generating functions are presented for the canonical transformation to action-angle variables.

Amplitude Death, Synchrony, and Chimera States in Delay Coupled Limit Cycle Oscillators

In this chapter we will discuss the effects of time delay on the collective states of a model mathematical system composed of a collection of coupled limit cycle oscillators. Such an assembly of coupled nonlinear oscillators serves as a useful paradigm for the study of collective phenomena in many physical, chemical, and biological systems and has therefore led to a great deal of theoretical and experimental work in the past [1–6]. Examples of practical applications of such models include simulating the interactions of arrays of Josephson junctions [7, 8], semiconductor lasers [9, 10], charge density waves [11], phase-locking of relativistic magnetrons [12], Belousov–Zhabotinskii reactions in coupled Brusselator models [2, 13–15], and neural oscillator networks for circadian pacemakers [16].

Models of driven relativistic magnetrons with nonlinear frequency-shift and growth-saturation effects

The driven van der Pol equation is widely used to model the behavior of regenerative electronic and microwave oscillators driven by an external locking signal. In the case of high-power microwave oscillators such as relativistic magnetrons nonlinear frequency-shift effects are believed to be important. They have been modeled by inclusion in the van der Pol equation of an additional cubic restoring force (Duffing) term. Use of the slowlyvarying amplitude and phase approximation to study the behavior of the driven van der Pol-Duffing equation has predicted stable single-valued locked behavior within a skewed range of values of the frequency mismatch. For parameter values consistent with the slowly-varying amplitude and phase approximation numerical solutions of the van der Pol-Duffing equation confirm this prediction. For oscillators with high growth rate such as the relativistic magnetron however the slowly-varying amplitude and phase approximation may be unjustified. Furthermore regardless of the satisfaction of the assumptions on which the approximation is based it may fail to predict the occurrence of complicated dynamical behavior of coupled oscillators with important implications for phase locking. Numerical solutions of the driven van der Pol equation show that the time of transient evolution to phase-locked states instead of depending solely on the frequency mismatch as conventionally assumed is also a function of oscillator growth rate and injection power level. Recent work suggests that the form of model oscillator equation appropriate for the magnetron may differ in both nonlinear growth-saturation and frequency terms from the van der Pol-Duffing equation. 1.

Pulse shapes for absolute and convective free electron laser instabilities

This paper contains an analysis of pulse shapes produced by a delta-function disturbance of the equilibrium state of a relativistic electron beam propagating through a constantamplitude helical magnetic wiggler field. Pulse shapes are determined by using the relativistic pinch-point techniques developed by Bers, Ram, and Francis. Two pulses are produced corresponding to a convective upshifted pulse (representing the production of the high-frequency radiation desired in a free electron laser) and a downshifted pulse. The downshifted instability may be convective or absolute, depending upon the beam density and momentum spread. Parameter regimes in which the downshifted instability is convective are investigated. It is found that momentum spreads sufficiently large to suppress the absolute instability reduce the growth rate of the upshifted pulse to negligible values. Pulse shapes computed by using the Raman and Compton approximations are compared with exact pulse shapes. It is found that the Raman approximation should be applied to the downshifted regime for most systems of practical interest.

Models of driven and mutually coupled relativistic magnetrons with nonlinear frequency-shift and growth-saturation effects

The driven van der Pol-Duffing equation has been used to model the behavior of a relativistic magnetron driven by an external locking signal. The authors continue the study of the driven van der Pol-Duffing equation and present initial results of the investigation of coupled van der Pol-Duffing equations as models of mutually coupled relativistic magnetrons. A method is presented for determining the amplitude and phase of a signal in the slowly-varying amplitude approximation in the case that both the signal, X(t), and its time derivative, X(t), are available. When second-order differential equations and coupled systems of such equations are used as models of driven and coupled nonlinear oscillators, both X(t) and X(t) are available. In this case, it is possible to determine the amplitude and phase without averaging over a fast time scale. Thus certain dynamical information is retained that is lost if it is necessary to average over a fast time scale. In the case of two oscillators linearly coupled with time delays in the mutual drive configuration, the slowly varying amplitude and phase approximation has been used in order to simplify the problem. In general, behavior of the oscillator amplitudes, as well as the phase difference between oscillators, must be considered. An essential step in studying this system is the determination of stationary amplitudes and phase difference. In the case of zero frequency mismatch between oscillators and optimum coupling delay phase, stationary amplitudes and coupled power are obtained analytically as functions of coupling quality factor and ratio of oscillator growth rate to natural frequency. Unless these parameters are sufficiently large, the potential increase in coherent power delivered due to coupling will not be realized.

Pulse shapes for absolute and convective cyclotron-resonance-maser instabilities

A linear analysis is presented for pulse shapes produced by a spatial and temporal delta-function disturbance of cyclotron-resonance-maser modes for the case where the initial equilibrium state is free of radiation. A pinch-point analysis based on the theory of R.J. Briggs (1964) and A. Bers (1983) is employed. Numerical and analytical techniques are developed for the straightforward calculation of pinch-point coordinates in a reference frame moving with arbitrary velocity in the axial direction. Examples analyzed include the absolute instability in the waveguide operating mode, in higher harmonics of the operating mode, and in lower-frequency waveguide modes when the operating mode is a higher-order waveguide mode. Effects of waveguide wall resistance on pulse shapes and the effectiveness of such resistance in suppressing or reducing the growth rates of absolute instabilities are also analyzed. >