Abstract
For autooscillators of different nature (e.g. active cells in a human heart under the action of a pacemaker, neurons in brain, spintorque nanooscillators, micro and nanomechanical oscillators, or generating Josephson junctions) a critically important property is their ability to synchronize with each other. The synchronization properties of an auto oscillator are directly related to its sensitivity to external signals. Here we demonstrate that a nonisochronous (having generation frequency dependent on the amplitude) autooscillator with delayed feedback can have an extremely high sensitivity to external signals and unusually large width of the phaselocking band near the boundary of the stable autooscillation regime. This property could be used for the development of synchronized arrays of nonisochronous autooscillators in physics and engineering, and, for instance, might bring a better fundamental understanding of ways to control a heart arrythmia in medicine.
Introduction
The systems with autooscillation properties are common in science and nature starting from a simple pendulum clock and ending with complex neural systems of animals and humans. In general, every autooscillator consists of an active element (source of energy) and a passive oscillating system, which converts the energy obtained from the active element to the energy of oscillations and determines the oscillation frequency. In electronics autooscillators are used for the generation of periodic signals and in digital systems as clocks that synchronize the system operation. Similarly, mechanical oscillators, such as quartz oscillators, play a dominant role in today's time keeping devices^{1}. Autooscillating systems are also well known in biology^{2}, chemistry^{3}, and even economics^{4}.
In applications, one of the most important characteristics of an autooscillator is its generation linewidth which is determined by the autooscillator interaction with noise^{1}. On the other hand, an equally important characteristic of an autooscillator is its ability to synchronize with other autooscillators. This property is important for the understanding of the behavior of coupled ensembles of autooscillators. In a broader sense, synchronization is a property of an ensemble of autooscillators that allows it to behave as a single entity^{5}. In particular, synchronization properties are very important in biological systems, where each cell of a tissue acts as a single autooscillator, while the population of these cells (e.g. the entire organism, or even a group of organisms) acts as a single autooscillating system consisting of an array of coupled and synchronized autooscillators. There are, also, many examples of autooscillator arrays in physics and microwave electronics (see e.g. generating Josephson junctions^{6} or spintorque nanooscillators (STNO)^{7,8,9,10}) for which synchronization properties are extremely important, because individual autooscillators have a relatively low output power and the action of a synchronized array is necessary for practical applications of these devices.
It should be noted, that the generation linewidth of an autooscillator, as well as its phaselocking bandwidth to an external periodic signal and its synchronization bandwidth with other autooscillators, are determined by the same property of an autooscillating system – its sensitivity to an external signal, be it an external noise or a periodic driving signal generated by an external driver or by other autooscillators in the ensemble.
To determine this sensitivity we need, at first, to qualitatively understand how a typical autooscillator reacts on the amplitude and phase fluctuations. In a steadystate autooscillation a flow of energy from an active element, producing a nonlinear negative damping, is compensated by the absorption of this flow by a dissipative element, providing a positive damping, which, in a general case, can also be nonlinear^{7}. Thus, the condition of a zero total damping or, in other words, the condition of a dynamic equilibrium between the energy taken from an active element and absorbed by a dissipative element of an autooscillator, defines the stable limit cycle of an autooscillation. In such a situation, the fluctuations of the autooscillation amplitude (or of the effective radius of the limit cycle) introduced into the system may shift the system trajectory from the limit cycle, but the above mentioned dynamic equilibrium condition will return the system back to the limit cycle after a finite time. The situation is drastically different with the phase fluctuations (or shifts along the limit cycle). Since the system has no mechanism to return to the initial phase, the phase fluctuations perform a random walk and accumulate over time. Thus, the autooscillating systems are particularly sensitive to the phase fluctuations, and the experimentally observed generation linewidth of a typical autooscillator is, for the most part, determined by the phase fluctuations^{11,12,13,14,15,16,17}. Analogously, when the external signal is periodic (e.g. when it is created by an external driver or by another autooscillator) the phase sensitivity mostly determines the phaselocking bandwidth of an autooscillator – the maximum deviation of the frequency of an external driving signal from the generation frequency of the autooscillator at which the autooscillation phase still follows the phase of the external signal^{5,18,19,20,21}.
As it was explained above, the stable limit cycle and, therefore, the steadystate amplitude in an autooscillating system is determined by the nonlinearity of its negative and positive damping. At the same time, the equilibrium autooscillation frequency is determined by the passive oscillating system, which can also be nonlinear, resulting in the nonisochronous property of the autooscillator – dependence of the autooscillation frequency on the autooscillation amplitude. Besides, in many cases the process of the energy transfer from an active element to a passive oscillating system may take much more time then one period of oscillation. In such a situation, typical for the systems with spatially distributed parameters, the dynamics of an autooscillating system can be strongly influenced by the time delay existing in it.
It turns out, that many of the practically interesting autooscillating systems are both nonisochronous and have a delayed feedback. One such example is an optoelectronic oscillator based on a lowloss delay line^{1}. Another example of such an autooscillating system is a heart of a mammal. In a bloodstream of a mammal the arterial pressure plays a role of a signal, while the arterial baroreceptors, vasculature, and central nervous system process this signal in the feedback lines with significant delays^{22}. Also, the frequency of the heartbeat is changing with the heartbeat amplitude, so this autooscillating system is also nonisochronous.
A concept of a nonisochronous autooscillator with delay is also widely used to describe the activity of neurons in the brain of animals. In the neurons inside the brain the large time delays in the feedback are caused by the signal processing in the whole neural network^{2,23,24}.
In biology, one of the most explicit examples of a nonisochronous autooscillating system with delayed feedback is the locomotion of fish and reptiles^{25,26}. The generator in the central nervous system (CNS) of a fish produces periodic oscillations that move through the spinal cord to muscles and resemble waves propagating in a transmission line. Specific receptors capture these waves and send a feedback signal to the CNS.
In all the above presented examples it is very important to understand the detailed behavior of the sensitivity of a nonisochronous autooscillating system with delay to an external signal, as the abnormal values of this sensitivity could cause serious irregularities in the system function, which in biological systems manifest themselves as diseases. For example, the abnormal increase of the phase sensitivity of a human heart causes arrythmia, while its abnormal decrease leads to serious medical conditions after the heart transplantation.
In the framework of a general theory of dynamical systems an autooscillator driven by an external force is described by the equation: where x(t) is the vector describing the state of an autooscillator, function f[x(t)] defines all the internal properties of an autooscillator, and ξ(t) is a weak external driving signal. This equation can be reduced to the socalled phase model: where ϕ(t) and ω_{0} are, respectively, the autooscillation phase and frequency, while X[ϕ(t)] is the socalled sensitivity function or phase response function, that describes the reaction of the autooscillator's phase on the external perturbation^{27,28}.
Unfortunately, this elegant scheme cannot be used directly to describe autooscillators with delayed feedback. Indeed, even in a simplest case of an oscillator with delayed feedback the function f in Eq. (1) becomes dependent on the delayed oscillator's state x(t − T) (T is the delay time), f = f[x(t), x(t − T)]. Thus, the equation for an oscillator with delay can be written in the form of Eq. (1) only by introducing an infinitedimensional state vector and an external force . Respectively, the vector sensitivity function X(ϕ) also becomes infinitelydimensional, which makes phase model Eq. (2) unsuitable for practical calculations.
This difficulty is the main reason why, despite a number of interesting results obtained in the theory of coupled oscillators with delay in the interoscillator coupling^{29,30,31,32}, in the majority of existing theoretical papers describing the dynamics of autooscillating systems (see e.g.^{11,12,13}) the time delay in the feedback loop of an autooscillator itself is either ignored or is considered as a small correction^{31}.
In our current paper we found an autooscillator model which allows one to formally take into account the time delay in an autooscillator feedback loop. In the framework of this model an autooscillator consists of two parts: a linear resonant part, which defines the autooscillating frequency and contains linear elements providing delay in feedback, and a nonlinear part containing an active element that provides negative damping and is responsible for the autooscillator's nonisochronous properties. Using this model it is possible to analytically calculate the sensitivity X[ϕ(t)] to an external signal of a nonisochronous autooscillator with delayed feedback and to evaluate all the nonautonomous properties of such an autooscillator determined by this sensitivity. Although our model is less general than the model (1), it, nonetheless, describes most of nonisochronous autooscillators with delayed feedback that are practically important in physics, biology and electronics.
Results
A model of a nonisochronous autooscillator with delayed feedback
The nonautonomous behavior of a wide range of autooscillators can be described by a simple model of a closed autooscillating loop (see Fig. 1) that consists of a nonlinear amplifier (active element), a linear resonance oscillating system (which may contain an element causing the signal delay, e.g. a delay line), and a source of a relatively weak (compared to the autooscillation amplitude) external signal, which may be either stochastic (noise) or/and periodic (driving signal).
The nonlinear active element, described by the function G(p), where p is the oscillation power, provides the energy flow into the system, while the passive linear oscillating system, described by the linear operator L (i d/dt), determines the autooscillation frequency and provides positive damping, or energy sink. In this approach, the active element is assumed to have an infinite frequency bandwidth, so the output signal of the active element depends only on the instantaneous value of the input signal. The corresponding operator equation written in the time domain and describing such a system has the form: where the function c(t) describes the complex amplitude of the autooscillation at the input of the nonlinear active element, p = c(t)^{2} is the signal power, and the function ξ(t) describes a weak external driving signal acting on the autooscillating loop. Please, note that the dimension of the external signal ξ(t) differs from the dimension of c(t) by the dimension of the linear operator L.
In the frequency domain the operator L (i d/dt) can be described by a transfer function L(ω) that is defined as a Fourier transform of the impulse response function of the oscillating system and can be directly measured experimentally.
We stress, that in the model equation (3) the operator L (i d/dt) describing the oscillating system with delay is linear, so that the nonisochronous properties of the autooscillator (i.e., the dependence of the oscillation frequency ω on the oscillation power p), as well as the nonlinear properties of the active element limiting the autooscillation amplitude and defining the limit cycle, are described by the nonlinear amplifier gain G(p).
The model (3) describes, either directly or after an appropriate change of variables, a vast variety of natural and artificial autooscillating systems. A few particular examples will be given below (see section “Examples”).
In the absence of external perturbations (ξ(t) = 0) we assume the stationary autooscillations at a limit cycle of the autooscillator Fig. 1 to be harmonic, and the stationary solution for the function c(t) has the form: where ϕ(t) = ω_{s}t + ϕ_{0}, p_{s} and ω_{s} are the stationary freerunning autooscillation power and frequency, respectively, and ϕ_{0} is an arbitrary initial phase of the autooscillation. The parameters p_{s} and ω_{s} of the stationary autooscillation are determined by the following amplitude and phase conditions, which directly follow from equation (3): where n = 1, 2, 3, … is an integer number.
In a general case the conditions(5) for stationary autooscillations define many different autooscillation modes which reflects the multimode nature of the autooscillator loop Fig. 1 (see the mode spectrum shown in Fig. 2). Which one of the possible autooscillation modes will be actually generated in the autooscillator is determined by the relative stability properties of the stationary solutions of equation (5), which can be analyzed using the central equation of our model (3).
Sensitivity of an autooscillator to external perturbations
In the presence of a weak external signal ξ(t), both the frequency and the power of the autooscillation will experience slight shifts from their stationary values, and the nonautonomous properties of the perturbed autooscillator will depend on the behaviour of the functions L(ω) and G(p) in the vicinity of the limit cycle, which corresponds to the stable autooscillation mode (p_{s}, ω_{s}). The parameters determining the nonautonomous dynamics of the autooscillator can be obtained by expanding the functions L(ω) and G(p) in a Taylor series near the limit cycle (p_{s}, ω_{s}): Both parameters T and β are, in general, complex quantities that have a simple physical meaning. The real part of T (Re(T)) acts as an effective delay time in the feedback loop. The imaginary part of T (Im(T)) characterizes the slope of the transfer function L(ω) at the frequency ω_{s} of the stationary autooscillation (see Fig. 2). The real part of β (Re(β) = β_{r}) – amplifier nonlinearity, characterizes the dependence of the gain of the active element on the oscillation power. In most cases β_{r} is positive (β_{r} > 0) which means that the amplifier gain decreases with power and this effect limits the autooscillation amplitude. The imaginary part of β (Im(β) = β_{i}) – nonlinear frequency shift, determines the dependence of the autooscillation frequency on power, and, therefore, describes the nonisochronous properties of the autooscillator.
Under the influence of an external timedependent signal the phase ϕ_{0} in equation (4) becomes a function of time. Assuming that the external signal ξ(t) is weak and, therefore, can only slowly change the phase and slightly vary the amplitude of the autooscillation, we derived the following approximate equation describing the phase dynamics of the autooscillation in time: where can be defined as phase sensitivity of a nonisochronous autooscillator to an arbitrary external signal ξ(t). Such a result for the phase model of equation (2) was, of course, expected due to the simple form of equation (4), which means the circular shape of the limit cycle on the phase plane.
In the framework of the approximate phase equation (7) the finding of all the nonautonomous characteristics of an autooscillator, such as the generation linewidth, phaselocking bandwidth to the external periodic signal, or the synchronization frequency band with other similar autooscillators, can be reduced to finding the autooscillator sensitivity . The solution of equation (3) near the autooscillation limit cycle (see “Methods”) leads to the following expression for the autooscillator sensitivity : where β* is the complex conjugate of β.
When the autooscillator sensitivity is known, it is easy, using equation (7) and equation (8), to evaluate the frequency phaselocking bandwidth of an autooscillator to an external periodic signal of the amplitude A: Similarly, using equation (7) and equation (8), it is easy to evaluate the generation linewidth of an autooscillator. In the presence of thermal noise the autooscillator phase ϕ performs a random walk with the phase variance linearly increasing with time and the generation line of the autooscillator has a Lorentzian shape with the linewidth 2Δω (full width at half maximum (FWHM)) evaluated as: where is the spectral density of the noise power calculated at the frequency of the stationary autooscillation. The spectrum of the thermal JohnsonNyquist noise could be written as (see equation (92) in^{7}) where p_{n} is the noise power (power of the oscillations in the system in the thermal equilibrium state) and Γ is the damping parameter of the loop.
One interesting and nontrivial consequence of the general expression (8) is the fact that the autooscillator sensitivity , and, therefore, the phaselocking bandwidth and the generation linewidth, diverge when Re (β*T) = 0. Thus, near this point the autooscillator is very sensitive to the external signal and will easily phase lock to an external driver or easily synchronize with other similar autooscillators. Therefore, the study of the autooscillator behavior in the vicinity of the point where Re (β*T) = 0 is very important for practical applications, and, if such regimes of stable autooscillations exist, they can be used for the synchronization of arrays of autooscillators, even in the case when their autooscillation frequencies are not very close to one another.
Below, we present several examples of application of the general expression (8) to different autooscillating systems.
Applications to simple autooscillating systems
Isochronous (β_{i} = 0) autooscillator generating at resonance (ω_{s} = ω_{0})
The model equation (3) covers a wide range of autooscillators of different physical nature, and it can be shown that in simple particular cases the expression for the generation linewidth (10) following from this model can be reduced to wellknown results.
For the isochronous autooscillator where the oscillation system is a simple LCR circuit with a quality factor Q = ω_{s}/2Γ, where Γ is a damping parameter, the transfer function can be represented in a simple Lorentzian form In this case the generation frequency coincides with the resonance frequency of the oscillating system (LCR circuit), ω_{s} = ω_{0}. Thus, the effective delay time is real and is given by T = 1/Γ = 2Q/ω_{s}. If the phase shift of oscillations at the amplifier output does not depend on power, the nonlinearity β contains only the real part, β = β_{r}, and the function G(p) in the vicinity of the stationary point can be written as: where the amplifier gain K should be chosen to satisfy the amplitude condition of the stationary generation (the first condition in equation (5)), which leads to K = Γ. Thus, the general expression for the autooscillator generation linewidth (10) is reduced to a wellknown result^{7,11}:
Nonisochronous (β_{i} ≠ 0) autooscillator generating at resonance (ω_{s} = ω_{0})
In a nonisochronous autooscillator the nonlinearity parameter has both real and imaginary parts, β = β_{r} + iβ_{i}, and the imaginary part β_{i} describes the nonisochronous properties of the autooscillator. The typical example of such an autooscillator is a spintorque nanooscillator (STNO) which can be described by the equation^{7}: where Γ_{tot} contains both the natural positive linear damping Γ and the positive and negative nonlinear damping Γ_{nl}(p). Separating linear and nonlinear parts in both Γ_{tot}(p) and ω(p) one gets: Then, the linear transfer function L(ω) can be represented in the Lorentzian form (12) and the “amplifier” nonlinear function has the form: Calculating the values for β and T from the above equations, one can get the autooscillator generation linewidth from equation (10) in the form^{7}: where ν = β_{i}/β_{r} = (dω/dp)/(dΓ_{tot}/dp).
Nonisochronous (β_{i} ≠ 0) autooscillator generating off resonance (ω_{s} ≠ ω_{0})
The qualitatively new results are obtained from equations (8–10) when both parameters T and β are complex. A sample equation describing an autooscillator with complex T and β can be written in the form: In the autooscillating system described by equation (19) the linear transfer function L(ω), compared to equation (12), has an additional multiplier which explicitly contains the delay time τ and should be written as while the nonlinear amplifier function G(p) also contains an additional multiplier responsible for nonlinear frequency shift and has the form: Using the above presented expressions for L(ω) and G(p) it is easy to calculate the autooscillator parameters equation (6) at the stationary frequency ω_{s} ≠ ω_{0}: where δω_{0} = ω_{0} − ω_{s}. The amplifier gain K is chosen to provide the stationary generation at the frequency ω = ω_{s} satisfying conditions equation (5). The equation (19) is also convenient for the numerical analysis of the problem.
The condition Re(β*T) = 0 for the singularity in the expression (8) for the phase sensitivity of an autooscillator can be rewritten as an equation of a straight line on the plane (β_{r}, β_{i}): The condition Re(β*T) = 0 (see the solid red line in Fig. 3a) also defines the boundary of the autooscillation stability region in respect to lowfrequency modulation of the generated signal. The other boundaries of the stability region (shown by black solid lines in Fig. 3a) are the boundaries of the stability region in respect to the decay into different autooscillation modes (see Fig. 2). These boundaries are found from the numerical solution of equation (19).
We would like to note, that normally the negative sign of the amplifier nonlinearity β_{r} < 0, corresponding to the case when the amplification gain increases with the increase of the oscillation power p, automatically leads to the instability of the autooscillation regime. However, if β_{i} ≠ 0, the stable autooscillations can exist even if the product β_{i}β_{r} is negative. On the other hand, when the real part of the autooscillator nonlinearity is positive β_{r} > 0 the region of stability of the autooscillations is much wider, and the stable autooscillations are possible for any sign of the product β_{i}β_{r}.
The phaselocking bandwidth of the autooscillator to an external signal (relative to the linear damping Γ of the oscillating system) is shown in Fig. 4 in a logarithmic scale. It is clearly seen from Fig. 4 that as soon as the system gets close to the lowfrequency boundary (23) of the stability region (shown by the solid red line in Fig. 3), the phaselocking bandwidth drastically increases. The similar behavior is demonstrated by the graph of the autooscillator generation linewidth (see Fig. 5). The numerical calculations confirm that the oscillations remain stable when the system approaches the boundary of the stability region.
Two coupled autooscillators
To illustrate the qualitative influence of the increase in autooscillator sensitivity near the line Re(β*T) = 0 on the behavior of coupled autooscillator arrays we considered an elementary example of a coupled pair of practically identical autooscillators that have slightly different generation frequencies, separated by the interval δω_{0} = ω_{0} − ω_{s}, and different initial phases. Both autooscillators in a coupled pair are described by the same model(19), where the external signal in the righthandside part ξ_{1,2}(t) = a_{1,2}c_{2,1}(t) is proportional (with a coupling constant a_{1,2}) to the oscillation amplitude of the other oscillator.
We numerically calculated the phase difference Δϕ between the oscillations of the two coupled oscillators in two different cases (see Fig. 3a): when the parameters of the coupled autooscillators were close to the lowfrequency boundary of the autooscillation stability region (left blue point in Fig. 3a) and when autooscillators parameters were far from that boundary (right red point in Fig. 3a). The evolution of the phase difference in the pair of coupled oscillators with time in these two cases are shown in Figs. 3b and c, respectively, where different colors correspond to different initial phase differences between the oscillators. As one can see, in the first case (Fig. 3b) of large sensitivity any initial phase difference Δϕ between the coupled oscillators is rapidly reduced to zero demonstrating effective synchronization, while in the second case of relatively low sensitivity the initial phase difference grows with time and no synchronization is observed.
It should be noted, that in this particular case the presence of the delay time in the feedback loop, in contrast to the case of synchronization of two autooscillators with a delayed coupling^{29}, does not lead to the existence of several synchronization frequencies and corresponding phase shifts. However, in principle, the delay time in the feedback loop defines the eigenfrequencies of the autooscillator by the equation (5), and can lead to a multimode autooscillation regime.
Discussion
Figs. 4 and 5 demonstrate that the divergence of the phaselocking bandwidth or the generation linewidth of a nonisochronous autooscillator with delayed feedback takes place only near the lowfrequency boundary of the autooscillation stability region defined by the equation (23), while the behavior of the same quantities near all the other boundaries of the autooscillation stability region is regular. Thus, the lowfrequency boundary of the stability region has special importance for the nonautonomous behavior of an autooscillator.
If in an autooscillator we have means to control the magnitude or/and sign of the nonlinear frequency shift β_{i} (or/and the detuning δω_{0} at which the stable autooscillation takes place), we could control the phaselocking bandwidth and the generation linewidth of the autooscillator.
The situations when the magnitude and even the sign of the nonlinear frequency shift could be controlled by changing a certain external parameter of the autooscillator are not rare. In particular, in a spintorque nanooscillator it is possible to change the magnitude and the sign of β_{i} simply by changing the direction or/and the magnitude of the external bias magnetic field (see Fig. 6 in^{7}), or by changing the bias direct current driving the device (see Fig. 1d in^{33}). Of course, for a normal autooscillator with saturation nonlinearity (β_{r} > 0) the minimum linewidth is achieved when the autooscillator is isochronous, β_{i} = 0, which is wellknown from the standard autooscillator theory^{7} (see also Figs. 4 and 5), but if we need to synchronize many autooscillators with different oscillation frequencies or switch an individual oscillator from a coherent autooscillation regime to a regime of generation of noiselike signals having a wide frequency spectrum, we could use the above described regime of offresonance generation in a nonisochronous autooscillator and bring this device close to the lowfrequency boundary of the stability region defined by the condition (23).
In conclusion, we found that in a nonisochronous autooscillator with a timedelayed feedback it is possible to achieve an extremely high sensitivity to the external driving signal, and, therefore, to get an unusually large synchronization band near the lowfrequency boundary of the autooscillations stability region. The understanding of the above described behavior of the autooscillator sensitivity could be also fundamentally important for the understanding of synchronization properties in biological systems and for the better understanding of the nature of some diseases, for example – heart arrythmia.
Methods
Analytical calculations
To derive the effective phase equation (7) and expression (8) for the phase sensitivity, we represented the external signal ξ(t) as a continuous sequence of δkicks, ξ(t) = ∫ξ(t′)δ(t − t′)dt′, evaluated phase response of the oscillator to a single δkick ξ(t′)δ(t − t′), and summed up responses from different kicks. The solution of equation (3) under the influence of one weak δimpulse is close to the freerunning solution and can be easily found using standard perturbation methods, namely, by linearizing equation (3) near the unperturbed solution (4).
To calculate analytically the synchronization bandwith Δω_{PL} equation (9), we used the periodic external signal in the phase equation (7) and looked for stationary phaselocked solutions of the form ϕ(t) = −ω_{e}t + Δϕ, where Δϕ is an arbitrary constant phase shift. The condition for existence of such solutions can be written in the form ω_{e} − ω_{s} < Δω_{PL}, where the phaselocking bandwidth Δω_{PL} is given by the equation (9).
Similarly, substituting white gaussian noise with known statistical properties for the external signal ξ(t) one can get the generation linewidth Δω of the autooscillator (see Supplementary materials for details).
Numerical methods
To check the above derived analytical results the differential equation describing a nonisochronous autooscillator with delay (19) was solved for different forms of the weak external signal ξ(t) using the method of slowly varying amplitudes, where all the frequencies were normalized to the linear damping parameter Γ (or linear relaxation frequency). The equation was solved by a RungeKutta method of the 4th order.
To calculate the phaselocking bandwidth a purely sinusoidal periodic signal of the small amplitude has been added as the righthandside part ξ(t) of equation (19). The phaselocking bandwidth has been obtained by varying the frequency of the external signal and determining the frequency of the autooscillation for each value of the external driving frequency.
To calculate the generation linewidth external white noise ξ(t) in the righthandside part of equation (19) was simulated by random points at regular time intervals with normal distribution of the amplitude. These points were locally interpolated by a 3rd order polynomial for the continuity of the function. The upper cutoff frequency of the noise was chosen to be 4 times larger than the relaxation frequency (Γ) in equation (19), which is the largest characteristic frequency of the model. The correlation function for the phase of autooscillations was determined after solving (19) numerically, and the generation linewidth was obtained from this correlation function.
To find the region of parameters where autooscillations are stable we analyzed the possibility of decay of stationary autooscillations with the frequency ω_{s} into oscillations with the frequencies ω_{s} ± Ω under the action of small perturbations. As it follows from the Lyapunov criteria the autooscillations are stable when Im(Ω) < 0 (see Supplementary materials).
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Acknowledgements
This work is supported by DARPA MTO/MESO grant N660011114114, by the U.S. Army Contract from TARDEC, RDECOM, and by the grant DMR1015175 from the National Science Foundation of the USA.
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Affiliations
Department of Physics, Oakland University, Rochester, Michigan 48309, USA
 Vasil S. Tiberkevich
 , Roman S. Khymyn
 & Andrei N. Slavin
Institute of Magnetism, National Academy of Sciences of Ukraine 03142 Kiev, Ukraine
 Roman S. Khymyn
Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA
 Hong X. Tang
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Contributions
The statement of the problem belongs to A.N.S. and H.X.T. V.S.T. proposed the general framework and theoretical approach of the problem and made calculations of stability diagram. R.S.K. developed analytical results and numerical calculations of the generation linewidth and phaselocking bandwith. All authors analyzed the data and participated in the preparation of the manuscript. A.N.S. and H.X.T. supervised the project.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Roman S. Khymyn.
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