Autoresonance in nonlinear systems - davidar/scholarpedia GitHub Wiki
Autoresonance is a fascinating phenomenon of nonlinear physics, where a perturbed nonlinear system is captured into resonance and stays phase-locked with perturbing oscillations (or waves) continuously despite variation of system's parameters. The persistent phase-locking (adiabatic synchronization) means controlled excursion in system's solutions space and frequent emergence of coherent structures. For nearly half a century studies of autoresonance were limited to relativistic gyroresonant wave-particle interactions (starting from Veksler 1945 and McMillan 1945 for particle accelerators), but many new applications of the autoresonance idea and progress in the theory emerged since 1990 in atomic and molecular physics (Meerson and Friedland 1990, Liu et al. 1995, Marcus et al 2004, 2005, Maeda 2007), a variety of dynamical systems with a finite number of degrees of freedom (for a review see Fajans and Friedland 2001), nonlinear waves (Aranson et al 1992, Friedland 1992, 1992a, 1998, Friedland and Shagalov 2003, 2005, Ben-David et al 2006), plasmas (Deutsch et al 1991, Fajans et al 1999, Friedland et al 2006), fluid dynamics (Friedland 1999, Friedland and Shagalov 2000, 2002, Borich and Friedland 2008), and, most recently, superconducting Josephson junctions (Naaman et al 2008) and optics (Barak et al 2009). Some of these new developments in dynamics of resonantly driven nonlinear systems are described below.
How to strongly excite a pendulum by a small perturbation, but without a feedback? Autoresonance yields a simple answer to this question. Consider a driven pendulum described by <math>d^{2}u/dt^{2}+\sin u=\varepsilon \cos \varphi (t)\ ,</math> where the driving perturbation has constant amplitude <math>\varepsilon </math> and slowly varying frequency <math>\omega (t)=d\varphi /dt\ .</math> Fig. 1 shows the evolution of the energy <math>E=\frac{1}{2}\left( du/dt\right) ^{2}-\cos u+1</math> of the pendulum (starting in equilibrium <math>u=0</math> at <math>t=-300</math>), as the driving frequency varies in time, <math>\omega (t)=1-\alpha t</math> (<math>\alpha </math> being the frequency chirp rate), and passes the linear resonance at <math>\ t=0 \ .</math> The parameters in the Figure are <math>\alpha =0.001</math> and <math>\varepsilon =0.03\ .</math> One can see three main stages of evolution in this example: the transition through and capture of the pendulum into resonance, the autoresonant stage, where the driven system self-adjusts (increases) its energy to remain in resonance with the drive continuously, and the transition to chaos, as <math>u</math> approaches the value of <math>\pi </math> prior to possible formation of a rotating state. The chaos is illustrated by showing 10 trajectories with different initial phases of the drive, exhibiting the departure from autoresonance at high excitations and sensitivity to initial conditions. Importantly, prior to reaching the chaotic state the autoresonance is a reversible process and the pendulum can be returned to its nearly zero initial equilibrium by simply reversing the direction of variation of the driving frequency. Note oscillatory modulations of the energy of the system in autoresonance around the growing average. These slow oscillations comprise an important characteristic of autoresonance, indicate stability of the autoresonant state, and have frequency scaling as <math>\sqrt{\varepsilon }\ .</math> Additional important characteristic of autoresonance is the existence of a sharp threshold on the driving amplitude (in this case <math>\varepsilon_{th} =0.0185</math>) for capturing the system into autoresonance (the sub-threshold evolution with <math>\varepsilon =0.01625</math> is shown in Fig. 2 in red). This threshold (see theory below) was discovered in experiments with trapped electron clouds (Fajans et al 1999) and scales as <math>\varepsilon_{th}\thicksim\alpha ^{3/4}</math> with the driving frequency chirp rate. The threshold phenomenon has a number of physical applications. An important example is the estimate of the characteristic time-scale of early evolution of the solar system from the relative abundance of resonant (with Neptune) masses in the Kuiper belt (Friedland 2001).
In contrast to dynamical systems with a finite number of degrees of freedom,
understanding of continuous systems described by nonlinear partial
differential equations (PDEs) comprises a greater challenge because of the variety and complexity of possible solutions of underlying PDEs. For example, the
Korteweg-de-Vries (KdV) equation <math>\partial u/\partial t+6u\partial
u/\partial x+\partial ^{3}u/\partial x^{3}=0\ ,</math> describing many physical
phenomena, differs from the pendulum problem by the existence of many types
of solutions, a traveling wave <math>u=u(kx-\omega t)</math> being the simplest
example (see Aranson et al 1992 for autoresonant excitation of these waves). In addition to the traveling waves, possible solutions of the KdV
equation are solitons and multi-phase waves of form <math>u=u(\theta
_{1},\theta _{2},...)\ ,</math> which are nonlinear functions of several phase
variables <math>\theta _{i}=k_{i}x-\nu _{i}t</math> with different wave vectors and
frequencies. Similar multi-phase solutions exist for many other fundamental, integrable
nonlinear wave equations, such as the nonlinear Schrodinger equation, the
sine-Gordon equation etc. Controlled formation of these complex states in
physical applications is difficult, because it requires realization of very
complicated initial conditions.
The question is weather autoresonance can be
used for excitation of multiphase waves? The answer to this question is
positive (see Friedland and Shagalov 2003) and Fig. 3 shows an example, where a 3-phase solution of the
periodic, <math>u(x,t)=u(x+L,t)\ ,</math> Korteweg-de Vries equation is formed by
starting from zero and adding a driving term of form <math>\sum_{i=1}^{3}\varepsilon_{i}\cos \varphi_{i}(x,t)\ .</math> This drive is a
superposition of three simple plane waves of constant amplitude <math>\varepsilon
_{i}</math> and phase <math>\varphi_{i}=k_{i}x-\int\omega _{i}(t)dt\ ,</math> where the wave
vectors <math>k_{i}</math> are multiples of the fundamental wave vector <math>k_{0}=2\pi /L</math>
and the frequencies <math>\omega _{i}(t)=\omega_{i0}-\alpha t</math> are slow
functions of time, such that at <math>t=0</math> all frequencies pass the resonances <math>
\omega _{i}(0)=k_{i}^{3}</math> with the corresponding plane waves of
the linearized Korteweg-de Vries equation. The solution in Fig. 3 seems very
complex, but one observes growing average amplitude of the oscillations after passage through resonances. Figure 4 shows our diagnostics of the excited solution. The spectral approach of the Inverse Scattering Transform method (IST) was used
for finding the number of phases in the wave and their frequencies <math>\nu_{i}</math> in the
process of evolution. The IST method associates an
N-phase solution with two linear eigenvalue problems, yielding discrete
main and auxiliary spectra. The main spectrum values remain constant in the
unperturbed problem and evolve slowly in a driven weakly perturbed problem and, thus, yield a convenient tool for diagnostics. For example, for a trivial solution <math>u=0\ ,</math> the main spectrum includes infinite number of real degenerate pairs. When N such pairs become nondegenerate, creating N open gaps in the spectrum, one has an N-phase solution. Figure 4a shows the results of
spectral analysis for the example of Fig.3. One observes the opening of three gaps in the spectrum after passage through resonances, i.e. formation of a three-phase solution. The frequencies <math>\nu_{i}</math> of the three phases and the three linearly chirped
driving frequencies <math>\omega_{i}</math> in this example (calculated using the main spectrum) are shown in Fig.4b. One can see a
continuing phase-locking of all three degrees of freedom in the driven problem
beyond the linear resonance (t=0). This multiple resonance, persistent despite the variation of the driving frequencies, is the main
signature of autoresonance in the system. Similarly to the autoresonant
pendulum problem, the autoresonance threshold phenomenon is
characteristic of multi-component waves. For example, one finds that the
phase-locking in Fig.4b requires all driving amplitudes to exceed certain
threshold values, scaling as <math>\alpha_{i}^{3/4}\ ,</math> <math>\alpha_{i}</math> being the driving frequency chirp rate of the
corresponding driving component. One can see the departure of one of the
frequencies of the 3-phase solution from resonance in Fig.4b, when the
associated driving amplitude is slightly below the threshold.
The nonlinear Schrodinger (NLS, http://www.scholarpedia.org/article/Nonlinear_Schrodinger_systems:_continuous_and_discrete) equation <math>i\partial \Psi/\partial t+\partial^2\Psi /\partial x^2+|\Psi|^2\Psi=0</math> is another fundamental, integrable equation of nonlinear physics. As in the KdV equation case, NLS equation allows spatially periodic multiphase solutions. The latter have a form <math>\Psi =U(\theta _{1},\theta _{2},...)\exp \{i[\xi]\}\ ,</math> where <math>U</math> and <math>V</math> are <math>2\pi</math>-periodic real functions of <math>N</math>-phases <math>\theta_{i}=k_{i}x-\nu _{i}t\ ,</math> <math>i=1,...,N\ ,</math> wave numbers <math>k_{i}</math> and frequencies <math>\omega _{i}</math> are constant, while <math>\xi =k_{0}x-\omega _{0}t</math> (the external phase) appears in the complex exponential only. The Inverse Scattering Transform method for NLS is mathematically more complicated than for KdV (its main spectrum, for example, is generally complex). Nevertheless, adding a multifrequency driving term in the right hand side of the NLS equation and passage through resonances yields a realizable approach to excitation of multiphase solutions (Friedland and Shagalov 2005). Example of formation of a periodic NLS breather oscillation is illustrated in Fig. 1, showing the evolution of <math>|\Psi|</math> from zero, through a flat <math>|\Psi|=const</math> solution, to nearly a solitary waveform <math>|\Psi|\approx(2\omega)^{1/2}sech(\omega^{1/2}x)\ ,</math> using the driving perturbation of form <math>[\varepsilon_{0}+\varepsilon_{1}\cos(kx)]exp[i\phi(t)]</math> with slowly varying frequency <math>\omega(t)=d\phi/dt\ .</math> The wave number <math>k=2\pi/L</math> of the breather is given by the spatial period <math>L (=\pi</math> in Fig. 1) in the perturbation. The driving frequency <math>\omega(t)=3+4.5\sin(0.005t)</math> in the example in Fig. 1 is negative initially (at <math>t=-300</math>), but slowly increases and becomes positive at later times. The evolution involves two stages: (a) excitation of a flat (<math>x</math>-independent) solution when <math></math>0<\omega(t)<k^{2}/2</math></k^{2}/2</math> and (b) transition to the spatially modulated state when <math>\omega(t) > k^{2}/2\ .</math> The modulated state shown in a single spatial period window in Fig. 1 approaches a growing amplitude solitary waveform with the increase of <math>\omega(t)</math> (serving as the soliton parameter). Note a large amplitude of the excited wave despite the smallness of the perturbing amplitudes (<math>\varepsilon_{1,2}=0.05</math>).
Autoresonance yields a different approach to pattern
formation in extended nonintegrable systems. For example, two-dimensional
ideal fluid dynamics is governed by a system of Euler and Poisson equations for vorticity <math>w(x,y,t)</math> and stream function <math>\psi(x,y,t)\ :</math> <math>\partial w/\partial t+v_{x}\partial w/\partial x+v_{y}\partial w/\partial y=0\ ,</math> <math>\partial^{2}\psi/\partial x^{2}+\partial^{2}\psi/\partial y^{2}=-w\ ,</math> where the fluid velocity components are <math>v_{x}=\partial \psi\partial y\ ,</math> <math>v_{y}=-\partial \psi\partial x\ .</math> The same system of equations describes azimuthal dynamics of magnetized electron clouds (Fajans et al 1999), where <math>\psi</math> represents the electric potential, while <math>w</math> is proportional to the electron density. These systems allow a variety of nontrivial solutions. For example, Deem and Zabusky (1978) discovered stable m-fold symmetric, steadily rotating uniform vorticity states (V-states) in computer simulations. Can one create and control V-states by a small forcing? A positive answer to these questions was suggested via autoresonance approach by Friedland and Shagalov (2000). It was shown that m-fold symmetric V-states can be formed by subjecting an initially axisymmetric vortex patch to a weak external flow, having stream function <math>\psi_{ext}=\varepsilon(t) r^{m}cos(m\phi)</math> (<math>r,\phi</math> being polar coordinates) and oscillating external strain rate <math>\varepsilon(t)\ .</math> The frequency of oscillations of the strain was a slowly varying function of time, passing through a resonance with a small amplitude m-fold symmetric perturbation of the vortex boundary. This process led to formation of V-states staying in a continuous resonance with the driving perturbation. We illustrate this process in Fig.5, showing the evolution of the vortex boundary for m=3,4,5 as the frequency of the strain is increasing in time. Below some critical frequency of the strain rate (at which the driven vortex developed filaments), the excitation process was reversible and the V-state could be returned into its nearly circular shape by reversing the direction of variation of the frequency of oscillations of the driving flow. In addition to uniform V-states, a similar perturbing flow was also used in forming nonuniform V-states (Friedland and Shagalov 2002). Finally, by using the same driving flow and starting from a circular vortex patch as in previous examples, but decreasing the driving frequency and passing through a different resonance, it was possible to form autoresonant, m-fold symmetric vorticity holes (Borich and Friedland 2008), the process illustrated in Fig. 6.
The threshold for autoresonance mentioned above is a weakly nonlinear phenomenon. Here, we discuss this effect for a driven, weakly nonlinear pendulum problem: <math> d^{2}u/dt^{2}+u-\beta u^{3}=\varepsilon \cos \varphi \ ,</math> where <math>\varphi =t-\frac{1}{2}\alpha t^{2}</math> (chirped frequency drive). We seek solutions of this equation in the form <math>u=a\cos \theta \ ,</math> where both the amplitude <math>a(t)</math> and the frequency <math>\omega (t)=d\theta /dt</math> are assumed to be slow functions of time. Then, using single resonance approximation (Chirikov 1979), one can reduce the problem to a system of two equations for <math>a</math> and the phase mismatch <math>\Phi =\theta -\varphi +\pi \ :</math> <math>da/dt=(\varepsilon /2)\sin \Phi \ ,</math> <math>d\Phi /dt=\alpha t-(3\beta/8)a^{2}+(\varepsilon /2a)\cos \Phi \ ;</math> This system involves three constants, <math>\varepsilon\ ,</math> <math>\beta \ ,</math> and <math>\alpha \ ,</math> i.e. the driving amplitude, nonlinearity parameter of the pendulum, and the driving frequency chirp rate. Nevertheless, by introducing rescaled time <math>\tau =\alpha ^{1/2}t\ ,</math> new amplitude <math>A=(3\beta/8)^{1/2}\alpha^{-1/4}a\ ,</math> the dimensionless driving parameter <math>\mu=(3\beta/32)^{1/2}\alpha^{-3/4}\varepsilon\ ,</math> and a new complex dependent variable <math>\Psi =A\exp (-i\Phi )\ ,</math> one obtains a single parameter nonlinear Schrodinger-type equation <math>id\Psi /d\tau +(\left\vert \Psi \right\vert ^{2}-\tau )\Psi =\mu \ ,</math> describing the capture into autoresonance, i.e. transition to phase locked solution <math>\Phi \rightarrow 0\ ,</math> <math>A\rightarrow \tau ^{1/2}</math> as <math>\tau </math> passes from <math>-\infty </math> to <math>+\infty </math> through the linear resonance at <math>\tau =0\ .</math> One finds that when starting from <math>A=0</math> (the pendulum at rest) the transition to autoresonance is controlled by the single parameter <math>\mu </math> in the problem and takes place for <math>\mu >\mu _{cr}=0.41\ .</math> This, in turn, yields the following expression for the critical driving amplitude <math>\varepsilon _{cr}=1.34\beta^{-1/2}\alpha^{3/4}\ .</math> Note that the form of the characteristic NLS-type equation in this problem hints at a possibility of autoresonant phase-locking transition in driven extended systems in general and nonlinear waves in particular. Indeed, we find the autoresonance threshold phenomenon and a similar scaling of the critical driving amplitude with parameters in all examples presented above.
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