## Abstract

Autoresonant energy transfer has been considered as one of the most effective methods of excitation and control of high-energy oscillations for a broad range of physical and engineering systems. Nonlinear time-invariant feedback control provides effective self-tuning and self-adaptation mechanisms targeted at preserving resonance oscillations under variations of the system parameters but its implementation may become extremely complicated. A large class of systems can avoid nonlinear feedback, still producing the required state due to time-variant feed-forward frequency control. This type of control in oscillator arrays employs an intrinsic property of a nonlinear oscillator to vary both its amplitude and the frequency when the driving frequency changes. This paper presents a survey of recently published and new results studying possibilities and limitations of time-variant frequency control in nonlinear oscillator arrays.

This article is part of the themed issue ‘Horizons of cybernetical physics’.

## 1. Introduction

Resonance energy transfer from a source of energy to a receiver has been identified as one of the most effective methods of excitation and control of high-energy oscillations for a broad range of physical and engineering systems. An idea of control intended to sustain ‘resonance under the action of forces produced by the system itself’ was first suggested by Andronov and co-workers [1]. Feedback control schemes building on this idea and using self-sustained oscillations with predefined energy as a working process were implemented in a number of engineering systems [2]. Control devices in these systems include electronic and electromechanical positive feedback and a synchronous type actuator for self-excitation of resonant vibration in combination with negative feedback for its stabilization. The implementation of these schemes may be extremely costly and cumbersome in practical situations [2].

A large class of systems can avoid feedback, still producing the required state with the help of time-variant feed-forward frequency control, which employs an intrinsic property of a nonlinear oscillator to change both its amplitude and natural frequency when the driving frequency changes. This means that an oscillator may remain persistently captured into resonance with its drive if the driving frequency, being initially close or equal to the natural frequency of the oscillator, varies slowly in time to be consistent with the slowly changing frequency of the oscillator. The ability of a nonlinear oscillator to stay captured into resonance due to variance of its structural or/and excitation parameters is known as *autoresonance* (AR). It is important to note that the emergence of AR leads to persistently growing mean amplitude of oscillations, and thus this process may be employed to attain the required energy level.

After first studies in the fields of particle acceleration [3,4] and planetary dynamics [5,6], a large number of theoretical approaches, experimental results and applications of AR in different fields of natural science have been reported in the literature (e.g. [7–9] and references therein). The study of AR was further extended to multi-dimensional processes such as excitations of continuously phase-locked plasma waves [10], particle transport in a weak external field with slowly changing frequency [11], autoresonant excitation of solitons [12], control of nanoparticles and electron beams [13,14], etc. In most of these studies, AR in the forced oscillator was considered as an effective tool for exciting high-energy oscillations in the entire system. However, recent results [15–18] have shown that this principle is not universal because capture into resonance of a multi-particle chain is a much more complicated phenomenon than a similar effect for a single oscillator and the emergence of AR in all coupled oscillators directly depends on the structure of the chain. The purpose of this work is to investigate the dynamics of a multi-particle resonant Klein–Gordon chain subjected to a periodic forcing with slowly varying frequency and find the conditions of the emergence of stable AR in the entire chain due to controlling the forcing frequency.

The paper is organized as follows. In §2, we briefly discuss earlier obtained results about the emergence and stability of AR in a single Duffing oscillator subjected to a harmonic forcing with time-varying frequency. The asymptotic analysis of resonant oscillations is developed with the emphasis on the calculation of quasi-steady states. This analysis leads to the definition of a parametric domain, in which stable AR with persistently growing mean amplitude can occur. In §3, the results obtained for a single oscillator are extended to the nonlinear Klein–Gordon chain consisting of *n* ≥ 2 identical linearly coupled Duffing oscillators. Parametric criteria, which guarantee the emergence and stability of AR in the entire chain, are established. It is shown that an increase of the number of oscillators in the chain does not change the conditions of capture into resonance in comparison with a two-particle array. Furthermore, the emergence of AR in the chain entails the asymptotic equipartition of energy among the oscillators at large times. Numerical results in §4 are in a good agreement with the theory. Concluding remarks are collected together in the last section.

## 2. Limiting phase trajectories and the emergence of autoresonance in an anharmonic oscillator

For the purposes of further analysis, in this section we demonstrate main results concerning the emergence and stability of AR in a single Duffing oscillator (for details, see [19]). As shown in earlier works (e.g. [20]), AR in an adiabatic system can occur if the initial point is far enough from the domain of small oscillations. In contrast with the results reported in [20], we have proved that the emergence of AR in the Duffing oscillator being initially at rest is similar to the transition from small (quasi-linear) to large (nonlinear) oscillations in the oscillator with constant (resonant) excitation frequency, and AR occurs due to the loss of stability of the so-called limiting phase trajectory (LPT) of small oscillations [18,19]. This section demonstrates that critical forcing, which determines the transition from small (non-resonant) to large (resonant) oscillations in the time-invariant system, may be treated as a lower threshold of the emergence of AR in the oscillator with slowly varying parameters. Furthermore, the threshold parameters numerically obtained, e.g. in [21,22], are invalid in the problem under consideration.

### (a) Governing equations

As a basic example, we study the Duffing oscillator subjected to a periodic excitation with slowly varying frequency. The equations of the oscillator reduced to the unit mass are given by
2.1
where *u* denotes the absolute displacement of the oscillator; *γ* > 0 is the cubic stiffness coefficient. The oscillator is excited by a periodic force with amplitude *A* and time-dependent frequency *Ω*(*t*) = *ω* + *ζ*(*t*); *k*_{1} is the initial constant detuning, *k*_{2} > 0 is the detuning rate.

Taking into account resonant properties of the oscillator, we introduce the small parameter of the problem as *ϵ* = *k*_{1}/*ω* and redefine the system parameters as follows:
2.2

In these notations, the equations of motion are rewritten as
2.3
where *τ*_{0} = *ωt* is the fast time scale, *τ* = *ϵτ*_{0} is the leading-order slow time scale and the coefficient *β* represents the detuning rate. The system is assumed to be initially at rest, i.e. *θ* = 0, *u *= 0, at *τ*_{0} = 0. We recall that zero initial conditions correspond to the so-called LPTs corresponding to maximum possible energy transfer from a source of energy to a receiver [23,24].

Asymptotic solutions of equations (2.3) are derived with the help of the multiple time-scale formalism [25,26]. To this end, we introduce the complex-conjugate envelopes *Ψ, Ψ** and the dimensionless forcing amplitude by formulae
2.4
It follows from (2.4) that the functions *ã* = |*Ψ*| and represent the dimensionless real-valued amplitude and the phase of oscillations, respectively. Substituting (2.4) into (2.3), we obtain the following equation for the envelope *Ψ*:
2.5
and a similar equation for the complex-conjugate envelopes *Ψ**. The coefficient *G* includes fast harmonics with coefficients depending on *Ψ* and *Ψ** but an explicit expression of this coefficient is insignificant for further analysis (see [23,24]). In the next step, we introduce the asymptotic decomposition of the variable *Ψ*
2.6
Standard averaging transformations [25] yield the following equation for the slow envelope *ψ*(*τ*):
2.7
The polar representation transforms (2.7) into the equations for the real-valued envelope *a* > 0 and the phase Δ
2.8
with initial conditions *a*(0) = 0, Δ(0) = −*π*/2. The accuracy of asymptotic approximations for systems with slowly varying parameters has been discussed, for example, in [25,26]. Omitting details, we mention that the error of approximation |*ã*(*τ,ϵ*) − *a*(*τ*)| < *δ*(*ϵ*) → 0 as *ϵ* → 0. This estimate remains valid on the interval of interest, which is at least *τ* ∼ *O*(1/*β*) in the problem under consideration. At the same time, the phase difference (mod 2*π*). Numerical confirmations of these statements will be presented below.

For better understanding of the emergence of resonant modes, we consider the underlying *time-invariant* problem
2.9
with initial conditions *a*(0) = 0, Δ(0) = −*π*/2, which correspond to the LPTs of system (2.9) [23,24]. Figure 1 illustrates the ‘limiting’ property of the LPTs. These numerical results clearly indicate that the LPT represents an outer boundary for a set of closed trajectories encircling the stable centre in the phase plane (Δ, *a*).

It was proved [23] that there exist two critical excitation levels, which describe the boundaries between different types of the solutions of system (2.9), namely,
2.10
The phase planes in figure 1 demonstrate that the threshold *f*_{1} corresponds to the boundary between small and large oscillations: at *f* = *f*_{1} the LPT of small oscillations coalesces with the separatrix passing through the homoclinic point on the axis Δ = −*π*. This implies that the transition from small to large oscillations occurs due to the loss of stability of the LPT of small oscillations. At *f* = *f*_{2}, the stable centre on the axis Δ = −*π* vanishes due to the coalescence with the homoclinic point, and only a single stable centre remains on the axis Δ = 0. By definition [23], conditions *f* < *f*_{1}, *f*_{1} < *f* < *f*_{2}, and *f* > *f*_{2} characterize *quasi-linear*, *moderately nonlinear* and *strongly nonlinear* oscillations, respectively.

### (b) Analysis of autoresonance

The quasi-steady solutions of equations (2.8) can be found from the equations
2.11
with the ‘frozen’ parameter *ζ*_{0}(*τ*). The first equation in (2.11) yields . Recall that stability of the solution was demonstrated in earlier work [27]. Figure 2 depicts oscillations of the response amplitude *a*(*τ*) and the phase Δ(*τ*) near the corresponding quasi-steady states ,.

In the next step, we obtain from the second equation (2.11) that as *τ* → ∞. Thus, the quasi-steady solutions of (2.11) can be expressed as
2.12
The function can be interpreted as the backbone curve, which depicts the dependence of the quasi-steady amplitude on the slowly varying frequency (figure 2*a*).

The obtained numerical results motivate the derivation of the parametric boundary between the bounded and unbounded oscillations. It was found numerically [21,22] that AR in this system occurs at *ϕ* > *ϕ*_{cr} = 0.41, where *ϕ* = |*β*|^{−3/4}*f*. The disagreement of this result with the analytical estimate is demonstrated below.

The emergence of AR from stable bounded oscillations under the change of rate *β* is observed in figure 3. It is clearly seen that in the first half-cycle of oscillations the amplitude of small oscillations of system (2.8) is close to the LPT of quasi-linear oscillations, while the amplitude of AR is close to the LPT of the time-invariant system with moderate nonlinearity. This implies that the transition from bounded to unbounded oscillations in the system with slowly varying forcing frequency is of the same nature as the transition from small to large oscillations in the system with constant parameters, i.e. it occurs due to the destruction of the LPT of quasi-linear oscillations (see [23,24]). This implies that the inequality *f* > *f*_{1} can be considered as *the necessary condition* of the emergence of AR.

Figure 3 clearly shows that in the first half-cycle of motion the solution *a*(*τ*) of system (2.8) is very close to the LPT of the time-independent system (2.9). Besides, the LPT of the nonlinear time-invariant systems has a distinctive inflection at *τ* = *T** (figure 3*b*). In order to assess the critical rate *β*_{cr} which allows transitions from bounded to unbounded oscillations, we introduce the parameter *f̃*(*τ*) = *f*/(1 + *βτ*)^{3/2} such that *f̃*(0) = *f* > *f*_{1}. The numerical results presented in figures 2 and 3 allow one to conclude that an adiabatically varying system in which *f̃*(0) > *f*_{1} gets captured into the domain of small oscillations if *f̃*(*T**) < *f*_{1}. Under this condition, the critical rate is given by
2.13
If *β* < *β*_{cr}, the system admits AR. The correctness of formula (2.13) is verified by the calculation of the critical rate *β*_{cr}. We recall that the point of inflection of the function *a*(*τ*) is determined by the conditions d*a/*d*τ* ≠ 0, d^{2}*a*/d*τ*^{2} = 0. It follows from (2.9) that the latter condition corresponds to dΔ/d*t* = 0, i.e. the envelope *a*(*τ*) intersects its point of inflection when the phase Δ(*τ*) achieves its minimum. From figure 4, it is seen that *T** ≈ 6.5 for *f* = 0.274; this yields *β*_{cr} ≈ 0.00075, whereas figure 3*a* shows that 0.001 < *β* < 0.002. Then, *T** ≈ 5 and *β*_{cr} ≈ 0.004 for *f* = 0.28 but the numerical results in figure 3*b* give 0.006 < *β* < 0.007. Note that for *f* = 0.28, the threshold parameter *ϕ*_{cr} = |*β*_{cr}|^{−3/4}*f* = 0.41 gives *β*_{cr} = (*f*/*ϕ*_{cr})^{4/3} = 0.577, which is much more than the real critical rate. Similarly, it is seen in figure 4 that *β*_{cr} = 0.053 for *f* = 0.34, while the numerical simulation (figure 3*c*) gives 0.061 < *β* < 0.062. Note that at *f* = 0.34, the inflection of the curve *a*(*τ*) is indistinguishable but the phase has a pronounced minimum at *T** ≈ 3. It is easy to check that the analytical approximations [19] give the values of *T** slightly exceeding the numerical results in figure 4.

## 3. Autoresonance in Klein–Gordon chains

In this section, we discuss main results concerning the emergence and stability of AR in a Klein–Gordon chain of finite length consisting of *n* ≥ 2 identical linearly coupled Duffing oscillators. The chain is subjected to an external forcing applied to one of the oscillators (the actuator). This oscillator array models, in particular, a microelectromechanical system with a broad spectrum of engineering applications (see [28]). Preliminary results concerning capture into resonance of the chain with a linear attachment are discussed in [16].

### (a) Governing equations

The dynamics of the chain is described by the equations
3.1
In (3.1), the variable *u _{r}* denotes the absolute displacement of the

*r*th oscillator;

*κ*is the stiffness coefficient of linear coupling between the oscillators;

*ω*

^{2}=

*c*/

*m*,

*m*and

*c*being the mass and the linear stiffness of each oscillator;

*γ*> 0 is the cubic stiffness coefficient. Motion is induced by periodic forcing with amplitude

*A*and frequency

*Ω*(

*t*) =

*ω*+

*ζ*(

*t*) applied to the first oscillator.

Equations (3.1) can be reduced to the standard form by the same way as in §2. Taking into account resonant properties of the chain, we define the small parameter *ϵ* = *k*_{1}/*ω*, and then introduce the following rescaled parameters:
3.2
In addition, we define the dimensionless ‘fast’ time scale *τ*_{0} = *ωt* and the ‘slow’ time scale *τ* = *ϵτ*_{0}. In these notations, the equations of motion are reduced to the form
3.3

The system is assumed to be initially at rest, i.e. *θ* = 0, *u* = 0, at *τ*_{0} = 0 (*r* = 1, … ,*n*). As in the previous section, asymptotic solutions of equations (3.3) are derived using the multiple time-scale formalism [25,26]. First, we introduce the dimensionless complex-conjugate *n*-vectors *Ψ* and *Ψ** with components and define the needed dimensionless parameter as follows:
3.4
It follows from (3.4) that the real-valued dimensionless amplitudes and the phases of oscillations are given by formulae and , respectively. Inserting transformations (3.4) into (3.3), we then have the set of equations
3.5

The coefficients *G _{r}* in (3.5) include fast harmonics in

*τ*

_{0}with coefficients depending on

*Ψ*and

*Ψ** but explicit expressions of these coefficients are insignificant for further analysis. The asymptotic decomposition of the envelope

*Ψ*is constructed in the form similar to (2.6), namely, 3.6 where the term

_{r}*ψ*(

_{r}*τ*) depicts the slowly varying envelope. As in the previous section, the equations for the slow envelopes

*ψ*(

_{r}*τ*) can be obtained by straightforward averaging of (3.5) with respect to

*τ*

_{0}. The resulting averaged equations for the slow complex envelopes

*ψ*are given by 3.7 with initial conditions

_{r}*ψ*(0) = 0. The change of variables yields the following equations for the real-valued amplitudes

_{r}*a*and the phases Δ

_{r}*: 3.8 with initial amplitudes*

_{r}*a*(0) = 0 and indefinite initial phases Δ

_{r}*(0). It is important to note that these initial conditions make the system singular at*

_{r}*τ*= 0, thus yielding a number of complications in numerical simulations. To overcome this uncertainty, one needs to solve the non-singular equations (3.7) and then calculate the real-valued solutions by formulae

*a*= |

_{r}*ψ*|, Δ

_{r}*= arg(*

_{r}*ψ*). The accuracy of asymptotic approximations for systems with slowly varying parameters has been discussed, for example, in [25,26]. We recall that |

_{r}*ã*(

_{r}*τ,ϵ*) −

*a*(

_{r}*τ*)| <

*δ*(

_{r}*ϵ*) → 0 as

*ϵ*→ 0 but (mod 2

*π*).

Equations (3.8) are used to calculate the quasi-steady states of oscillators. As in §2, the quasi-steady solutions can be calculated by formulae
3.9
The last condition *P _{n}* = 0 implies . Inserting this equality into the condition

*P*

_{n}_{−1}= 0, we then have . Repeating this procedure for each equation

*P*= 0, one obtains ,

_{r}*r*> 1. Finally, the equation

*P*

_{1}= 0 yields . This means that either (mod 2

*π*) or (mod 2

*π*),

*r*= 1, … ,

*n*. At the second group of equations (3.9) is reduced to the form 3.10

One can conclude that the phases correspond to the stable AR, while the phases are unstable. Under these conditions, maximal quasi-steady amplitudes corresponding to AR in the entire chain are given by
3.11
where *ζ*_{0}(*τ*) = 1 + *βτ*. In analogy to a single oscillator, the slow functions can be interpreted as the backbone curves. It follows from (3.11) that the higher-order corrections may be ignored at large times; furthermore,
3.12

Formulae (3.12) clearly indicate that the energy of excitation tends to equipartition among all oscillators. This conclusion is illustrated below by the results of numerical simulations for the oscillator arrays with different number of particles.

### (b) Critical parameters

Let *ρ* = column(*f, μ, β*) be the vector of the constant parameters of the dimensionless chain (3.8). It was recently shown [18] that AR in two coupled oscillators may occur, if . Furthermore, the parameters of *O*(|*ρ*|^{2}) can be ignored in the asymptotic analysis. In this section, these assumptions will be used in the asymptotic analysis of multi*-*particle chains.

As the coupling response acts as an external perturbation with respect to the attachment, the emergence of AR in the forced oscillator can be considered as a necessary condition of the emergence of AR in the entire chain. This means that the threshold values of the parameters can be found assuming small oscillations of the attachment, i.e. the problem can be reduced to the analysis of the system with |*ψ _{r}*| ∼

*O*(

*μ*). Under this assumption, the leading-order approximations and are given by the following separated equations describing the dynamics of a single oscillator: 3.13 with initial conditions

*a*

_{1}(0) = 0, Δ

_{1}(0) = −

*π*/2 (the upper index ‘0’ is omitted for brevity). Note that the term

*μa*

_{1}reflects the effect of the coupling response with negligible amplitude

*a*

_{2}.

From figure 3*a* it is seen that, if the detuning rate is small enough, then the amplitude (3.13) within the first cycle of oscillations is very close to the amplitude of the identical oscillator subjected to forcing with constant frequency. Therefore, the first step towards analysing AR in the coupled system is the study of the transition from small to large oscillations in the underlying time-invariant system
3.14
with initial conditions *a*_{1}(0) = 0, Δ_{1}(0) = −*π*/2. Extending the results of earlier works [23,24] to (3.14), one can derive the following boundary between small and large oscillations:
3.15

The change of the forcing amplitude from *f* < *f*_{1μ} to *f* > *f*_{1μ} entails the transition from small to large oscillations (see [23,24] for details). The second parameter that determines the dynamical behaviour is the critical rate *β*_{cr} such as the transition from bounded to unbounded oscillations occurs at *β* < *β*_{cr}. Reproducing the transformation from [19], we deduce that AR in the actuator occurs if , or *β* < *β*_{cr}, where the critical rate *β*_{cr} is given by
3.16
Note that conditions (3.15) and (3.16) are obtained under the assumption of weak coupling between the actuator and the attachment, such that the energy transferred from the actuator is insufficient to produce AR in the attachment. As the coupling response plays the role of an external forcing with respect to the attachment, AR in the attached oscillator may be excited only if the strength of linear coupling between the oscillators is strong enough to provide the required level of excitation. Therefore, the next step is to evaluate the critical coupling coefficient *μ*_{cr}, such that all oscillators are captured into resonance at *μ *> *μ*_{cr}. To solve this problem, we begin with the analysis of the time-invariant analogues of equations (3.9) and (3.10). At *ζ*_{0}(*τ*) = 1, , equations (3.10) take the form
3.17

It is easy to conclude that the maximum solutions of equations (3.17) are given by formulae similar to (3.11), namely,
3.18
Ignoring higher-order corrections, we consider the approximations *ā*_{1} = 1 + *f/*2, *ā _{r}* = 1 in further analysis.

Equations (3.17) and (3.18) may be used to determine a parametric region, wherein AR may exist. We first define the coefficient *μ*, which yields the coupling response sufficient to sustain resonance in the *n*th oscillator under the condition of resonance in the previous oscillator. Under this assumption, the last equation in (3.17) is rewritten as
3.19
with . The roots of equation (3.19) are analysed through the properties of the discriminant [29]. If *D _{n}* < 0, then equation (3.19) has three different real roots; if

*D*= 0, two real roots merge, if

_{n}*D*> 0, there exists a single real and two complex conjugate roots [29]. In analogy to [23,24], one can show that the latter case corresponds to the maximal root . Substituting

_{n}*φ*= 1 into (3.19) and ignoring higher-order corrections, we transform the condition

_{n}*D*> 0 into the inequality . Thus, the critical coefficient of linear coupling is given by 3.20

_{n}If *μ* > *μ*_{cr}, then equation (3.19) has a single (maximum) root. In the next step, we analyse the roots of the *r*th equation in (3.17) assuming resonance in the previous oscillator and small oscillations of the subsequent oscillators. Under these assumptions, the equation for *a _{r}* is rewritten as
3.21
where . The roots of (3.21) are analysed through the properties of the discriminants [29]. It is easy to prove that

*D*> 0 at

_{r}*μ*> 0.18 for all attached oscillators from

*r*= 2 to

*r*=

*n*− 1. This implies that parametric thresholds for a multi-particle chain are determined by condition (3.20), and thus, the number of oscillators in the chain does not change (in the main approximation) the conditions of capture into resonance.

Boundaries (3.15) and (3.20) are depicted in figure 5. It follows from (3.15) and (3.20) that the chain with parameters from the domain below *f*_{1μ} executes small quasi-linear oscillations; if the parameters belong to the shaded domain *D*, then the actuator is captured into resonance but the dynamics of the attachment should be investigated separately; if the parameters *f* and *μ* lie within the dotted domain *D*_{0}, then the entire chain is captured into resonance.

Note that expressions (3.15) and (3.20) have been obtained for the time-invariant system. However, numerical results in §4 indicate that these inequalities adequately describe a boundary between small oscillations and AR in a system subjected to a periodic forcing with a slowly increasing frequency (). A detailed discussion for a pair of coupled oscillator is given in recent work [18].

## 4. Numerical results

The numerical results presented in this section confirm the existence of the parametric thresholds, beyond which the entire chain be captured into resonance. For clarity, we first reproduce the results earlier obtained for a pair of coupled Duffing oscillators [18]. Figure 6 depicts the amplitudes of oscillations in the cells with coupling strength *μ* = 0.08 subjected to a periodic force with detuning rate *β* = 0.01 but with different forcing amplitudes. Figure 6*a*,*c* illustrates the transition from bounded oscillations to AR in both oscillators when the forcing amplitude changes from *f* = 0.235 < *f*_{1μ} to *f* = 0.3 > *f*_{1μ}, while the critical threshold *f*_{1μ} = 0.24. Figure 6*b* shows that in the first cycle of oscillations the amplitude *a*_{1}(*τ*) is close to the LPT of the oscillator subjected to forcing with constant frequency (*β* = 0); the time instant *τ* = *T** determines to the point of inflection of the ‘large’ LPT at *f* = 0.3.

Figure 7 confirms that the coupling response in the weakly coupled system may be insufficient to excite growing oscillations in the attachment even in the presence of AR in the excited oscillator. In figure 7*a*, one can observe the convergence of the backbone at large times.

Figure 8 depicts the amplitudes of oscillations for the five-particle chain with parameters *f* = 0.5, *μ* = 0.25 and *β* = 0.01. It is clearly seen that chaotic oscillations observed in the initial interval (figure 8*a*) turn into small regular oscillations about the backbone curves (3.11) (figure 8*b*).

Figure 9 demonstrates similar results for the eight-particle chain with parameters *f* = 0.25, *μ* = 0.25 and *β* = 0.001. For brevity, we show only amplitudes of oscillations for the first and the last oscillators. It is easy to check that the gap between the backbone curves and (*r* > 1) in figures 8 and 9 is close to the theoretical value .

## 5. Conclusion

It was shown in early works on particle acceleration that AR could potentially serve as an effective tool to excite and control the required high-energy regime in a single oscillator. However, the behaviour of coupled oscillators may drastically differ from the dynamics of a single oscillator. In particular, capture into resonance may not exist, or AR in the excited oscillator may be insufficient to enhance the response of the attachment. In this work, these effects have been investigated for a nonlinear Klein–Gordon chain consisting of identical linearly coupled Duffing oscillators, one of which is subjected to periodic forcing with a slowly varying frequency. It has been shown that capture into resonance of the entire chain may occur if both forcing amplitude and coupling stiffness exceed certain threshold values but detuning rate is small enough. Asymptotic approximations of the critical parameters and the quasi-steady amplitudes of oscillations have been obtained. It has been demonstrated that AR in the entire chain results in the asymptotic equipartition of incoming energy among all oscillators at relatively large times. Theoretical findings have been validated by numerical simulations.

## Competing interests

I declare I have no competing interests.

## Funding

Support for this work received from the Russian Foundation for Basic Research (grants 14-01-00284 and 16-02-00400) is gratefully acknowledged.

## Footnotes

One contribution of 15 to a theme issue ‘Horizons of cybernetical physics’.

- Accepted November 24, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.