1. Авторезонанс в цепочке осцилляторов А.С. Ковалева (ИКИ РАН)

2. INTRODUCTION Autoresonance (AR) is the tendency of a weakly driven nonlinear system to stay in resonance with its drive when the forcing and/or system frequencies slowly vary in time. Example: the Duffing equation 0 0, b > 0,  > 0. The system is assumed to be initially at rest; u  0, v  du/dt  0 at t  0. In the linear system, the change of the driving frequency entails the exit from resonance. In the NL system, the slow change of the driving frequency entails a gradual increase of the oscillations amplitude along with an increase of the instant natural frequency. If these processes are synchronized to some extent, the resonance growth of the amplitude is preserved.

3. AR was first employed in the problem of particle acceleration (V.I. Veksler, 1945, E.M. McMillan, 1946) and reported as nonlinear phase-locking between the system and the driving signal. Many new applications of AR in atomic and molecular physics, fluid dynamics, nonlinear waves, plasmas, superconductivity, planetary dynamics have appeared ever since. The analysis was first concentrated on the study of AR in the basic nonlinear oscillator but then it was extended to two- or three- dimensional systems. In most of these studies, AR was considered as an effective method of excitation and control of high-energy oscillations in the entire system.

4. In this work we demonstrate that this conclusion cannot be applied universally because AR in the multi-dimensional system is a much more complicated phenomenon than AR in a single oscillator, and the behavior of each element in the multi-dimensional array may drastically differ from the dynamics of a single oscillator. For example, the capture into resonance may not exist or AR in one part of the system may be unable to enhance the response of other oscillators. We illustrate this effect by an example of the multi-dimensional linear chain attached to an externally forced nonlinear actuator (the Duffing oscillator).

5. Two types of systems are considered: (1) harmonic forcing with a constant frequency is applied to the actuator with the slowly time-decreasing linear stiffness; (2) the nonlinear actuator with constant parameters is subjected to harmonic forcing with the slowly increasing frequency. In both cases, stiffness of the linear oscillators as well as linear coupling remains constant, and the system is initially engaged in resonance. The parameters of the system and forcing are chosen to guarantee autoresonance (AR) in the nonlinear actuator. As this paper demonstrates, in the first case AR occurs in the entire chain but in the second case AR occurs only in the actuator while the response of the linear chain remains bounded. This means that the systems that seem to be almost identical exhibit different dynamical behavior. The difference in the dynamical behavior follows from different resonance properties of the systems.

6. CONTENT 1.AR in the Duffing oscillator: criteria of the emergence of AR. 2.AR in the 2DOF system with the time-varying parameters of the nonlinear oscillator. 3.AR and irregular oscillations in the 2DOF system with the slowly changing forcing frequency. 4.AR in the chain.

7. AUTORESONANCE IN THE DUFFING OSCILLATOR The nonlinear dynamics is governed by the equations: 0 0, b > 0,  > 0. The system is assumed to be initially at rest; u  0, v  du/d  0  0 at  0  0. In the linear system, the change of the driving frequency results in the exit from the resonance domain. In the NL system, a slow change of the driving frequency maintains a gradual increase of the oscillations amplitude along with an increase of the instant natural frequency, thereby preserving the resonant growth of the amplitude.

8. ASYMPTOTIC SOLUTIONS

9. Constant detuning Constant detuning (  = 0)

10. Time-varying detuning (   0) Fig. 2. Autoresonance and saturation for different values of the parameters f > f 1 and  ; the cycle of oscillations in the conservative system is shown by the dashed line. The amplitude of the first cycle of oscillations looks like the deformed amplitude of the conservative system with a distinct inflection at  = T *. An increase of the detuning rate  results in the transition from AR with increasing amplitude to quasi-linear bounded oscillations.

11. CRITICAL PARAMETERS

12. THE 2DOF MODEL WITH TIME VARYING PARAMETERS The model of coupled oscillators: where u 0 and u 1 - absolute displacements of the nonlinear and linear oscillators; m 0 and m 1 - their masses; c 1 and k - the coefficients of linear stiffness and cubic nonlinearity, respectively; c 12 - the linear coupling coefficient; C(t) = c 0  (k 1 + k 2 t), k 1,2 > 0 – time-varying linear stiffness of the Duffing oscillator; A and  - the amplitude and the frequency of the periodic force. The system is assumed to be initially at rest but the nonlinear oscillator is launched externally, while the attached oscillator is excited through coupling.

13. The small parameter  is defined through weak coupling: c 12 /c 1 = 2  << 1. Resonance: c 1 /m 1  c 0 /m 0 =  2 The dimensionless parameters:  0 =  t,  1   0, k 1 /c 0 =  s, k 2 /c 0 =  2 b  (slowly-varying stiffness), k/c 0  8  (weak nonlinearity), c 12 /c 0 = 2  0, c 12 /c 1  2  1, 1 < 0 (weak coupling), A =  m  2 F (weak forcing) The dimensionless equations of motion: Slow modulation:  (  1 )  s  b  1, The system is initially at rest: u r = 0, v r = du r /d  0 = 0 at  0 = 0.

15. Rescaling of the variables and the parameters:   s  1,  r   r / ,   (s/3  ) 1/2, r = 0, 1, f  F/s , µ r = r /s,   b/s 2,  0 (  ) = 1 + . The rescaled equations of the slow dynamics : The real-valued amplitudes and phases of oscillation: a r = |  r |,  r = arg  r, a r (0) = 0,  r (0) =  π/2.

16. Strong asymmetry: m 1 << m 0  m 1 =  m 0,  = O(1)   µ 0 = c 1 µ 1 /c = m 1 µ 1 /m =  µ 1 << µ 1. The truncated system: The truncated system is more convenient for further analysis, as the nonlinear oscillator can be investigated separately. The obtained NL response is considered as external forcing for the linear oscillator. The real-valued amplitudes and phases of oscillation: a 0 = |  (0) |, a 10 = |  1 (0) |,  0 = arg  (0),  10 = arg  1 (0).

17. Fig. 3. Amplitudes of oscillations in the original (red) and truncated (blue) system; dashed black lines depict the backbone curves AMPLITUDES OF AUTORESONANCE Example :  = 0.05,  = 0.05 <  *  0.06, µ = 0.02 << µ 1 = 0.25; f = 0.34 Regular AR in the Duffing oscillator is observed. The superposition of free and forced oscillations in the linear oscillator entails irregular response at the initial stage of motion but then forcing with increasing amplitude dominates and motion is transformed into regular oscillations The close proximity of exact and approximate solutions implies that the conditions of the occurrence of autoresonance for a single oscillator can be extended to the weakly coupled system.

18. STABLE PHASE LOCKING Fig.4. Phase-locking in the 2DOF system

19. The resonance state of the system is efficiently controlled via the change of the linear stiffness in the Duffing oscillator, and a required state can be maintained by terminating the change of the parameter at the prescribed energy level (Fig. 5). Fig. 5. Transitions from AR in the system with the time-varying parameter  0 (  ) to oscillations with the prescribed energy in the system with the fixed parameter  0 * =  0 (T * ) in the nonlinear (a) and linear (b) oscillators

20. Quasi-steady states and fast oscillations

22. SATURATION IN THE 2DOF SYSTEM

25. THE SYSTEM WITH CONSTANT PARAMETERS AND THE TIME-VARYING FORCING FREQUENCY

26. The rescaled equations of the slow dynamics: The change of variables, r = 0, 1, transforms the system to the equations with the slowly-varying coefficients and the constant right-hand side: This results in the drastic change in the dynamics of the linear oscillator.

28. 2. The linear oscillator After calculating the solution  0 (  ), the response of the linear oscillator  1 (  ) is directly found from the linear equation for  1 (  ). Ignoring the effect of small fast fluctuations, we obtain The limiting value K 0 (  ) can be explicitly evaluated, and equals K 0 (  ) = (2  ) 4/3  (¾)e 3iπ/8, where  is the gamma function. Hence, a 1 (  ) = |  1 (  )|  µ 1 (2  ) 1/3  (¾),  . that is, AR in the nonlinear oscillator is unable to generate oscillations with permanently growing energy in the attached oscillator but it suffices to produce linear oscillations with bounded amplitude.

29. Example:  = 0.05,  = 0.05 >  *  0.06, µ = 0.02 << µ 1 = 0.25; f = 0.34. Fig. 7. Amplitudes of oscillations for the nonlinear and linear oscillators; red dashed lines correspond to the backbone curve The amplitude of the nonlinear oscillator is similar to its analogue in the first case but the behavior of the linear oscillator differs from oscillations with growing energy in the system of the first type. The shape of the amplitude a 1 (  ) is similar to the resonance curve with a distinct resonance peak at the initial stage of motion, where the effect of the time-dependent detuning is negligible, but then it turns into small oscillations with the limiting amplitude tending to a 1 

30. A key conclusion from these results is that in the system of the second type the energy transferred from the nonlinear actuator is insufficient to produce oscillations with growing energy in the attached linear oscillator. The different dynamical behavior follows from different resonance properties of the systems. In the system with a constant forcing frequency both oscillators are captured in resonance; if the forcing frequency slowly increases and the parameters of the system remain constant, resonance in the nonlinear oscillator is still sustained by an increase of the amplitude, while the frequency of the linear oscillator falls into the domain beyond the resonance. This conclusion is by no means trivial, as the linear oscillator is actually driven by the coupling response with permanently increasing amplitude.

31. THE AUTORESONANT CHAIN 1. The chain with the time-dependent nonlinear actuator The dimensionless equations of motion: Slow detuning:  (  1 )  s  b  1,  1 =  0, Initial conditions: u r = 0, du r /dt = 0 at t = 0, r = 0, 1,..,n.

32. ASYMPTOTIC APPROXIMATIONS

34. Example: the four-dimensional array consisting of the Duffing oscillator with an attached chain of 3 equal linear oscillators:

35. Fig. 8. Amplitudes of autoresonant oscillations The frequencies:  k = µ  k,  1 = 0.2,  2 = 1.56,  3 = 3.25. The period of the dominant low-frequency component T 1 = 2π/µ  1 = 314 is close to the exact (numerical) value T  350. The amplitudes of the low harmonics: a 11 = 0.63; a 21 = 0.97; a 31 = 1.22 are close to the exact (numerical) results The parameters:  = 0.05,  = 0.05, f = 0.34, µ 0 = 0.015, µ = 0.1.

36. II. The chain with constant parameters and the slowly-varying frequency of the driving perturbation The dimensionless equations of motion:

38. The change of variables  k (  ) =  k (  )e i  (  ) transforms the system to the form with the slowly-varying coefficients and the constant right-hand side: The analytical solution demonstrates AR in the nonlinear oscillator and small bounded oscillations in the linear chain.

39. Fig.9. Amplitudes of oscillations in the 4D chain The numerical results (Fig. 9) confirm the bounded response in the linear chain despite AR with the permanently growing amplitude in the forced Duffing oscillator. The 4D chain with parameters  = 0.05,  = 0.05, f = 0.34, µ 0 = 0.015, µ =0.1

40. The significant difference in the dynamical behavior of the two types of systems results from their different resonant properties. In the system with a constant excitation frequency all oscillators are engaged in resonance, because the constant frequency of the linear oscillator is always close to the excitation frequency, and the nonlinear oscillator sustains the resonant frequency due to an increasing amplitude. However, if the forcing frequency slowly increases, AR in the nonlinear oscillator is still sustained by an increasing amplitude of oscillations while the frequencies of the linear oscillators fall into the domain beyond the resonance.

41. CONCLUSIONS The AR analysis earlier developed for the Duffing oscillator have been extended to a multi-DOF system with a dominant resonant frequency. Two classes of system with different types of excitation have been considered. In the systems of the first class, a periodic force with constant frequency is applied to the Duffing oscillator with slowly time-decreasing linear stiffness; in the systems of the second class, the nonlinear oscillator with constant parameters is excited by a force with slowly increasing frequency. In both cases, the parameters of the attached linear oscillator and linear coupling are constant, and the system is initially engaged in resonance. It is shown that in the system of the first type AR occurs in both oscillators. However, in the system of the second type AR occurs only in the nonlinear oscillator whereas the linear oscillators exhibit irregular oscillations of diminishing intensity. Considering slow detuning, we have obtained explicit approximations of the solutions close to exact (numerical) results.

42. APPLICATIONS ?