Nonlinear Dyn (2019) 98:1795–1806https://doi.org/10.1007/s11071-019-05286-x

ORIGINAL PAPER

Bifurcation phenomena and statistical regularitiesin dynamics of forced impacting oscillator

Sergii Skurativskyi · Grzegorz Kudra · Krzysztof Witkowski · Jan Awrejcewicz

Received: 6 March 2019 / Accepted: 9 October 2019 / Published online: 19 October 2019© The Author(s) 2019

Abstract The paper is devoted to the study of har-monically forced impacting oscillator. The physicalmodel for oscillator is a cart on a guide connectedto the support with springs and excited by the step-per motor. The support also is provided with lim-iter of motion. The mathematical model for thissystem is defined with the second-order piecewisesmooth differential equation. Model’s nonlinearity isconnected with the incorporation of dry friction andgeneralized Hertz contact law. Analyzing the clas-sical Poincare sections and inter-impact sequencesobtained experimentally and numerically, the bifur-cations and statistical properties of periodic, multi-periodic, and chaotic regimes were examined. Thedevelopment of impact-adding regime as a new nonlin-ear phenomenonwhen the forcing frequency varieswasobserved.

Keywords Forced impacting oscillator ·Bifurcations ·Impact · Chaos

S. SkurativskyiSubbotin Institute of Geophysics, Palladin av.32, Kyiv,Ukrainee-mail: skurserg@gmail.com

G. Kudra (B) · K. Witkowski · J. AwrejcewiczLodz University of Technology, Stefanowski St. 1/15,Lodz, Polande-mail: grzegorz.kudra@p.lodz.pl

1 Introduction

A large number of artificial devices and natural sys-tems contain oscillating elements, dynamics of whichessentially affect [1,2] the parent system. Therefore,one of the important problems is the classification ofthe dynamics types of oscillating systems, sources oftheir appearance, evolution scenario, etc. To constructvibration systems, springs and dash-pots as structuralelements are used.Tomodel their dynamics, the smoothmathematical models are applied, for studies of whichthe powerful analytical and numerical tools have beendeveloped. But there are a wide range of observed phe-nomena [3–7] which cannot be described within theframework of smooth models.

Incorporation of dry friction and impact rheologicelements into vibration systems allows one to describea wider range of physical processes. Correspondingmathematical models are governed by the nonsmoothequations of motions [5,8–10]. The detailed analyticaltreatment is admitted mainly by piecewise linear mod-els. If nonlinearity is taken into account, the nonsmoothmodels exhibit essentially nonlocal dynamics [11–15]for the study of which numerical methods are involved.The attractors inherent in the smooth models can beendowed by new properties. Moreover, such modelscan demonstrate special types of evolution includingborder-collision bifurcations [10,15–18] and chatter-ing phenomena [19].

The present studies are focused on revealing thepeculiarities of nonlinear regimes occurred in nons-

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1796 S. Skurativskyi et al.

(b)(a)

Fig. 1 Experimental rig (a) and physical model (b) [20]

mooth models. In particular, it is investigated a one-degree-of-freedomoscillator incorporating dry and vis-cous resistance, the generalized Hertz contact law[14,20] during an impact and subjected to the harmonicloading. In [20], the multi-parametric mathematicalmodel for the impact oscillator was developed. Dur-ing the model validation, several sets of model param-eters providing the best agreement of theoretical andexperimental data were obtained. Validated mathemat-ical submodels allowed one to estimate impact dura-tion time and coefficient of restitution, and to classifyroughly the vibration regimes based on the bifurcationdiagrams. As shown in [21], additional information onmodels’ regimes can be obtained from the inter-impactsequence analysis which was performed for the peri-odic and chaotic attractors.

The investigations mentioned in [20,21] do notcover the whole interval of forcing frequency variation,although a cursory glance at the bifurcation diagramindicates that no less interesting regimes are expectedto be revealed in another part of the diagram.

Thus, the question arises: What else new regimesand their bifurcations can be caught in selected intervalof forcing frequency? To answer the question, the clas-sical Poincaré section technique, top Lyapunov expo-nents (TLE), inter-impact (spike) sequences, and infor-mation entropy are used.

The detailed description of the studies is providedbelow. Thus, the next section contains the experimentalrig introduction and corresponding equation of motion.Section 3 is devoted to the presentation of experimental

results when the frequency of forcing varies. Sections 4and 5 are concerned with the numerical investigationof mathematical model and identification of observedregimes. The final section ends up with the concludingremarks.

2 Experimental stand and mathematical model

To begin with, let us give brief description of the exper-imental stand and proper mathematical model (theadvance discussion can be found in [20]). In Fig. 1a,there is presented part of the larger stand used in thepresent work as an experimental model for investiga-tions of dynamics of one-degree-of-freedom harmoni-cally excited oscillator with one-sided obstacle [20]. Itconsists of a cart (1) on a guide (4) with the integratedlinear ball bearing and Hall effect sensor (2). The cartis connected to the support (7) with the springs (3). Astepper motor (9) is mounted on the cart with a disk(10) and an unbalance (5) in order to realize harmonicforcing. Impacts are realized by limiters of motion inthe form of stops (6) mounted on the support and thecart. A transoptor (8) is situated on the cart behind thedisk in order to control absolute angular position of thestepper motor. Position of the cart is measured by theHall effect sensor, while angular position of the diskis obtained from an encoder integrated with the step-per motor. Measurement data are then registered byNational Instrument data acquisition system.

Figure 1b exhibits physical concept of the exper-imental rig. As was shown in the work [20], it is

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Bifurcation phenomena and statistical regularities in dynamics of forced impacting oscillator 1797

described by the following differential equation ofmotion

mẍ + kx + FR(x) + FI(x, ẋ) = m0eω2 sinωt, (1)where

FR(ẋ) = cẋ + T ẋ√ẋ2 + ε2 ,

FI(x, ẋ)

={kI f (x, ẋ) for x − xI ≥ 0 and f (x, ẋ) ≥ 0,0 otherwise,

f (x, ẋ) = (x − xI)n1 + b(x − xI)n2 sgn ẋ |ẋ |n3 .In these equations, x denotes position of the cart;

m—total mass of the cart; k—total stiffness of thesprings; FR—resistance force acting on the cart;FI(x, ẋ)—impact force;m0—an unbalancedmass; e—radius of the unbalancedmass;ω—angular speed of thedisk; c—linear damping coefficient; T—magnitude ofrolling resistance (assumed to not depend on the nor-mal load); ε—special parameter of regularization of thesign function (ẋ/

√ẋ2 + ε2 ≈ sgn ẋ); xI—position of

the obstacle; kI—stiffness coefficient of the obstacle;b—damping coefficient of the obstacle; n1, n2, n3—parameters of the contact. Note that impact is modeledas compliant based on extended Hertzian contact stiff-ness and damping.

To achieve good agreement between experimentaland simulated data, the estimation of the parametersbased on the numerical optimization was carried out[20]. The procedure consists in minimization of objec-tive function

F0(μ) =∫ t2t1

(xexp − xsim)2dtt2 − t1 ,

where xexp and xsim are the cart positions obtainedexperimentally and during numerical simulation on thetime interval [t1, t2], while μ = (k, c, T, kI, b, n2,3)denotes the vector of the estimated parameters. Mini-mization of the function F0 is provided by the Nelder–Mead multi-dimensional optimization algorithm real-ized with the help of the f minsearch MATLAB func-tion. Using the parameters measured directly m =8.735 kg, m0e = 0.01805 kg · m and fixed quanti-ties ε = 10−6 m/s and n1 = 3/2, the application ofF0 minimization procedure results in several groupsof parameters defining different submodels, concern-ing different mathematical descriptions of resistancein the linear bearing and various versions of compliantimpact model, including different models of the related

nonlinear damping. From these groups, the optimal (inthe sense of simplicity and correspondence betweenexperimental and simulation data) submodel (1) waschosen with rather small F0 = 0.00997 mm2, forwhich the set of estimated parameters is as follows:k = 1418.9 N/m, c = 6.6511 N · s/m, T = 0.63133 N,kI = 2.3983 · 108 N/m3/2, b = 0.8485 m−n3sn3 ,n2 = 3/2, n3 = 0.18667. Note that small deviationsfrom these derived values or certain small modifica-tions of model of nonlinear damping during the impact(compare for example the models denoted as BC andBD, Table 2, [20]) do not affect essentially the resultsof simulations.

In the reference [20], the correspondence betweenthe mathematical model (1) and real behavior of thesystem for its different operation regimes is also inves-tigated, including free and forced bifurcation vibrationsfor a wider range of forcing frequencies and differentpositions of the obstacle. In all cases, a good agreementbetween experiment and simulation is observed.

Further analysis presented in the paper correspondsto the obstacle position xI = −2.086 · 10−3 m. Sinceits value is negative, at the starting moment the cartis located in the permanent contact with the obsta-cle. Therefore, considerations always deal with thetwo component equation (1) and do not encounter thepurely linear oscillations. This essentially complicatesthe study of model when, for instance, the first impactone is interested in.

3 Experimental and preliminary simulation results

Let us briefly recall that the key point of paper [20] is thecomprehensive analysis of admissible models accom-panied by the comparison of bifurcation diagramswith-out studies of their fine structure. Therefore, it was con-structed the bifurcation diagram for ω ∈ [0, 50] rad/scontaining the interval ω ∈ [35, 50] rad/s where theclassical period-doubling scenario leading to a chaoticattractor creation is observed. To find out the propertiesof these chaotic solutions, the inter-impact sequenceswere constructed numerically and their statistical prop-erties were analyzed in [21]. But, besides the chaotic“window” at rather large ω, the bifurcation diagramcontains another chaotic zone ω < 25 rad/s with oscil-lations whose amplitudes are five times smaller.

It turns out that at these values of ω the scenarioof oscillations development is much more complicated

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1798 S. Skurativskyi et al.

Fig. 2 Bifurcationdiagrams for increasingvalue of excitationfrequency ω constructed onthe base of classicalPoincare sections (leftcolumn) and inter-impactsequences (right column).Figures a, b correspond toexperimental data, whereasc, d concern the simulationresults

14 16 18 20 22 24

−7

−6

−5

−4

−3

ω [rad/s]x[m

m]

14 16 18 20 22 24

0.1

0.2

0.3

0.4

ω [rad/s]

Tj[s]

(b)(a)

14 16 18 20 22 24

−7

−6

−5

−4

−3

ω [rad/s]

x[m

m]

15 20 25

0.05

0.1

0.15

0.2

0.25

0.3

0.350.4

ω [rad/s]

Tj[s]

(d)(c)

and richer. In the present research, to consider theproperties of the regimes, in addition to the classi-cal Poincare section technique the new experimentaldata concerning the inter-impact events are extractedand compared with the results of simulation of sys-tem (1).Moreover, scaling features of bifurcational dia-grams, the TLE and entropy index behavior and newimpact-adding phenomenon are investigated. Estima-tion of the moment of first grazing bifurcation basedon the approximate solution is discussed as well.

Let us consider each step of the studies in detail andstart from the construction of classical Poincare sec-tions, based on the points at time moments t = 2π i/ω,i = 1, 2, . . .. The time moments of impacts also arederived. These moments define the inter-impact (inter-spike) intervals or inter-impact sequence Tj [22]. Thissequence can be regarded as a sort of Poincare section[23,24].

Thus, Fig. 2 exhibits experimental bifurcation dia-grams for increasing value of excitation frequency ωconstructed on the base of classical Poincare sections(Fig. 2a) and inter-impact sequences (Fig. 2b). They arebuilt for the continuous change of bifurcation parame-ter with rate 8.0905 × 10−3 rad/s2. Since the impactsare very weak, a threshold below the obstacle is intro-

duced and impacts are detected during crossing thelevel xI − 0.01 mm. Moreover, the experimental dataare slightly smoothed by passing them through a fil-ter of transfer function (θs + 1)−1, where θ = 10−3 s.To clarify the observed regimes and prove physical andmathematical models consistency, the analogical bifur-cation diagrams (Fig. 2c, d) are constructed numeri-cally. Since experimental data are blurred because ofcertain additional oscillations not taken into account inthe model and noise in the measurement system, it isnot possible to observe some small details on experi-mental bifurcation diagrams. However, one can noticegood general agreement between corresponding bifur-cation diagrams.

Amongothers, the sequenceof impact-adding eventsfor frequencies ω ∈ [14; 17.5] rad/s is observed, bothexperimentally and numerically, while period of theattractor remains equal to the period of external forcing.It is the scenario related to transition between attrac-tors corresponding to permanent contact between thecart and obstacle and solutions with impacts [25]. Thiscase will be investigated in more detail in Sect. 5. Itshould be, however, noted that in the case of experimen-tal bifurcation scenario, the impact-adding sequence isconfirmed at least up to two-impact periodic trajec-

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Bifurcation phenomena and statistical regularities in dynamics of forced impacting oscillator 1799

tory. For higher frequencies, there are observed jumpsbetween the extensions of the branches correspondingto the long inter-impact intervals (upper parts of bifur-cation diagrams in Fig. 2c, d). It can be explained byappearance and loss of some impacts induced by someunpredictable factors present in the real system.

Analyzing Fig. 2, it is experimentally confirmed thescenario of impact-adding events detected later numer-ically, at least up to two-impact periodic trajectory. Itshould be, however, noted that in the case of experi-mental bifurcation diagram, there are observed jumpsbetween the extensions of the branches correspondingto the long inter-impact intervals (upper part of bifurca-tion diagrams), for the periodic trajectories with morethan two impacts. It can be explainedby appearance andloss of some impacts induced by some unpredictablefactors present in the real system.

When ω reaches 20 rad/s, the system goes out tothe attractor with essentially different amplitude. Atincreasing ω, the system undergoes several grazingbifurcations which are identified clearly in the inter-impact sequence diagram (Fig. 2c, d). There is a criticalvalue of ω corresponding to the multi-periodic regimedegeneration. Instead, a periodic trajectory describingthe single-impact regime of oscillations develops.

4 Bifurcation analysis of the mathematical modelfor the oscillator with impacts

Let us now investigate the phenomena admitted by themathematical model (1) in more detail.

Figure 3 exhibits bifurcation diagrams presentingthe system dynamics for frequencies of the exter-nal forcing lower than 25.0 rad/s. They are obtainedbased on numerical simulations for increasing (redpoints) and decreasing (green points) value of bifur-cation parameter ω. Bifurcation diagrams consist ofPoincare maps defined in standard way, i.e., they arebased on sampling the position of the cart every periodof external forcing (Fig. 3a) and inter-impact intervals(Fig. 3b).

Analysis of these diagrams (Fig. 3) for ω ∈(20.5; 25) shows that several grazing bifurcations andattractor coexistence (ω ≈ 22.71) take place. Visualfigure examination suggests the presence of some sortof self-similar diagram structure. In particular, the cas-cade of grazing bifurcations can manifest the scalingproperties. To consider this, let us estimate the fre-

Fig. 3 Bifurcation diagrams constructed on the base of classi-cal Poincare section (a) and interspike intervals (b) when thefrequency ω ∈ [20.5; 25.0]

quencies at which such bifurcations occur. The setf = (20.59; 20.8586; 21.5085; 23.0702) (Fig. 3b) isobtained. The distances between them D = fi+1 −fi = (0.2686; 0.6499; 1.5617). Evaluation of the frac-tions (D2/D1; D3/D2) gives (2.419; 2.403). Derivedquantities are close enough. Hence, it can be assumedthat D2/D1 ≈ D3/D2 meaning that Di form the geo-metric sequence with the common ratio about 2.4.

No less interesting is the bifurcation diagram at ω ∈[17.5; 20.5] (Fig. 4). The classical period-doubling cas-cade exists in the window ω ∈ (20; 20.5). It is alsorevealed the intervals near forcing frequencies whichare equal to 17.9, 18.5, and19.7 rad/swith the hystereticbehavior of bifurcation diagrams. This can imply thecoexistence of at least two attractors.

Note that at the diagram in Fig. 4b several similarobjects can be distinguished. They lay in the intervals(17.6931; 17.8200), (17.9024; 18.2418), (18.3233;19.2300). Their lengths are D = (0.1269; 0.3394;0.9067) and corresponding fractions are (D2/D1; D3/D2) = (2.6745; 2.6714) which are rather close quan-tities. From this, it follows that highlighted parts in the

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1800 S. Skurativskyi et al.

17.5 18 18.5 19 19.5 20 20.5

−5

−4.5

−4

−3.5

−3

ω [rad/s]

x[m

m]

ωω

(a)

(b)

Fig. 4 Bifurcation diagrams constructed on the base of classi-cal Poincare section (a) and interspike intervals (b) when thefrequency ω ∈ [17.5; 20.5]

bifurcation diagram form geometric sequence with thecommon ratio about 2.67.

Top Lyapunov exponents To consider the trajec-tory stability, the TLE [14,26] are derived. In Fig. 5,the dependence of the TLE λ1 on the frequency ω ispresented. One corresponding to the excitation phase,which is always equal to zero, is omitted. Period ofnormalization was chosen as equal to the period ofexternal forcing. The presented bifurcation diagramsare performed for increasing (red line) and decreas-ing (green line) values of bifurcation parameter witha step equal to ±0.005 rad/s. 100 and 50 periods ofexternal forcing were ignored as transient motion afterstarting simulation and after each further jump of bifur-cation parameter, respectively. For each value of exter-nal forcing, the TLE is computed for 103 periods. Inthe diagram of Fig. 5, the positive values of TLE cor-responding to the chaotic attractor existence are dis-tinguished. There are also frequency intervals, e.g.,ω ∈ [18.4; 18.5] or ω ∈ [20; 20.5], which are typ-ical for the period-doubling cascades. In these inter-vals, the TLE vanishing means the appearance of thelimit cycle of double period. Concerning the TLE anal-

18 19 20 21 22 23

−15

−10

−5

0

ω [rad/s]

λ1[1/s]

ωω

Fig. 5 The TLE dependence on the frequency ω

ysis, it should be note that the inter-spike sequencescan provide the possibility to extract the TLE esti-mation [27,28]. It shows promise in addressing thedirectTLEmeasurement from impact sequences duringexperiments.

Statistical analysis Some additional information onthe observed regimes can be obtained from the sta-tistical analysis of interspike sequences [23,29]. Thesequence Tj at fixed ω is considered as a samplearranged in the normalized histogram. Such a sampleallows one to estimate the probability distribution. Thisdistribution is characterized within the framework ofthe Tsallis entropy approach [30,31]. This generalizedentropy ST is defined as follows

ST = kq − 1

(1 −

N∑i=1

pqi

),

N∑i=1

pi = 1,

where the parameter q is the entropy index, N is thetotal number of microstates, k is Boltzmann’s constant(k = 1 can be assumed).

The value of q allows one to characterize the struc-ture of distribution. Namely, when q < 1 the rareevents are important, whereas at q > 1 the frequentevents are more weight [31]. Therefore, the estimationof q is required. This can be done on the base of max-imum entropy principle [32]. It can be proved that theentropy ST gains its maximum STmax at the equiprob-able microstates when pi = 1/N .

STmax = k 1 − N1−q

q − 1 .

The deviation of ST from the STmax can be character-ized by the redundancy function RT = 1− ST/STmax.This function depends on the entropy index q. Thederivation of RT maximum with respect to q provides

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Bifurcation phenomena and statistical regularities in dynamics of forced impacting oscillator 1801

Fig. 6 The normalizedhistogram and theredundancy function RT.The RT maximum isachieved at q = 0.62

Fig. 7 The dependence of the entropy index q on the frequencyω

the estimation of entropy index. Thus, consider theq estimation at fixed ω = 20.465 and arrange thesequence Tj in the normalized histogram (Fig. 6a) at theinterval (0.05; 0.3)with the step 0.001. Some zeroes inthe frequencies sequence should be eliminated. Usingresulting distribution, pi , N , and finally the redundancyfunction RT can be identified. The profile of RT isdepicted in Fig. 6b.

The numerical procedure of “Mathematica” pack-ages gives the maximum estimation q = 0.62 < 1.This means that the distribution describes a large num-ber of rare events which affect the system behav-ior. Doing in the same manner, the dependence ofq on the frequency ω is studied. The ω interval(20.433; 20.515), when the chaotic regimes are devel-oped, is chosen. Estimating the entropy indexes q withthe step 0.005, the graph in Fig. 7 is plotted. From theanalysis of this graph, it follows that chaotic attrac-tors are characterized by q < 1 which are changedweakly. At the edges of frequency interval, when thechaotic attractors are reduced to the almost periodictrajectories, the entropy index grows abruptly to 1.

This means that the distribution contains fewer vari-ants which occur more frequently.

5 The system behavior at low frequencies

The regimes created at low frequencies deserve to themore detailed analysis. For the best of our knowledge,this type of dynamics is not widely known [25]. It isworth noting that the obstacle grazing at frequenciesfrom the interval ω ∈ [14; 18] does not cause the tra-jectory bifurcation as opposed to the phenomena atω ∈ [20.5; 23]. Let us construct the phase portraitsof typical orbits forming the so-called impact-addingscenario (do not confuse with the grazing-inducedimpact-adding bifurcation [10]) presented in Fig. 2d.For ω = 14 rad/s, one can observe a periodic attractor(Fig. 8) corresponding to the cart being in permanentcontactwith the stop. Position of the obstacle is denotedby red line, while red dot corresponds to Poincare sec-tion t = 2π i/ω, i = 1, 2, . . . .

For ω = 14.1 rad/s, a phase portrait with a seg-ment of trajectory detached from the obstacle and oneimpact is seen. The intersections of increasing positionx of the cart with the level of obstacle xI (instancesof impacts) are denoted by blue dots. Further increasein the forcing frequency leads to separation from theobstacle of further segments of the orbit and adding theconsecutive impacts. For ω = 17.5 rad/s, one can seean orbit with five impacts and of period equal to theperiod of external excitation. Finally, an irregular orbitand a periodic stable solution of double periodwith fiveimpacts for forcing frequency equal to 18 and 19 rad/s,respectively, are shown.

As it is shown in the first phase portrait of Fig. 8at ω = 14 rad/s, the oscillations exist in the region,

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1802 S. Skurativskyi et al.

Fig. 8 Phase portraits presenting bifurcation scenario of impact-adding corresponding to bifurcation diagram exhibited in Fig. 2d forthe interval ω ∈ [14; 19] rad/s

where the equation of motion is defined by the singleexpression

mẍ + kx + cẋ + T sgn(ẋ)+kI(x − x1)n1(1 + b · sgnẋ |ẋ |n3)

= m0eω2 sinωt. (2)Since k � kI, then the nonlinear term prevails andequation (2) is closer to the strongly nonlinear equa-tion. The profile (Fig. 9a) of periodic solution testifies,

furthermore, about the strongly nonlinear character ofoscillations.

In order to understand the nature of this regime, letus consider the Fourier spectrum of solution (Fig. 9b).It is seen that the signal contains only two dominantfrequencies, i.e., ω and 2ω. Therefore, the main fea-tures of solution can be approximated by the followingexpression

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Bifurcation phenomena and statistical regularities in dynamics of forced impacting oscillator 1803

x = a0 +2∑

k=1(ak cos kωt + bk sin kωt) . (3)

Using the numerical procedure from the “Mathemat-ica” packages, the solution (3) can be derived in theform

x = −2.0813 × 10−3 − 1.3136 × 10−6 cosωt +3.9328×10−6 sinωt+6.6538×10−7 cos 2ωt+2.915×10−7 sin 2ωt .

But due to the presence of power and discontinuousfunctions, one cannot get the analytical expressions forthe required coefficients of Fourier series. Instead, thefirst harmonics can be treated.

Consider the linearized counterpart [33] of equation(2). This equation can be considered in the vicinityof steady state x = X0 = const which satisfies thealgebraic equation kX0 + kI(X0 − xI)n1 = 0. Fromthis, it follows X0 = −2.0807 × 10−3.

Next let us suppose that the dissipation is small andconsider the deviation from the steady state y = x+X0.Equation (2) reduces to the following one

mÿ + k(y + X0) + cẏ+ kI(y + X0 − xI)n1(1 + �b · sgnẏ|ẏ|n3)

= m0eω2 sinωt − �T sgn(ẏ),where � is a small parameter prescribing formally thesmallness of nonlinear friction.

Then, the simple linear approximation gives thesolution x = X0 + A1 sinωt + A2 cosωt , where

A1 = m0eω2Δ

(ωc)2 + Δ2 , A2 = −cm0eω3

(ωc)2 + Δ2 ,Δ = k + 1.5kI

√X0 − xI − mω2.

Thus, the solution at the parameters fixed above is

x = −2.0807 × 10−3 − 4.7779 × 10−10 cosωt+ 4.2606 × 10−6 sinωt. (4)

This form of solution allows one to estimate the valueof frequency at which grazing the x = xI is possible.Namely, let us derive ω when x = xI, i.e.,

xI = X0 − m0eω2√

(ωc)2 + Δ2 .

From this, it follows that ω = 15.657. This value isoverestimated in comparison with the exact value offirst grazing.

To take into account the equation structure undermanipulations more effectively, another method of

approximation for the y amplitude is applied. Here,the approach used in [34,35] is referred. If the viscos-ity takes into account, then the approximate solution ofEq. (2) with the external force F = F0 sinωt reads asfollows y = A sin(ωt − α). If instead the solution hasthe form y = A sinωt , then the force can be written inthe form F = F0 sin(ωt +α). Since the equation holdsfor all ωt , and hence for ωt = π/2. But in this casey = A, dy/dt = 0, d2y/dt2 = −Aω2, F = F0 cosα.

Analogously, for ωt = 0 when y = 0, dy/dt =Aω, d

2ydt2

= 0, F = F0 sin α. This allows us to writethe algebraic system with respect to A and α:

− mAω2 + k(A + X0) + kI(A + X0 − xI)n1= m0eω2 cosα,

cAω = m0eω2 sin α.Then the approximated solution reads

x = X0 + A(ω) sin(ωt − α). (5)Substituting the parameters’ value, the specified solu-tion is obtained in the following form

x = −2.0807 × 10−3 − 3.5707 × 10−10 cosωt+ 3.6833 × 10−6 sinωt. (6)

The comparison of “exact” solution of system (2),numerically derived solution (3) and (6) at ω = 14,is presented in Fig. 9a. From the analysis of this figure,it can be concluded that the last method gives betterresults than the method of simple linearization.

Using the solution (5), the estimation of grazingphenomenon is possible. Solving the system xI =X0 + A(ω), one gets ω = 12.789. It is evident thatthis value is undervalued with respect to exact value ofthe first grazing. Analyzing Eq. (2), it is worth notingthat the deviation from the fixed point X0 is not smallenough. Therefore, the omitting of high-order terms,appearing in the Taylor series for the power function(y + X0 − xI)n1 expanded near X0 − xI, contributes anotable inaccuracy in the approximated equation.

Between the first grazing (ω ≈ 14 rad/s) and chaoticmode (ω ≈ 18 rad/s) oscillations manifest someinteresting features. Analyzing the profiles of solution(Fig. 10), one observes that trajectories contain smalloscillations near stopper position. This regime lookslike an incomplete chattering [11,19,24]. The finitenumber of small oscillations and the length of shelves(Fig. 10a) are caused by dry friction incorporation.

It can be checked the Fourier spectrum containsfour dominant frequencies jω, j = 1 . . . 4. At ω ≈

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1804 S. Skurativskyi et al.

Fig. 9 The numericalsolution (black points) ofsystem (2), numericallyapproximated solution (3)(green curve), the profile offunction (4) (blue curve)and the graph of function (6)(brown curve). The Fourierspectrum for the solution ofsystem (2) depicted in theleft panel with black points.For both figures ω = 14.(Color figure online)

0.2 0.4 0.6 0.8 1.0 1.2t

-0.002086

-0.002084

-0.002082

-0.002080

-0.002076x

20 40 60 80 100 120

0.000028

|F|

(b)(a)

0 0.2 0.4−2.2

−2.18

−2.16

−2.14

−2.12

−2.1

−2.08

t [s]

x[m

m]

ω = 15 [rad/s]

0 0.2 0.4

−2.6

−2.5

−2.4

−2.3

−2.2

−2.1

t [s]

x[m

m]

x[m

m]

ω = 16.7 [rad/s]

0 0.5 1 1.5−3.2

−3

−2.8

−2.6

−2.4

−2.2

t [s]

ω= 18 [rad/s]

(c)(b)(a)

Fig. 10 The profiles of the solution x-component at different ω

16.7 rad/s the number of micro-oscillations increases,and their maximal values are almost linearly decreas-ing (Fig. 10b). The number of excited frequencies inFourier spectrum reaches dozen of commensurate val-ues. Exploring the micro-oscillations behavior in thechaotic mode at ω = 18 rad/s (Fig. 10c), it should benoted that three most prevailing peaks remain almostunchanged during time increasing, whereas the small-est peak undergoes the essential deviations, althoughfour hits in the chattering window are still observed.The Fourier spectrum possesses clearly visible severaldominant frequencies, and its smooth part is also essen-tially excited. Since these regimes possess the commen-surate frequencies in Fourier spectra, one can considerthem as nonlinear quasiperiodic oscillations forming inthe phase space closed orbit on the torus surface.

6 Concluding remarks

Summarizing aforementioned results, the researchpresents newexperimental and theoretical studies of the

physical oscillating systemwith impacts. The observeddynamics of this system is managed to play with thehelp of nonsmooth mathematical model which takesinto account the dry and viscous friction, generalizedHertz contact law during impacts. It should be empha-sized that the system is considered under the conditionswhen pure linear dynamics is absent and the most ofanalytical methods are not applicable. Therefore, theinformation on the systemproperties ismainly obtainedfrom the numerical integration, the results of which areused in the qualitative and statistical analysis. Deal-ing with impacts, it was shown that the inter-spikesequences analysis has significant capability to exper-imental and theoretical investigation of impacting sys-tems.

The main results of the work can be enumerated asfollows:

– In the considered interval of frequency ω, the sys-tem possesses the periodic, multi-periodic, andchaotic attractors. Among bifurcation scenarios,the period-doubling cascades leading to chaotic

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Bifurcation phenomena and statistical regularities in dynamics of forced impacting oscillator 1805

attractor creation and stiff attractor appearance anddestruction are revealed. The windows with hys-teretic behavior andperiod-adding cascades are dis-covered as well. These dynamical phenomena areendowed by scaling properties.

– It is shown that in the system the impact-addingscenario is realized which does not show up in theclassic Poincare diagram, whereas the inter-impactsequence technique allows us to identify main itsfeatures.

– The inter-impact sequences admit analysis via theq-entropy concept. This approach tells us that thedistribution of impacts characterizes the set ofrare events. The parameter of this distribution, i.e.,entropy index q, is sensitive to the approaching to abifurcation point. Therefore, it can be regarded as anatural identificator of chaotic attractor variability.

– the estimation of the moment of first grazing bifur-cation is carried out with the help of approximatesolution derivation. In particular, despite the sat-isfactory estimation of bifurcation value of fre-quency, the improvement can be achieved by theincorporation of additional terms in the approxi-mation. However, then the derivation of bifurca-tion value becomes more complex task. On theother hand, to improve the approximation proce-dure, instead of linear generating system the non-linear one can be chosen. This way of studies isvery promising but can be realized for that param-eter values when the generating system possessessolutions written in the closed analytic form.

Theoretical studies thus clarified the experimentallyobserved phenomena, shed the light on the specific sta-tistical properties of chaotic regimes, allowed one toidentify novel bifurcation events and open up new chal-lenges in the field of nonlinear dynamics.

Acknowledgements This work has been supported by the Pol-ish National Science Centre under the Grant OPUS 14 No.2017/27/B/ST8/01330.

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

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Bifurcation phenomena and statistical regularities in dynamics of forced impacting oscillatorAbstract1 Introduction2 Experimental stand and mathematical model3 Experimental and preliminary simulation results4 Bifurcation analysis of the mathematical model for the oscillator with impacts5 The system behavior at low frequencies6 Concluding remarksAcknowledgementsReferences