Prediction and diagnostics of crises and critical states in an unusual vibro-impact system with soft impact

In nonlinear dynamical systems, whenchanging control parameters, critical states may occur. To predict these undesirable events, it is necessary to study the system dynamic behavior with changing in different control parameters. Two-body 2-DOF vibro-impact system is under consideration. The huge mass of the upper body, which can break away from the lower one and be in “free flight” for a short time, may be attributed to its specific and is a cause to consider it unusual. Some changes in the parameters of both external loading and the system itself can increase the efficiency of the machine, the mathematical model of which is the considered vibro-impact system. But the same changes lead to appearance of unwanted nonlinear phenomena, such as permanent sustained chaos, transient chaos, crisis-induced intermittency, and coexisting regimes with larger oscillatory amplitudes. The article shows their occurrence, traces the precursors, carries out their diagnostics by both traditional and less common methods. The description of many numerical experiments is accompanied by numerous figures and tables that clearly and convincingly demonstrate the results obtained.


Introduction
The vibro-impact system is a strongly nonlinear nonsmooth discontinuous system. In such systems, undesirable motion modes can occur when various control parameters are changed. These may be dangerous critical states that should be avoided in both design and operation. The most important parameters that strongly affect the system dynamics should be chosen as control ones. These can be parameters of both external loading and the system itself. To predict the critical states and avoid them in a particular system, it is necessary to study its dynamical behavior when changing different control parameters.
The system under consideration is a two-body 2-DOF vibro-impact system, which is a mathematical model of a platform-vibrator with shock. It is employed in the construction industry to compact and mold large concrete products using shock-vibration technology. The asymmetric accelerations of the mold with concrete that arise during its application increase the concrete compaction efficiency. The problems associated with the efficiency of the platform-vibrator and its parameters choice are still of interest today [1,2].
The platform-vibrator with shock uses shockvibration technology. The efficiency of molding reinforced concrete products using this technology largely depends on the selected equipment parameters and, as a result, on the oscillatory modes of action on the compacted concrete mixture. The successful choice of the parameters of the oscillatory system "platform tableelastic element-mold with concrete" can significantly increase the efficiency of vibroforming products when concrete products are formed in the molds that are freely installed on the table without fastening through elastic elements. Appropriate modes of action on the compacted concrete mixture make it possible to obtain concrete products with high density and uniformity. Experiments carried out in laboratories with different layouts [3] showed a great influence of both the ratio of the masses of the platform table and the mold with concrete, and the stiffness parameters on the implementation of the oncoming movement of the table and the mold. The gasket stiffness is one of the most important parameter. It is determined by the shape, sizes and modulus of elasticity of the gasket material. In the factory, it is usually recommended to use the sheets of rubber or plastic for gaskets. One of the main evaluation criterion for the efficiency of the machine operation is the maximum acceleration of the mold with concrete at the moment of its impact on the platform table. Experiments show that light molds using provides the larger acceleration.
The numerous numerical experiments presented in this paper for a model with parameters values close to real ones clearly show the influence of the main machine parameters on its dynamics. Their results allow you to choose the optimal parameters values that can ensure the efficient machine operation and will not lead to undesirable and harmful events.
The results obtained help in the development and improvement of vibroforming machines without mold fastening, since this eliminates the need for fastening means, simplifies the design and reduces the metal consumption.
We call vibro-impact system unusual because of tearing off a heavy mold with concrete from the plat- form table and its subsequent fall with an impact on the  rubber gasket attached to the table. In modern scientific literature, there is an enormous amount of publications on vibro-impact systems, in which various aspects of their dynamics are discussed. The papers present both analytical and experimental results, as well as numerical simulations. In [4][5][6][7], the authors discuss the possibilities of analytical and semianalytical methods for predicting and determining peri-odic motions to chaos. In [8], the authors propose an approach to predicting vibro-impact processes, which, in their opinion, is a major step forward in making vibro-impact processes predictable and will be useful for the predictive design of a multitude of engineering applications. The prediction is computationally demanding, but challenging; the proposed approach is described on the example of two cantilever beams with impacts at the ends under harmonic excitation. The emergence of hyperchaos is discussed in the latest recent work [9]. A sudden large increase in amplitude accompanies hyperchaos; but the occurrence of the large-amplitude events is a rare phenomenon; three models exhibiting it are presented. The difference in dynamic behavior of vibro-impact system under ideal and non-ideal excitation was studied in the works [10][11][12][13]. The authors study in [10] what changes occur in the dynamics of a vibro-impact system when an ideal excitation in form of a harmonic force becomes nonideal one. In particular, they observe the Sommerfeld effect. The authors believe that the use of a model with a non-ideal excitation source makes it possible to obtain more realistic and consistent results. In this work, an impact is simulated by a linear force by introducing a linear elastic spring with a high stiffness. In [11], the authors investigate the effect of a low-mass impact damper on the dynamics of a nonlinear oscillator under non-ideal excitation. They observe two coexisting chaotic attractors and show that impact dampers of small masses are effective at suppressing chaos, but make the dynamical system less predictable. In [12], the authors study the suppression of chaotic motion by a smart damper, which changes the damping coefficient according to the sign of the relative velocity. They also investigate a non-ideal system and the influence of the power supply on attractors. The coexisting attractors, both periodic of various periods, and chaotic, are identified. In this paper, there are presented examples of controlling chaotic dynamics of vibro-impact and nonideal oscillators. The used control procedure may help avoiding undesirable behavior of mechanical systems with practical applications. An impact rule in [11,12] is the reset of the velocity after an impact using the restitution coefficient in accordance with Newton's law of inelastic impact. The article [13] illustrates dynamics of a non-ideal mechanical system with an electrical DC motor as a non-ideal energy source. It analyzes the vibrations due to impact of the non-ideal mechanical system with a rigid wall. The Sommerfeld effect is numerically obtained for the studied system. The collisions between bodies can be described with Hertzian contact theory. The authors believe that method given in this work can be extended for other problems of vibro-impact. The dynamic behavior of vibro-impact systems with non-ideal excitation is also studied in the works [14][15][16]. In [14], the results found for mechanical systems with ideal and non-ideal excitation are compared mutually; the differences between them are pointed out. The region of two coexisting solutions is shown, each of which depends on the initial conditions. The chapter [16] analyzes the steady-state and transient motions when the system starts as a vibro-impact one and switch to non-impact motion. Regions of impact, non-impact and multiple solution are pointed out as the positions where jumps are occurring. The impact model in these works is inelastic, where the coefficient of restitution describes how much the oscillator speed is changing after every impact. The dynamic behavior of 3-DOF vibro-impact systems under harmonic excitation was studied in the articles [17,18]. In [17], the authors observed four scenarios for the occurrence of multistability with coexisting attractors. The paper [18] describes the loss of stability by periodic motion with two symmetric impacts and arising of a sequence of period-doubling bifurcations and a chaotic motion in a relatively narrow range. The responses of SDOF systems with one or two soft constraints of different types with a change in selected parameters were studied in [19][20][21]. Elastic linear springs simulate a soft impact; the numerical model assumes the Kelvin-Voigt model in [21]. There is a good survey and bibliography in this work. In recent article [22], the authors reconsider and look more closely at soft and hard impact models applied to vibro-impact systems. They emphasize that the specific dynamic behavior of the system is obviously related to its physical model in general and to impact model in particular.
The creation and detailed description of a mathematical model of a platform-vibrator with shock were given in [23][24][25][26]. In previous works, we studied the dynamic behavior of a platform-vibrator with shock when both the system parameters and the parameters of the external exciting force change. In [23], we examined the effect of the exciting frequency, in [24], the effect of the technological mass of the mold with concrete, in [25], the effect of the stiffness parameters. In all these cases, interesting nonlinear phenomena that are inherent and often unique to strongly nonlinear non-smooth discon-tinuous systems were observed. Chaotic motion, interior crisis, crisis-induced intermittency, transient chaos and a hysteresis zone with coexisting regimes obtained for different initial conditions were watched. These phenomena are widely discussed in the scientific literature today [27][28][29][30][31]. The chapters in the books [32][33][34] "facilitate a better understanding of the mechanisms and phenomena in nonlinear dynamics and develop the corresponding mathematical theory to apply nonlinear design to practical engineering." In this article, undesirable modes and critical states are singled out among the previously obtained results and systematized. Particular attention is paid to finding precursors of critical states and indicating the parameters ranges in which unwanted regimes may occur.
Thus, the goals of this paper are: • To show the dynamic behavior of a specific vibroimpact system (a platform-vibrator with shock) with varying different control parameters and highlight its critical states; • to show the parameters ranges where the undesirable movements can occur; • to show the possibility of forecasting crises and critical states; • to add information to fundamental knowledge about phenomena that occur in nonlinear dynamical systems.

Mathematical model of platform-vibrator with shock
The mathematical model of the dynamical vibroimpact system is based on a low-frequency platformvibrator with shock without fastening the mold. The platform-vibrator consists of four blocks equipped with vibrators that generate the harmonic exiting force. The model, in which four identical blocks are combined into one block, takes the form shown in Fig. 1. It is two-body 2-DOF vibro-impact system. Its lower body-platform table with mass m 1 is connected to the base by a linear vibro-isolating spring with stiffness k 1 and a damper with damping coefficient c 1 . A rubber gasket, which we accept as elastic, is attached to the table; its thickness is h, and stiffness is k 0 . Its upper body is a mold with concrete, it has a huge mass m 2 since a platformvibrator with shock is used for compaction and molding of large products, up to 15,000-18,000 kg. The stops are attached to the table, they do not allow turning and slipping of the mold and permit only vertical movement. The technical literature have given the numerical parameters [35], but some of them were absent in the literature and were chosen in such a way as to ensure model verification.
The movement starts from an equilibrium state with the beginning of the exciting force action generated by vibrators located under the table. First, both bodies move together, then the upper body tears off from the lower one, since it does not fasten to it. The bodies move separately for a short time, then the upper body falls on the lower one, impacts the rubber gasket, and bounces off it again. The collision with the rubber gasket is soft due to its softness and flexibility. The motion process is repeated; we see the vibro-impact movement of the model, which is described by the following equations: The exciting force is harmonic F(t) = P cos(ωt + ϕ 0 ); its period is T = 2π/ω. The initial conditions are: The static deformation of the gasket is: λ 0 = m 2 g/k 0 , g is the acceleration due to gravity. Here, the following notations are introduced: H (z) is Heaviside step function relatively bodies' rapprochement z = h − (y 2 − y 1 ). F con (z) is contact interactive force that simulates an impact and acts only during an impact, when (y 2 − y 1 ) ≤ h.
The damping forces are taken to be proportional to the first degree of velocity: in the rubber gasket F damp0 = c 0ẏ1 , in the vibro-isolating spring F damp1 = c 1ẏ1 . The influence of the concrete mixture can be taken into account as some additional damping F damp2 = c 2ẏ2 .
The movement is vertical along the y axis, the origin of which is taken at the table mass center in a state of static equilibrium. The coordinates y 1 and y 2 correspond to the displacements of the mass centers of the lower and upper bodies.
The problem of impact simulation, especially soft impact, is very important when considering the vibroimpact movement. We have discussed this issue in our previous papers. In particular, in [25], we performed a detailed comparison of soft impact simulation by a linear force and a nonlinear Hertzian force. Since the soft impact is not instantaneous and its duration is quite long, it is advisable to simulate it by a nonlinear contact Hertzian force in accordance with his quasi-static contact theory [36] and take the contact interactive force according to following formula: Here, ν i and E i -Poisson's ratios and Young's moduli of elasticity for both bodies; A, B, q are constants characterizing the local geometry of the contact zone. The gasket surface is flat, but it can be considered as a sphere of the large radius R to use Hertz's formulas. Then, in the collision of a plane (mold) and a sphere (rubber gasket) A = B = 1/2R, q = 0.318. It should be noted that arbitrariness in the choice of R has practically no effect on the results. As given in Table 1, a difference of one unit appears only in the fourth decimal place for the platform table oscillation amplitude and in the third for mold with concrete. The shock-vibration technology implemented in platform-vibrator work allows creating asymmetric accelerations of the mold with concrete, that is, the accelerations of different magnitudes when the mold moves up and down. The mold acceleration in the uppermost position is the upper acceleration w U , the mold acceleration in the lowest position is the lower acceleration w L . Their ratio w L /w U is the coefficient of the asymmetry. Exactly the lower acceleration w L is considered as acceleration that realizes the mix com- paction. The prevalence of w L over w U accelerates the compaction process. The maximum acceleration of the mold with concrete in the lowest position w L , that is, at the moment of collision with the platform table, is one of the main criteria for the technical assessment of the vibration machine efficiency. The asymmetric vibrations use makes it possible to increase the value of this compaction acceleration up to several g(2 . . . 4 . . . 6g).
Verification and validation of this model are described in [25]. Its adequacy is confirmed by experimental data obtained during design and operation of this low-frequency machine [1,25,37]: • A steady-state T -periodic regime with one impact per cycle is set in the platform-vibrator after short transitional process. • In this regime, the mold with concrete has an oscillatory amplitude of 0.79 mm with the required 0.8 − 1 mm. • The coefficient of asymmetric acceleration is w L / w U = 3.6 with the required ∼ 4.
It should be noted that the oscillatory amplitude for non-harmonic vibrations is calculated by a simple formula after direct numerical integration of stiff differential Eq. (1).

Control parameters
The control parameters choice is important when studying the dynamical behavior of a vibro-impact system. It should be taken into account that changing some parameters can increase the lower impact acceleration w L and improve the quality of concrete compaction. However, these same changes give rise to unwanted nonlinear phenomena such as interior and boundary crises, chaotic motion, crisis-induced intermittency, transient chaos that generates a chaotic regime, and a hysteresis zone with coexisting regimes with larger amplitudes obtained for different initial conditions.  Figure 2 shows the acceleration of the mold with concrete for different values of the rubber gasket stiffness k 0 . Figure 3 depicts the dependence of the impact lower acceleration of the mold w L on the parameters chosen as control ones. These are the stiffness of the rubber gasket k 0 , and the technological mass of the mold with concrete m 2 . Impact acceleration w L increases with an increase in the gasket stiffness k 0 and a decrease in the mold mass m 2 . It also increases with a decrease in the exciting frequency ω and spring stiffness k 1 and with an increase in the gasket modulus of elasticity E 1 , which are also chosen as control parameters. But changing these parameters can lead to critical states, so it is necessary to study the dynamical behavior of the platform-vibrator under such changes.

Frequency of exciting force ω is a control parameter
The recommended frequency of exciting force is ω = 157 rad · s −1 (25 Hz), since the platform-vibrator with shock is a low-frequency machine. A decrease in this frequency can increase the lower impact acceleration of the mold w L , but at the same time lead to the emergence of critical states such as chaotic motion and the coexisting modes in the hysteresis zone. The two graphs in Fig. 4 represent a picture of motion over a wide range of the control parameter ω. The coexisting regimes in the hysteresis zone in a narrow frequency range are clearly seen in both graphs. The curves in Fig. 4a are the amplitude-frequency responses for both bodies in main (1,1)-regime (black and gray curves) and in coexisting regimes (yellow and red curves), which were obtained for other initial conditions. Note 1 (n, m)-regime is the periodic mode with period nT and m impacts per cycle.
The lower black curve in Fig. 4b shows the largest Lyapunov exponent λ max with a negative sign for the main periodic (1,1)-regime. The upper red curve gives it for coexisting modes, where it changes sign. It has a positive sign in a narrow frequency range, which indicates the possibility of a chaotic regime existence. Indeed, in this frequency range, there is a chaotic motion, its characteristics are shown in Figs. 5 and 6.
Poincaré map is neither a finite set of points nor a closed curve; this is an infinite number of pointssome smear. A continuous Fourier spectrum has many frequencies of low intensity in its broad part, which is typical for chaotic movement. Figure 6 shows the fractal structure of the Poincaré map obtained at three magnifications for 71,000 points in the Poincaré map. The presence of a fractal structure in the Poincaré map, that is, the nesting of a structure within structure also confirms the chaotic nature of the motion.
As shown in Fig. 4b, chaotic movement can be implemented in a narrow frequency range in coexisting regimes. The routes to chaos and its harbingers can be seen if we trace the modes alternation at the borders of this range, as shown in Table 2. We see a period doubling on the left border, and it may be considered as a chaos precursor; this is Feigenbaum's route. A short period doubling is also observed on the right border. Table 2 shows a global picture of the complexity of the  Table 2 Global picture of the complex dynamics of the system with a change in the exciting frequency ω system and the richness of its dynamics when exciting frequency ω changes. The frequency ranges in which unwanted events can occur are highlighted in red.
Thus, at low exciting frequencies, an undesirable chaotic regime occurs. Period doubling can serve as a prediction of its arising.

Technological mass of the mold with concrete m 2 is a control parameter
Heavy molds with concrete are commonly used when operating a platform-vibrator. The use of lighter molds, on the one hand, can increase its lower impact accel-eration, and on the other hand, lead to emergence of undesirable regimes and critical states. Three figures clearly show the overall motion picture in a wide range of the control parameter ( Figs. 7 and 8). The coordinates of the points on the Poincaré maps depending on the control parameter values are shown in the bifurcation diagram (Fig. 8). Separate dots show periodic regimes, their number on one vertical is n in the nTperiodic mode, and solid vertical lines correspond to the chaotic regime. Red dots and lines, as in Fig. 7b, conform to coexisting regimes in the hysteresis area. The routes to chaos in main "black" regime on the left and right are clearly visible on the bifurcation diagram. These are sudden arising of a chaotic movement on the left, that is, a boundary crisis, which is also called a blue-sky catastrophe, and a cascade of period-doubling on the right. The modes alternation to the left and right of chaos is shown in Table 3. Chaotic regimes also exist in the hysteresis zone, when the largest Lyapunov expo-nent has a positive sign. Table 3 also shows the mode alternation in this region. In general, Table 3 shows a global picture of the complexity of the system and the richness of its dynamics when mold mass m 2 changes. The ranges of mold mass in which unwanted phenomena occur are highlighted in red.
On the left border at m 2 =3200 kg, a periodic attractor vanishes and a chaotic attractor suddenly arises in a boundary crisis [39]. But a closer look says that it is Type I intermittency by Pomeau and Manneville ( Fig. 9) [30]. Here, Floquet multiplier modulus becomes greater than 1 (Reμ =+1.16, I mμ =0 ), and it crosses the unit circle at point +1. Figure 10 presents the surface of wavelet coefficients and the projection of wavelet surface for this motion, which clearly show bursts of chaos among an almost periodic signal. The wavelet characteristics were obtained using Continu- Table 3 Global picture of the complex dynamics of the system with a change in the technological mass of the mold with concrete m 2 Fig. 9 Time history for the mold with concrete in intermittency at m 2 =3200 kg ous Wavelet Transform (CWT) with Morlet wavelet and MATLAB software [40,41].
After Type I intermittency at m 2 =3200 kg, crisisinduced intermittency with interior crises gradually develops (Fig. 11) and finishes with periodic movement at m 2 =5600 kg. We observe an interior crisis [42] where we watch a sudden discontinuous change in a chaotic attractor size as the control system parameter varies.
In a narrow control parameter range, transient chaos arises in the hysteresis zone among the periodic windows. Transient chaos is a chaos with a finite lifetime in difference from permanent chaos that has an infinite lifetime. With the same value of the control parameter, the motion changes its character. If initially it was chaotic, then after a while it becomes periodic [43]. This abrupt changing is clearly seen in the time series and contact force graphs in Fig. 12.
The transient chaos lifetime is not constant, its shape and lifetime strongly depend on both the initial conditions and the control parameter value [44][45][46]. Therefore, it is advisable to find the average lifetime, which quantitatively describes how long the transient chaos exists, and is its general characteristic [28,47,48]. Averaging is recommended to be carried out over a large ensemble of realizations, since often the difference in lifetime values for the same control parameter meaning and different initial conditions is very large. The dependence of the average transient chaos lifetime T obtained over 12 realizations on the mold mass m 2 is shown in Fig. 13. As we see, the average transient lifetime T increases dramatically with m 2 reducing, since it obeys an exponential law T ≈ Ce −κm 2 where κ > 0. In Fig. 13, this dependence is presented as a linear-linear graph and as a logarithmic versus the linear graph in the inset, where it is depicted as a straight line with a slope −κ, κ = 0.089. The red curve and straight line correspond to the equations of an exponential law and a straight line. The slope κ, called the escape rate, is a quantity that measures the duration of the transient chaos existence. It is considered to be a more convincing characteristic of the breakup process than the average lifetime T , since T depends on many details, for example, on the initial conditions chosen for realizations and on their number taken for averaging [28].
Thus, at low values of technological mass of the mold with concrete m 2 , that is, when using light molds, undesirable regimes occur. Period doubling can predict the arising of chaotic motion, if this mass is reduced. If it is increased, the boundary crisis occurs, followed by intermittency, and then crisis-induced intermittency with interior crises. The transient chaos occurs in the hysteresis zone among the periodic windows. Therefore, the use of lightweight molds may lead to unwanted modes, although their use may improve the efficiency of concrete compaction.

Stiffness parameters are the control parameters
The article considers three stiffness parameters, namely the stiffness of vibro-isolating spring k 1 , the rubber gasket stiffness k 0 , and Young's modulus of elasticity for rubber gasket E 1 . All of them have an influence on the dynamic behavior of a platform-vibrator; their change, on the one hand, can increase the impact acceleration of the mold with concrete and improve the efficiency of concrete compaction, but at the same time, on the other hand, lead to arising of undesirable motion modes and critical states.
6.1 Effect of the vibro-isolating spring stiffness k 1 Three plots clearly show the overall motion picture in a wide range of the control parameter (Fig. 14, Fig. 15).
In Fig. 15a, the oscillatory amplitudes in main regime are depicted by black A A curve; the larger amplitudes in coexisting regimes obtained under different initial conditions are shown by red B B , blue CC , and green D D curves for small values of the spring stiffness k 1 . Figure 15b shows that a chaotic regime can occur in a very narrow range of low stiffness values, where the largest Lyapunov exponent λ max has a positive sign. At these small stiffness values, the crisis-induced intermittency and interior crises were observed. In Fig. 16, we present them for a lighter mold with mass m 2 =9000 kg, where they are pronounced. The interior crisis, also known as an explosive bifurcation, leads to the sudden widening of a chaotic attractor; one can see a sudden increase (or decrease) in the size of a chaotic attractor [28]. Following an interior crisis, a crisis-induced intermittency is observed. This type of intermittency is characterized by permanent jumps between two chaotic attractors [49].
The very impressive wavelet characteristics of this movement, presented in Fig. 17, clearly show the frequent alternation of various chaotic regimes.
The general picture of the complex dynamics of the system with a change in the stiffness of the vibroisolating spring k 1 is given in Table 4. Two graphs in Fig. 18 show the overall motion picture in a wide range of the control parameter. One can see the huge jump in both graphs, which shows the abrupt change of the movement nature at this moment.
With an increase in the gasket stiffness k 0 , the sustained chaos with an infinite lifetime arises in the sys-tem. But transient chaos with a finite average lifetime T exists in a narrow control parameter range and precedes the birth of a permanent one; in this case, it is a harbinger of chaotic motion. Figure 19 shows the exponential growth of the average transient lifetime T , calculated by averaging over 11 realizations, with increasing gasket stiffness k 0 , since the change in the average lifetime is described by the exponential law T ≈ Ce κk 0 , where κ > 0, C = 5.47 · 10 −45 , escape rate κ = 6.47 · 10 −8 . In Fig. 19, this dependence is presented as a linear-linear graph and as a logarithmic versus the linear graph in the inset. The red curve and straight line correspond to the equations of an exponential law and a straight line.

Effect of Young's modulus of elasticity for rubber
gasket E 1 Figure 20 and Fig. 21 show a general movement picture with a change in the elastic modulus of the rubber gasket. Vertical lines with arrows indicate the dynamical system "jumps" from one regime to another. All three graphs show how dramatically the movement  Table 4 Global picture of the complex dynamics of the system with a change in the vibro-isolating spring stiffness k 1 Fig. 18 Dependence on the stiffness of rubber gasket k 0  changes at the moment when they make an abrupt huge jump. In the hysteresis zone between points B and C at E 1 = 1.2 · 10 10 N ·m −2 , there is a T -periodic regime with short transitional process on the AB branch, when zero initial conditions are taken from the resting state. On the C D branch, there is a chaotic mode, more precisely, transient chaos. Its lifetime depends on the initial conditions. Figure 22 shows these modes for the platform table.
Thus, a change in stiffness parameters, namely a decrease in the stiffness of vibro-isolating spring, an increase in the stiffness of the rubber gasket and in its modulus of elasticity, which could improve the efficiency of concrete compaction, leads to undesirable motions, such as coexisting regimes with large amplitudes, crisis-induced intermittency and interior crises, transient and permanent chaos.
The general picture of the complex dynamics of the system with a change in the stiffness parameters of the rubber gasket k 0 and E 1 is presented in Table 5.

Conclusions
The considered vibro-impact system with soft impact is a mathematical model of a platform-vibrator with shock, which is used in construction industry for concrete compaction and concrete products molding. Some changes of the system parameters could improve the efficiency of concrete compaction. But these same changes can lead to undesirable and even dangerous nonlinear phenomena, such as interior and boundary  table   Table 5 Global picture of the complex dynamics of the system with a change in the parameters of the rubber gasket stiffness crises, chaotic motion, crisis-induced intermittency, transient chaos that generates a chaotic regime, and a hysteresis zone with coexisting regimes with large amplitudes obtained for different initial conditions. The performed analysis of these phenomena allowed us to indicate the parameters ranges where crises and critical states take place. The constructed Tables 2-5 clearly show a global picture of the complexity of the system and the richness of its dynamics. The parameters ranges in which crises and critical states occur are highlighted in red. Some signs, such as period doubling, transient chaos, interior crisis, also point out that chaotic motion gets closer, that is, they forecast it. However, the most reliable way to avoid undesirable, even harmful phenomena is to choose the system parameters outside the "bad" ranges. It is the set of control parameters values, which lead to unwanted events, that is the criterion of the predictive rule.
In addition, in authors' opinion, the study of the manifold of nonlinear phenomena in a specific vibro-impact system adds information to fundamental knowledge about phenomena that occur in nonlinear dynamical systems.
Funding The authors did not receive support from any organization for the submitted work. No funding was received to assist with the preparation of this manuscript. No funding was received for conducting this study. No funds, grants, or other support was received.

Data availability
The authors declare that all data supporting the finding of this study are available within the article.