Escape of two-DOF dynamical system from the potential well

We consider the escape of an initially excited dynamical system with two degrees of freedom from a potential well. Three different benchmark well potentials with different topologies are explored. The main challenge is to reveal the basic mechanisms that govern the escape in different regions of the parametric space and to construct appropriate asymptotic approximations for the analytic treatment of these mechanisms. In this study, numerical and analytical tools are used to classify and map the different escape mechanisms for various initial conditions and to offer the analytic criteria predicting the system’s behavior for those cases.


Introduction
Many common and widely explored phenomena, such as dynamics of molecules and absorbed particles [1,2] in chemistry or celestial mechanics and gravitational collapse [3,4] in physics, exemplify the escape from a potential well. This fundamental problem served as the foundation for a variety of engineering issues and applications, including energy harvesting [5], phenomena in Josephson junctions [6], capsizing of ships [7,8], and dynamic pull-in in microelectromechanical systems (MEMS) [9,10]. More details concerning the dynamic pull-in and its relationship to the escape dynamics can be found in [25]. Many of the above applications involve the escape under external forcing. In a recent paper on escape from a potential well via vortex-induced vibration [11], approximate escape criteria were proposed for a simplified 1DOF model. For this purpose, harmonic balance, along with exploration of the bifurcations, were utilized. Other works concerning the escape of the 1DOF system by an external harmonic force introduce additional analytical approaches, such as approximation of isolated resonance based on canonical action-angle transformations [12][13][14]. An extension for two-coupled-DOF systems under harmonic forcing was also studied [15]. Under some alternative periodic forcing laws [16], the escape reveals fractal boundaries between the escape and non-escape basins in the phase space. Fractalization is quite typical for systems characterized by strong sensitivity to initial conditions. Thus it is usually related to the chaotic scattering field. An extended overview of the analytical framework, numerical approaches, and applications of chaotic scattering can be found in [17]. Another well-studied variant of the escape phenomenon is a noise-activated escape. Noise is usually used to describe the coupling of the model to the environment or internal irregularities of the system. The energy fluctuations generated by the noise transform the bounded motion into a chaotic scattering [18]. Moreover, the system's response to the same initial conditions can exhibit different asymptotic behaviors in the presence of noise [19]. In these works, the escape scenarios have been studied in terms of classical chaotic scattering features, such as escapetime, and newer concepts, such as probability basins. Evidence of fractal behavior has also been found in the escapes governed by different regions of initial conditions and system control parameters [20]. Clearly, predicting the dynamical behavior for any region in the IC space is of crucial importance for all the above applications, as well as for design and even optimization purposes [21]. An extended numerical investigation was performed to map the escape region for 3DOF models. Color-coded diagrams, Poincaré maps, and the study of normally hyperbolic invariant manifolds were used [22]. For the simple case of a 1DOF model, an analytical prediction was established, using approaches of multiple scales and the Melnikov criterion [23].
This study considers the escape of initially excited 2DOF systems from the potential wells. For a better understanding of the problem and for a more reliable model, in our work, three one-dimensional benchmark potentials are considered, and two sets of initial conditions are studied. The main challenge is to reveal the basic mechanisms governing the escape of the system for different ranges of the system parameter and to construct suitable asymptotic approximations for the analytical treatment of these mechanisms. One observes that the exploration of quite narrow set of initial conditions allows one to identify the basic escape mechanisms relevant for the complete state space. In Sect. 2, the model is presented, and in Sect. 3, an extended numerical investigation is performed to reveal the main assumptions regarding the dynamical behaviors of the system. Section 4 is devoted to the analytical assessment of the results, using the previous assumptions to simplify the analytical expressions and to derive the asymptotic approximations of the initial energy required to escape. Section 5 is devoted to the concluding remarks.

Model description
In this paper, we will investigate the escape dynamics of 2DOF systems under different time-independent potential fields. The general setting contains two identical particles, described by the coordinates q 1 ; q 2 , and coupled via a linear spring (cf. Fig. 1).
The analysis is conducted for three benchmark potentials: biquadratic potential (V b ), cubic potential (V c ), and hyperbolic potential (V h ), all given by the following forms (1) and presented in Fig. 2.
Here, V 0 and a À1 are the hyperbolic potential depth and width, respectively.
Equations of motion are presented as: The transition to dimensionless variables for the hyperbolic potential is performed in the following way Here,ẽ refers to the physical stiffness value, e refers to the dimensionless linear coupling, and m is the mass of the particles. The benchmark potentials allow us to explore a variety of escape mechanisms and assess their universality. The escape of the system is naturally registered when both particles leave the potential well in the same direction. Alternative scenario is detected when two particles leave the well in opposite directions; the increase of the coupling energy is thus balanced by the gain in potential energy of each individual particle. We refer to this escape scenario as dissociation of the coupled system. Out of the benchmark models we explore, obviously only the biquadratic potential can support the aforementioned dissociation, allowing one to demonstrate the coexistence of the two escape scenarios in the parametric space.
To figure out the fundamental dynamical mechanisms of the escape, we pay special attention to two different types of initial conditions: set I, when the initial velocity is supplied to one of the particles and set II, when the particles initially obtain the same velocities in the opposite directions. For convenience, other coordinates were used in this case: center of mass R and interparticle displacement W. To examine an additional escape mechanism, we also add minor disturbance to the center of mass velocity.
Here, d \ \ 1 and E 0 is the initial energy of the system.

Poincaré section
The systems under consideration are 2DOF and conservative, so Poincaré sections are natural tools for the exploration of the global dynamics. _ q 2 ¼ 0 was chosen to be the first Poincaré condition. For brevity, only the results of the biquadratic potential are presented since the same essential properties were found for the other benchmark potentials.
The Poincaré sections in Fig. 3 reveal a clear distinction between the case of weak coupling (cf. Fig. 3a, b) and strong coupling (cf. Fig. 3c, d). In the case of weak coupling, abundant dynamical behaviors, quasi-periodic orbits, and chaotic trajectories coexist at energy levels below the energy required to escape (cf. Fig. 3a). As we add more energy to the system (cf. Fig. 3b), one can identify a new region of instability (red dots) where the system escapes for any IC.
The case of strong coupling reveals different escape scenarios, described in more detail in the next section. Chaotic responses in the considered energy range appear primarily in the case of weak coupling.
Another Poincaré condition studied in this section is _ R ¼ 0. Thus, each IC in these portraits satisfies _ q 1 0 ð Þ ¼ À _ q 2 0 ð Þ for different locations of the particles. The variety of Poincaré sections for all ranges of spring stiffness values shows only one clear escape mechanism. The evaluation of the instability region for the symmetry case can be seen in Fig. 4.
One can observe that as more energy is added to the system, the stability region narrows to the vicinity of R ¼ 0. Therefore, the ideal symmetric ICs require the highest energy to escape. In addition, the boundaries of the stability region are determined by the location of the center of mass (CM), while the interparticle distance has less influence on the system's escape. In the following section, an additional escape mechanism has been revealed by adding disturbance to the CM velocity for the case of the symmetry set of IC. Therefore, further investigations were performed on the following Poincaré condition _ R ¼ 1 Á 10 À3 ; only a slight difference from the previous case was found, as expected for any IC that is not described by ideal symmetry.

Identification of the typical escape mechanisms
While examining various system responses for all IC sets and coupling values, one reveals some common phenomenological features. For the asymmetric IC set, a clear distinction between the cases of weak and strong coupling has been revealed. The typical response for the case of the weak coupling is described by a high amplitude periodic motion of the exited particle, while the other particle stays approximately at the bottom of the well, as shown in Fig. 5a. Thus, the escape threshold scenario occurs when the exited particle has enough energy to escape from the well and pull the other particle from it (cf. Fig. 5b). When the coupling is strong, the response exhibits two distinct time scales, the one rapid small-amplitude interparticle oscillations and relatively slow but large-amplitude motion of the CM (cf. Fig. 5c).
For the case of symmetric IC, one encounters the single escape mechanism, again characterized by two time scales-rapid, possibly high-amplitude interparticle oscillations and slow CM evolution (cf. Fig. 6d). As discussed in the previous section, only the biquadratic potential allows the dissociation mechanism (cf. Fig. 6b) under the symmetric IC, while any breach of the symmetry, as a disturbance in the CM velocity, led to the former escape scenario. For the square hyperbolic secant potential, in which the force the well extract goes to zero as the particle diverges, the coupling energy of the symmetric IC will overcome the influence of the external potential; thus, escape is not possible for this scenario.

Lyapunov exponents
To study the chaotic properties of the system under the different potential fields and circumstances, the Lyapunov exponents (LE) were calculated. For this purpose, we used the Anton O. ATLAB function. For the asymmetric IC, the maximum LE (MLE) as a function of the spring stiffness value is shown in Fig. 7.
One may note that the MLE values above 1 Á 10 À3 were found only in the hyperbolic potential for cases of weak coupling, indicating chaotic behavior in these circumstances. Smaller MLE values may be governed by numerical inaccuracy and therefore are not classified as chaotic. No MLE values above 1 Á 10 À3 were found for the symmetric IC. All these features As mentioned, a clear distinction between the cases of weak and strong coupling has been revealed for the asymmetric IC. For the case of weak coupling, one can easily derive the force exerted by the well on the particle: From expressions (7) and (8), it is clear that the force exerted by the potential on the particles increases the farther they are from the maximum point of the well. Therefore, the escape thresholds of V b and V c determined when the exited particle passes the maximum point of the well, as shown in Fig. 8. In contrast, the force exerted by the hyperbolic potential approaches zero as the particle approaches infinity. Therefore, the escape threshold scenario is defined at the maximum distance between the particles, where the elastic force is maximal and equal to the maximum force exerted by the well. An illustration of the escape scenarios and the analytical description of the cases are presented in Fig. 8 and (10)-(12), respectively.
Here Considering the energy conservation, we obtain the following asymptotic approximation for the minimal energy required for escape B for the three benchmark potentials functions: For the case of strong coupling, we take into account the aforementioned time scale separation (obtained in the numerical investigation cf. Figure 5) and consider the system in terms of the interparticle displacement and the CM. The potential for each of the cases is written as: Since we are referring to high stiffness values, the elastic energy strongly influences the system dynamics. Thus, the fast interparticle oscillations for each of the benchmark potentials can be approximately Fig. 7 The MLE as a function of the coupling values for the asymmetric IC, a hyperbolic potential, b cubic potential, c biquadratic potential Fig. 8 Representation of the potential escape scenarios for the asymmetric IC. Here, the horizontal axis refers to the q coordinates described by omitting all terms of the potential besides the linear stiffness: Here, E 0 is the initial energy of the system. Averaging of the potentials according to the interparticle oscillations period yield the following effective potentials As discussed, the escape threshold scenarios for the V c V b potentials are defined by the cross of the CM at the maximum point. Therefore, the conservation of energy for these cases is as follows: For the hyperbolic potential, the system's escape, with both particles diverging to infinity, sets the total potential energy to zero. Therefore, we are only interested in the value of the effective potential at the bottom of the well The averaging for the hyperbolic potential was performed using the Taylor series of the argument of the integral: The CM initial velocity for the case of the asymmetric IC can be written by the initial energy One can insert the values of the mean potentials together with (26) into the conservation of energy (23)-(24) and obtain: From Eqs. (27)-(29), one can easily derive the asymptotic approximation of the energy required to escape B as a function of the spring stiffness e: To validate our analytical results, we compare the analytical relations to the numerical simulations. The error bars represent the range of energies in which the stability (when the system does not escape) is inconclusive and strongly depends on the simulation time scale (cf. Fig. 9).
By dividing the parameter space and treating each region based on its specific phenomenological characteristics, we were able to outline the analytical predictions for almost all ranges of stiffness values, yet the intermediate region remains unattended. The lower boundary for the escape energy (found in Appendix) shows great agreement with the numerical simulation for low stiffness values. One can notice that the appearance of the chaotic behavior is limited to the case of the hyperbolic potential for small coupling values, as expected from the calculated LE (Fig. 7).

Analytic prediction of the escape energysymmetric IC
This section presents the analytical treatment developed for the asymptotic approximation of the cubic and biquadratic potentials (which allow the symmetry IC to escape). We continue to consider the system in terms of R and W variables. The aforementioned properties of the response (cf. Sect. 3.2), in which the interparticle displacement oscillates rapidly while the CM approximately remains at the origin, allow us to simplify the interparticle distance EOM by disregarding the distribution of the CM: For the cubic potential, one can substitute the symmetric IC and easily derive the estimated solution For the case of the biquadratic potential, based on the oscillations obtained for the interparticle displacement (cf. Fig. 6), we guess a solution with the form (36) as suggested by the harmonic balance method. By substituting the estimate solution into the simplified EOM (33), we obtain We can neglect the high-frequency term from (38) and reduce an analytical relation between the amplitude and the frequency of the estimated solution: After substituting the symmetry IC, one can determine the amplitude of W by the following equation To further explore the problem, we obtain the form of the equivalent CM equation of motion for each potential by substituting the approximate solution for W(t). For the case of the biquadratic potential, as mentioned, the CM remains in the vicinity of the origin. Hence, a high order of R coordinates can be neglected. (Eqs. (41) -(48) refer to the biquadratic potential) Fig. 9 Numerical simulation results and analytical prediction for the minimal initial energy required to escape as a function of the coupling value for the asymmetry IC, a biquadratic potential, b cubic potential, c hyperbolic potential We denote the following transformation and obtain: One can identify (44) as the well-known Mathieu Eq. (45) with the following parameters: Next, we could use the Ince-Strutt stability diagram of the Mathieu equation, shown in Fig. 10. For this purpose, the transformation between the Mathieu parameters (46) and the system parameters (39) and (40) was studied. First, it was found that the image of the relevant domain in the parameter space is on the left half-plane of the Mathieu-Ince-Strutt diagram, which can be formally assigned to the inverted pendulum mechanism with the following stability boundary [24] d ¼ À One can substitute to (47) the transformation between the Mathieu parameters d ; E to the system parameters _ W 0 ; e and isolate the minimal separation velocity required to escape - The equivalent EOM for the case of the cubic potential is given by (49) (Eqs. (49)-(56) refer to the cubic potential) The effective potential can be defined as - For the symmetry IC, where R(0) = 0, the effective potential value is zero, and the total initial energy is given by the separation velocity. Thus, for the escape of the system, we require that the value of the potential at the maximum point be zero (cf. Fig. 11). We denote (51) and obtain: From (51), one can easily derive (55) By equating (55) and (54), one can find the single amplitude value satisfied the discussed requirement to be A ¼ ffiffiffiffiffiffi ffi 1:5 p . Using the transformation (35), one can determine the asymptotic approximation of the minimal separation initial velocity required to escape: As presented in the numerical investigation, the dissociation mechanism is defined by the diverged displacement between particles while the CM remains at the origin. Therefore, the equivalent of W EOM (33) is also valid for this scenario. It is clear that for W values above the critical value (57), the interparticle displacement diverges Hence, the escape threshold scenario will satisfy (57) and zero separation velocity. One can apply the above requirements to energy conservation and obtain:  Figure 12 presents the simulation results and analytical prediction for each escape mechanism (48), (56), and (59). Serval CM initial velocity disturbance was simulated along with the ideal symmetric IC.
As expected for the biquadratic potential, the two mechanisms, escape and dissociation, coexist (cf. Fig. 12a). The cubic potential (cf. Fig. 12b) shows no sensitivity to the breach of the symmetry of the IC. One can observe the satisfactory agreement of the numerical simulations with the analytical predictions for all cases analyzed.

Concluding remarks
The findings from the last sections can be generalized for the whole parametric space. For this purpose, one divides the total kinetic energy T of the system into the kinetic energy of CM and the kinetic energy of the interparticle distance, as follows: As mentioned earlier, the escape mechanism (where both particles diverge in the same direction of the well) requires the escape of the CM. Thus, as more of the initial energy is granted to the interparticle distance DOF, the more total energy is required to ensure the system's escape. Therefore, the symmetric IC studied in this work requires the highest energy to escape, in agreement with the findings from the Poincaré section above (cf. Fig. 4). Using the last relation between the IC sets, we can partition the parameter space as follows (cf. Fig. 13).
Below the lower boundary of the escape energy (the gray region), the escape is not possible, while for any IC that corresponds to the red area, an escape is guaranteed. The escape boundaries for any asymmetric IC can be found in the blue area, such as the specific asymmetric IC studied in this work (the broken line). The properties of the biquadratic potential lead to a unique additional region of instability (the bright region (cf. Fig. 13a)) between the escape and dissociation boundaries, where the ideal symmetric IC cannot escape. However, any violation of the system's symmetry leads to the system's escape. Some uncertainty remains about the system's behavior for IC in the blue region. Further work could use the analytical treatment performed in this paper to generalize the analytical prediction for the intermediate area.
The system under consideration is non-integrable-one can claim it with a confidence, since positive Lyapunov exponents are observed. Consequently, the complete analytic exploration is beyond the reach, and it is common to expect rather complex behavior. Moreover, if the escape dynamics is considered, the nonlinearity cannot be treated as small perturbation in any sense, since the system has to overcome the barrier or to diffuse to infinity. Therefore, one commonly expects rather ''complex'' dynamics. Somewhat surprisingly, for rather wide subset of the state space, and for the different potential shapes, it is possible to describe the escape dynamics Fig. 13 Parametric space (escape energy as a function of stiffness) division for a biquadratic potential and b cubic potential. By the lower boundary of the escape energy (Appendix)-' No escape,' the analytical prediction for the asymmetric IC-'AS IC' and the symmetric IC-'S IC,' and the dissociation threshold in terms of simplified tractable models, and to predict the escape thresholds in the parametric space. Thus, the escape phenomenon turns out to be rather simple and predictable, despite the non-integrability.
As for the systems with more degrees of freedom, one should expect even more complicated dynamics. Still, it is possible to hope that some of the simplification possibilities revealed in the paper will be helpful also in these cases.
Funding The authors are very grateful to Israel Science Foundation (Grant 2598/21) for financial support.
Data Availability Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Declarations
Conflict of interest The authors have no relevant financial or non-financial interests to disclose. The authors declare no conflicts of interest.

Appendix
The lower boundary of the escape energy level The escape of a dynamical system from a potential well is impossible if the initial energy is less than the value of the potential at its saddle points (E sp ). For any initial energy E 0 \E sp the iso-potential curves are closed, and the is bounded, while for the exact E 0 ¼ E sp the closed counters of the iso-potential curves are broken, and escape channels are formed (cf. Fig. 7). Therefore, the lower limit of the escape energy level can be derived from the analysis of the total potential of the system.