# Minimal Model for Sprag-Slip Oscillation as Catastrophe-Type Behavior
## 1 Introduction
Nonlinearities, such as social and behavioral sciences and natural phenomena where continuous changes in parameters can lead to discontinuous changes in the resulting variables, are described as cusp catastrophes [18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28]. [PERSON] et al. [19] studied the necessary and sufficient conditions leading to landslides. The bifurcation point is a turning point from stability to potential instability, showing that chaos can be formed. It also shows that the chaos phenomenon is related to the mechanical parameters of the medium along the sliding surface. In a similar study case, [PERSON] et al. [23] studied the dynamical behavior of water or gas flow in a broken rock. They used the catastrophe theory to obtain the fold catastrophe model of the stability of flow system and predicted that dynamic disasters could occur near bifurcation. In this way, the catastrophe theory was used to analyze bifurcation curve in a nonlinear equilibrium. In another study, cusp catastrophe interpretation was also studied in the analysis of dynamic instability due to friction. [PERSON] et al. [24] simulated the problem of tangential force with the perspective of a micromechanical contact model. It has also been shown that reducing the applied normal force results in energy release due to snap-back instability associated with tangential force and sliding displacement. This result provides an explanation of the stick-slip phenomenon according to the catastrophe theory of tangential load, which showed similar to cusp-catastrophe instability. [PERSON] et al. [28] expressed equilibrium surfaces using cusp catastrophe theory in harmful algae blooms (HABs). The static equilibrium surface consists of two stable areas and a folded area, which can be projected as a bifurcation set. When entering the bifurcation set on the equilibrium surface, the state variable suddenly jumps and causes cusp catastrophes of the system. The basic concept of the cusp catastrophe theory is the 'potential' of the system, and it tends to produce certain sudden results. Catastrophic phenomena undergo periods of equilibrium at a minimum potential and, conversely, periods of sudden changes at a maximum potential. The potential of the system is not a structurally stable function, but the critical point (singularity) is degenerated. In the field of multi-body dynamics, the nonexistence of solutions under the Coulomb friction on rigid contact is well known as [PERSON]'s paradox [29; 30; 31; 32; 33]. Under the assumption of the impenetrability where the gap between the rigid system and rigid ground is not allowed negative, the occurrence of inconsistency and indeterminate can be found in the parameter space. Frictional hopping motion has been suggested as the similar kind of sprag-slip oscillation by the Painleve paradox [29]. [PERSON]'s paradox was well introduced in a review study by [PERSON] et al. [33]. The author described the [PERSON]'s paradox through the difference between pulling and pushing force by inducing chalk's sprag-slip vibration. In addition, various minimal models were reinterpreted to describe the uncertainty, multiple stability, and instability associated with [PERSON]'s paradox. In compliant contact for which the negative gap is allowed, the indeterminate of the steady-sliding equilibrium was found in the parameter space [16; 17]. However, the static and dynamic phenomenon at a nonlinear equilibrium of a dynamic system has not been investigated when a sudden jump occurs in friction-induced vibration. The previous study [17] presumed that the so-called \"sprag-slip oscillation\" occurs when a singular condition such as no steady-sliding situation is met. The purpose of the present work was to resolve this singularity problem and explain the sprag-slip oscillation in a continuous manner. For this, the more and realistic nonlinear modeling was introduced and adopted. In our model, one mass was attached by an inclined spring where the inclined spring angle was allowed varying.