## 1 Introduction  
The problem of predicting chaotic dynamics has been studied for a long time in dynamical systems [1; 2; 3; 4; 5]. Chaos is a non-periodic solution with unstable parameters in a dynamic system, but it has a limitation within a strange attractor. This has been introduced into widely known nonlinear equations, such as [PERSON], [PERSON] [1], and [PERSON]. The classical method to analyze a dynamic system uses a time series and a phase portrait. On the other hand, after the Cooley-Tudkey algorithm in the 1960s, research and development of many dynamic systems have led to studies, such as higher-order spectrum analysis (HOSA), logistic maps, the Lyapunov exponent, and recurrence plots to analyze methods, including high-dimensional nonlinearity. Among them, the Lyapunov exponent has been developed and studied in many papers as the most popular method for quantifying chaos. [PERSON] et al. [6] developed a numerical algorithm determining the Largest Lyapunov exponent. On the other hand, this algorithm cannot be applied to dynamic systems including discontinuities, and can only be used in smooth dynamic systems. [PERSON]'s algorithm, developed to overcome this problem, can be used in a nonsmooth dynamical system, making it possible to calculate the Lyapunov exponent by applying it to impact and friction problems, including discontinuities [7]. In mechanical vibration, chaotic signals are a phenomenon that occurs in various applications, such as friction-induced vibration, impact noise, and fault diagnosis. In particular, mechanical vibrations can generate a chaotic phenomenon because they contain strong nonlinearity stemming from the discontinuity of friction and impact. Vibration problems occurring in general friction systems, including nonlinearity, were analyzed for stability via linearization [8; 9; 10]. On the other hand, after frictional vibration occurs, the nonlinear signal of the vibration increases and can become chaotic. [PERSON] [11] examined the chaos phenomenon caused by friction-induced vibration using a 2-DOF model that includes the negative slope of a continuous friction curve. In particular, the negative slope was found to be a control parameter capable of expressing chaotic characteristics. In addition, studies on the parameters that express the chaotic characteristics occurring in dynamic systems, including friction, have been investigated in many papers [2; 3; 4; 5]. In the impact condition, the collision force is extremely discontinuous at the moment of impact. The resulting kinetic phenomena for impact motion have been studied extensively to assess the chaos phenomenon. [PERSON] et al. [12; 13] investigated the chaotic characteristics by calculating the Lyapunov exponent for a dynamic system with discontinuities through [PERSON]'s procedure for [PERSON]'s and [PERSON]'s contact models. [PERSON] [14] calculated the Lyapunov exponent for the truncated number of modes of the impact beam under distributed contact using the continuous beam model. In addition to the theoretical approach, the vibration signal can be analyzed using the visualization method of the time series. The analysis method of visualization of the vibration signals, including high-order nonlinearity, was developed in studies, such as Gauss wavelets [15; 16] and recurrence plots [17]. [PERSON] et al. [17] introduced various recurrence plot analyses as a visualization method of dynamic characteristics occurring in a complex system and analyzed the dynamic characteristics. The phase space is reconstructed using the time delay method. The embedding dimension and time delay must be determined to express the reconstructed phase space by the time delay method. The time delay method is a dynamic problem that can be determined using the Mutual information method [18]. The embedding dimension is a geometric problem that is being studied to obtain the minimum physical embedding dimension [18; 19]. In this study, the embedding dimension was determined using the method of false nearest neighbors (FNN) proposed by [PERSON].