# Dynamic Analysis of Modified Duffing System via Intermittent External Force and Its Application
## 1 Introduction  
Since the pioneering work of [PERSON], the nonlinear equation describing an oscillator with cubic nonlinearity has been called the Duffing equation, and it is used in its various forms to describe many nonlinear systems [1]. With external force, this equation exhibits an enormous range of well-known behaviors in nonlinear dynamical systems, for example, the chaos phenomenon. It is also one of simplest equations to describe chaotic behavior. During the past half-century, chaos phenomenon and chaos theory have been widely investigated, and many chaotic systems have been proposed [2; 3; 4; 5]. The bifurcation of a Duffing oscillator is observed by keeping all parameters fixed except for the amplitude of the driving force; then, bifurcation value is determined [6]. In [7], the bifurcation structure and synchronization of a double-well Duffing oscillator coupled to a single-well one and subjected to periodic forces were studied. On the basis of the method of switching chaos on/off, a strategy of adjusting the shape of the time-dependent forcing was proposed to suppress or generate chaotic behavior in the Duffing oscillator [8]. Bifurcation diagrams and phases of two coupled periodically driven identical Duffing oscillators were presented in [9]. In [10], an electronic model of a Duffing oscillator with characteristic memristive nonlinear element is proposed instead of the cubic nonlinearity. For a two-degree-of-freedom mass-spring-damper model on a moving lubricated belt, the system's dynamic response behavior was studied for various parameter combinations, and the averaging method was used to solve nonlinear coupled equations of motion for qualitative analysis inorder to find the amplitude and frequency of steady-state vibration [11]. Furthermore, many research works investigated the Duffing oscillator and new chaotic systems [12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. For some applications, high sensitivity to initial conditions and complex chaotic dynamics make chaotic systems useful, e.g., fluid mixing, and estimation of weak signal and information encryption [22]. In [23], a new signal detection and estimation method based on intermittency transition was developed by using chaotic Duffing oscillators. A novel parallel image-encryption algorithm was also presented based on chaotic Duffing oscillators [24]. A secure communication scheme based on chaotic Duffing oscillators and frequency estimation for the transmission of binary-coded messages was presented by a Lyapunov-based observer [25]. In [26], some general trends of the climatic response to time-dependent parameters were identified on the basis of a generalized stability criterion, and theoretical predictions were validated by the evolution equations of relevance in atmospheric and climate dynamics. Dynamic characteristics of climate systems were also discussed in some studies [27; 28; 29; 30]. Meanwhile, there are many references on the dynamic analysis of chaotic systems and the applications of chaos-based encryption [31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42]. The Duffing oscillator with continuous external force, such as continuous sinusoidal function or cosine function, has attracted many researchers' attention and been widely studied in the past. However, what happens to the Duffing system if the controller has intermittent external force? Does it behave with periodic or chaotic motion? In order to answer these problems, this paper studies the dynamics of Duffing systems with intermittent external force and their applications. The modified Duffing (MD) system is very different from other references, in that external force is usually continuous, such as continuous cosine function. The main contributions of this paper are summarized as follows: (1) Intermittent external force, with work time and rest time, was designed for Duffing oscillator. In the work period, the external force was applied to control the Duffing oscillator.