Sleep, in contrast to wakefulness, is characterized by reduced awareness and responsiveness. A basic model of sleep homeostasis is based on the concept of sleep-wake transition [1]. Conventionally, sleep stages in humans are classified as wake, rapid eye movement (REM) sleep, and an approximate continuum of depth during non-REM (NREM) sleep based on electroencephalographic patterns, which comprises about 80% of the entire sleep [2]. This cycling model of wake/NREM/REM sleep switches has been the primary focus of sleep research for decades. However, this reductionist type of approach is over-simplified and has limitation in understanding pathophysiological mechanisms in sleep disorders. Sleep is not simply a succession of human invented stages, but a delicate and sophisticated nonlinear symphony played by the brain in a democratic and mutual interaction with the rest of the body [2]. Real sleep stages are dynamic transitions between multiple physiological states swinging between the dual condition of stability and instability to warrant environmental adaptations and achieve physical and mental restoration [3]. Quantification of sleep stages via the analysis of electroencephalography (EEG) signal has been a challenge for years. Conventional visual sleep stage scoring is arbitrary and does not fully capture intrinsic EEG activity [4]. Fourier-based spectral analysis can quantify frequency compositions in EEG signals and is the most commonly used EEG analysis; however, it has intrinsic limitations to capturing underlying dynamics of the brain oscillations. First, fast Fourier transform (FFT)-based analysis assumes that complex oscillations embedded in the EEG signal are comprised of sine waves with different frequencies [5]. In this context, EEG signal can be decomposed into frequency components such as beta, alpha, theta, or delta frequency bands. However, it has long been known that brain oscillation is not a linear combination of these arbitrary frequency components, a property called “nonlinearity” [۶]. Second, FFT-based spectral analysis assumes that none of these frequency components change in amplitude or shape as time evolves, which is clearly against what has been observed in complex brain oscillations, a property called “nonstationarity” [۵]. It has long been observed that physiologic output of human body is nonstationary and nonlinear. Controls of physiological systems and outputs such as heartbeat, respiration, and brain wave oscillations are extraordinary complex [7]. Such complexity is believed to arise from nonlinear interactions among multiple control nodes in different physiological body systems that operate at multiple time scales. It has been hypothesized that the complexity of a biological system should be related to the system’s capacity to adapt and function in an ever changing environment [7]. Conventionally, scientists employ a reductionist approach to disassemble the complex system into constituent pieces, examine each component, and, finally, reassemble them to recreate the original entity. However, this approach is often unrealistic. In most circumstances, we can observe only the macroscopic output of physiological functions, such as an EEG, heart rate, or respiration. In the language of complex systems, the composite behavior cannot be fully understood by “adding up” the components. Instead, one needs new approaches to measuring a system’s integrative behavior. Thus, the understanding of the complex dynamics of the physiologic output, such as changes in EEG dynamics observed across different sleep stages, will be improved by applying nonlinear dynamical approaches to the analysis of EEG signal.