## 1 Introduction  
The applications of supercritical fluids with their peculiar characteristics have been considered in many technical fields, such as renewable energy system [1], refrigeration system [2], next generation nuclear technology [3] and commercial ultra-supercritical coal power plant. In addition, a supercritical water nuclear reactor (SCWR) with the merits of comparatively high thermal performance, compact and inherent safety design, and economic competitiveness is listed as a promising type in the international development program of the Generation-IV advanced nuclear reactor [4]. At a supercritical pressure, the heating fluids exhibit dramatic changes in fluid properties near their critical or pseudocritical points. These features can substantially affect the thermal characteristics and stabilities of the systems with supercritical fluids, which should be addressed in the development and applications of such systems. At supercritical pressures, flow boiling phenomenon will not appear during the heating process. For a supercritical heated channel, the fluid behaves in a liquid-like manner below the corresponding pseudocritical point; however, it presents gas-like behavior when the fluid temperature is beyond the pseudocritical value [5]. Therefore, some researchers suggested three-region methodology to separate the supercritical flow into heavy fluid region, heavy and light fluid mixture region, and light fluid region, while considering the investigations in the supercritical heating systems [6,7]. Upon the drastic variation of flow properties in the pseudocritical region, the supercritical fluid systems could be subject to various kinds of instability, particularly the typical type of density-wave instability [8,9]. Since the instability problems may lead to the fluctuations of working fluids and system components, and thus affect the operation safety of a supercritical heated system, much more attentions need to be paid on the stability issues. In the literature, a linear model in the frequency domain and a non-linear model in the time domain are two major categories for carrying out theoretical studies of stability. Linear analysis methodology is a familiar tool to explore the stability threshold of a supercritical heated system due to its simplicity [8,9]. However, such systems are inherently non-linear, which cannot be explored by the linear methodology. This suggests the need of a proper non-linear model to clarify the instability phenomena of supercritical heated systems. Several researchers had predicted non-linear characteristics of supercritical heated systems by the tools of computational fluid dynamics (CFD) [10] or the modifications of the existing large system codes from two-phase flows [11,12]. However, because their complexity and time-consumption, these tools are, in general, not favorable for the detailed investigation of non-linear phenomena. Thus, our recent studies developed simple non-linear dynamic models to analyze the stability problems of supercritical water uniformly heated in a single channel [13] and parallel channels [14], respectively, particularly for that after the occurrence of instability. The results indicated that complex and abundant non-linear phenomena, such as supercritical Hopf and period-doubled bifurcations, and their evolutions to chaotic oscillations, could appear in a uniformly-heated channel with supercritical water [13]. On the other hand, parallel supercritical heated channels essentially presented out-of-phase mode of oscillations if total flow rates were kept constant. Channel-to-channel interactions due to the more difference among radial power distributions would have unstable effects on these systems [14]. Such out-of-phase instabilities have also been identified by experimental studies in parallel two channel systems with supercritical water [15,16].