Nonchaoticity of Ordinary Differential Equations Describing Autonomous Transcriptional Regulatory Circuits

Gene transcriptional regulation （TR） processes are often described by coupled nonlinear ordinary differential equations （ODEs）. When the dimension of TR circuits is high （e.g. n≥3） the motions of the corresponding ODEs may, very probably, show self-sustained oscillations and chaos. On the other hand, chaoticity may be harmful for the normal biological functions of TR processes. In this letter we numerically study the dynamics of 3-gene TR ODEs in great detail, and investigate many 4-, 5-, and lO-gene TR systems by randomly choosing figures and parameters in the conventionally accepted ranges. And we find that oscillations are very seldom and no chaotic motion is observed, even if the dimension of systems is sufficiently high （n≥3）. It is argued that the observation of nonchaoticity of these ODEs agrees with normal functions of actual TR processes.

Numerical Investigation of the Superimposed Effects on Stator Wake Oscillation in an Axial-radial Combined Compressor

The superimposed influences of the blade rows in a multistage compressor are important because different matches of upstream and downstream blades can result in significant differences in the stator wake oscillation. Numerical investigation of the axial stator wake oscillation, which is affected upstream by the axial rotor and downstream by the radial rotor, was performed in an axial-radial combined compressor. Many configurations with different blade numbers and locations, which influence axial stator wake oscillation were investigated. When rotors have equal blade numbers, the axial stator wake oscillates periodically versus time within time T(moving blade passing 1/3 revolution). In contrast, stator wake oscillates irregularly within T when rotors have different blade numbers. A model-split subtraction method is presented in order to separate the influences of the individual blade rows on the wake oscillation of the axial stator. Analysis from the rotor-stator configuration showed that the unsteady flow angle fluctuation response is caused by the upstream rotor. For the rotor-stator-rotor configuration, the unsteady flow angle fluctuations are influenced by upand downstream blade rows. With the model-split subtraction method, the upand downstream influences on the flow angle fluctuation could be clearly separated and quantified. Low amplitudes could be observed when the influences from upand downstream moving rows were superimposed with the "positive peaknegative peak" type wave. Clocking investigations were carried out to change the relative superimposed phase of influences from the surrounding blade rows in order to modulate the amplitudes of the axial stator wake oscillation. However, the amplitudes did not reach the maximum when they were superimposed with "positive peak-positive peak" type wave due to the impact of the interaction between the two moving blade rows.