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This paper introduces a new practical method for distinguishing chaotic, periodic and quasi-periodic orbits based on a new criterion, and apply it to investigate the local bifurcations of the Chen system. Conditions for supercritical and subcritical bifurcations are obtained, with their parameter domains specified. The analytic results are also verified by numerical simulation studies.

This paper focuses on the bifurcation and chaos of an axially accelerating viscoelastic beam in the supercritical regime. For the first time, the nonlinear dynamics of the system under consideration are studied via the high-order Galerkin truncation as well as the differential and integral quadrature method (DQM & IQM). The speed of the axially moving beam is assumed to be comprised of a constant mean value along with harmonic fluctuations. The transverse vibrations of the beam are governed by a nonlinear integro-partial-differential equation, which includes the finite axial support rigidity and the longitudinally varying tension due to the axial acceleration. The Galerkin truncation and the DQM & IQM are, respectively, applied to reduce the equation into a set of ordinary differential equations. Furthermore, the time history of the axially moving beam is numerically solved based on the fourth-order Runge–Kutta time discretization. Based on the numerical solutions, the phase portrait, the bifurcation diagrams and the initial value sensitivity are presented to identify the dynamical behaviors. Based on the nonlinear dynamics, the effects of the truncation terms of the Galerkin method, such as 2-term, 4-term, and 6-term, are studied by comparison with DQM & IQM.

This paper focuses on the Turing patterns in the general Gierer–Meinhardt model of morphogenesis. The stability analysis of the equilibrium for the associated ODE system is carried out and the stability conditions are obtained. Furthermore, we perform a detailed Hopf bifurcation analysis for this system. The results show that the equilibrium undergoes a supercritical Hopf bifurcation in certain parameter range and the bifurcated limit cycle is stable. With added diffusions, we then show that both the stable equilibrium and the Hopf periodic solution experience Turing instability with unequal spatial diffusions and obtain the instability conditions. Numerical simulations are given to illustrate the theoretical analysis, which show that the Turing patterns are of either spot or stripe type.

We prove that all Hopf bifurcations in the Nicholson’s blowfly equation are supercritical as we increase the delay. Earlier results treated only the first bifurcation point, and to determine the criticality of the bifurcation, one needed to substitute the parameters into a lengthy formula of the first Lyapunov coefficient. With our result, there is no need for such calculations at any bifurcation point.

We investigate a system of partial differential equations modeling supercritical multicomponent reactive fluids. These equations involve nonideal fluid thermodynamics, nonideal chemistry as well as multicomponent diffusion fluxes driven by chemical potential gradients. Only local symmetrization of the resulting system of partial differential equations may be achieved because of thermodynamic instabilities even though the entropy function is globally defined. Local symmetrized forms are explicitly evaluated in terms of the inverse of the chemical potential Hessian and local normal forms lead to global existence and asymptotic stability of equilibrium states as well as decay estimates. We also discuss the deficiency of the resulting system of partial differential equations at thermodynamically unstable states typically associated with nonideal fluids.

In this paper, by utilizing a newly established variational principle on convex sets, we provide an existence and multiplicity result for a class of semilinear elliptic problems defined on the whole ${ℝ}^N$ with nonlinearities involving linear and superlinear terms. We shall impose no growth restriction on the nonlinear term, and consequently, our problem can be supercritical in the sense of the Sobolev embeddings.

It is experimentally shown that, unlike evaporation drying, supercritical CO₂ drying makes it possible to obtain free-standing single-coil InGaAs/GaAs nanotubes and other nanoshells with wall thicknesses down to 1 nm without their deformation by capillary forces. The performed process optimization has allowed us to reduce tenfold the duration of the process for long nanotubes. For the first time, etching of sacrificial AlAs layers in supercritical media was performed.

Zinc oxide (ZnO) is a compound with versatile applications. The compound can be synthesized in multiple sizes and shapes. The actual final product shape or morphology depends largely on the solvent used. The understanding of the actual interaction amongst the solvent and solute molecules is a very promising way to predict the shape of ZnO. Further, the operating conditions used during synthesis of ZnO also influence the final size of the product. This review paper specifically focuses on synthesis of zinc oxide through solvothermal synthesis with salt of zinc oxide with single solvent and with different operating conditions. The literature on effect of mixture of solvents is also reported in one of the subsections. The aim of this review is to understand the actual interaction amongst single salt and solvent without presence of any other compound in it. The presence of more than one compound in solvent or mixture of solvent increases the complexity of the problem. Therefore, there is a need to focus on this specific collection of literature to understand the basics, while future studies can be devoted to more complex studies. This review covers the effect of solvents like alcohols, alkanes, glycols, ring compounds and water on zinc salt. Also, it reports the effect of operating parameters like effect of precursor, temperature and time based on the available literature.

The work investigates steady-state responses of a pipe conveying fluid with a harmonic component of flow speed superposed on a constant mean value in the supercritical regime. If the flow speed exceeds a critical value, the straight configuration of the pipe becomes unstable and bifurcates into two stable curved configurations. The transverse motion measured from each of the curved equilibrium configurations is governed by a nonlinear integro-partial-differential equation. The Galerkin method is employed to discretize the governing equation into a set of coupled nonlinear ordinary differential equations with gyroscopic terms. For the pipes in the supercritical regime, the method of multiple scales is used to determine the steady-state in the vicinity of two-to-one resonance. In the presence of the internal resonance, the subharmonic, the superharmonic and the summation, and the difference resonances exist due to the pulsating fluid. The amplitude–frequency relationships are established with the emphasis on the effects of the viscosity, the pulsating amplitude, the nonlinearity, and the mean flow speed. Some nonlinear phenomena such as the appearance of the peak or jumps pertaining to modal interaction are demonstrated. The numerical integration results are in agreement with the analytical predictions.