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The purpose of the study is to obtain explicit formulas to determine the stability of periodic solutions to the new system and study the extent of the stability of those periodic solutions and the direction of bifurcated periodic solutions. More than that, the authors did a numerical simulation to confirm the results that the authors obtained and presented through numerical analysis are the periodic and stable solutions and when the system returns again to the state of out of control.

The authors studied local bifurcation and verified its occurrence after choosing the delay as a parameter of control in Zhou 2019’s dynamical system with delayed feedback control. The authors investigated the normal form theory and the center manifold theorem.

The occurrence of local Hopf bifurcations at the Zhou's system is verified. By using the normal form theory and the center manifold theorem, the authors obtain the explicit formulas for determining the stability and direction of bifurcated periodic solutions. The theoretical results obtained and the corresponding numerical simulations showed that the chaos phenomenon in the Zhou's system can be controlled using a method of time-delay auto-synchronization.

As the delay increases further, the numerical simulations show that the periodic solution disappears, and the chaos attractor appears again. The obtained results can also be applied to the control and anti-control of chaos phenomena of system (1). There are still abundant and complex dynamical behaviors, and the topological structure of the new system should be completely and thoroughly investigated and exploited.

This study aims to perform the surface treatment of synthetic α-Fe₂O₃ red iron oxide pigment with hydrolysate 3-aminopropyl silane (A) and colloidal silica (CS) and investigate the effects of surface-treated pigment on the styrene acrylic (SA) emulsion and polyurethane (PU) dispersion.

For this purpose, firstly red iron oxide particles were modified with A and CS separately in an aqueous medium. After isolation of the modified iron oxide were characterized by Fourier transform infrared spectroscopy (FTIR), X-ray photoelectron spectroscopy (XPS) and scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS). Moreover, the degree of the dispersion stability of the modified pigment in coatings with SA emulsion and PU dispersion was investigated by using an oscillation rheometer. Loss (G''), storage (G') modulus, loss factor [tan(δ)] and yield stress (τ⁰) values were determined by performing amplitude and frequency sweep tests.

The τ⁰ in SA coatings decreases with the amount of used A and increases with the amount of used CS. The τ⁰ decreases as the amount of used A and CS in PU coatings increases. The use of CS on red iron oxide pigments causes storage modulus to increase in SA coatings at low angular frequencies, while it causes a decrease in PU coatings.

To the best of the authors’ knowledge, for the first time, the suspended state of the iron oxide hybrid pigment formed with CS in the coating was investigated rheologically in this study.