arXiv subjects

Renhao Tian

Publications and source records attributed to Renhao Tian.

4 recordsLinked to original sources

arXiv2026

On the Number of Limit Cycles in Generalized Abel Equations with Coefficients Having the Chebyshev Property

This paper concerns the maximum number of limit cycles of generalized Abel differential equations $dx/dt = A(t)x^p + B(t)x^q$, where $A$ and $B$ belong to the linear span of a family of functions having the Chebyshev property. Motivated by a recent open problem posed by Huang et al. (Nonlinearity, 2026), we investigate whether this maximum number can be bounded in terms of $p$, $q$, and the structure of the family. Under some natural hypotheses and by means of first- and second-order analyses using Melnikov functions, we provide lower bounds for this maximum number. In contrast to previous work, no specific form for the coefficients is assumed. We then apply these estimates to Abel equations with trigonometric polynomial, polynomial, and hyperbolic coefficients. In the trigonometric polynomial case, we reestablish the results of \'{A}lvarez et al. (J. Math. Anal. Appl., 2008) and Huang et al. (SIAM J. Appl. Dyn. Syst., 2020), while in the polynomial case, we improve the classical lower bound given by Lins-Neto.

math.DS↗

arXiv2026

Interstellar Object 3I/ATLAS Observed from Mars by China's Tianwen-1 Spacecraft

China's Tianwen-1 Mars orbiter successfully imaged the third interstellar object, 3I/ATLAS, during its close encounter with Mars using the onboard HiRIC CMOS camera. This is China's first deep-space observation of an astronomical object. These observations constitute the first imaging of this object from a vantage point significantly out of its orbital plane, providing a unique constraint on dust dynamics. Three observing epochs between 2025 September 30 and October 3 reveal clear changes in coma and tail morphology driven by the rapidly evolving viewing geometry. Comparison with Finson-Probstein dust dynamical models indicates that the coma is dominated by large grains with solar radiation pressure parameter $\beta \approx 10^{-3} $ - $10^{-2}$, corresponding to grain sizes of a few 100s $\mu$m. The extent of the sunward coma implies dust ejection velocities of $3$ - $10$ m s$^{-1}$. Despite the morphological evolution, the azimuthally averaged surface brightness profile remains nearly unchanged through the three epochs, transitioning from a radial slope near -1 close to the nucleus to slightly steeper than -1.5 at larger cometocentric distances, consistent with steady-state dust outflow accelerated by solar radiation pressure. Photometry yields an average $Af\rho \sim (2.0\pm0.2)\times10^4$ cm and a corresponding dust mass loss rate of $\dot{M} \sim 10^3$ kg s$^{-1}$.

astro-ph.EP↗

arXiv2025

A Chebyshev criterion for at most two non-zero limit cycles in Abel equations

In this paper, we investigate the maximum number of limit cycles of the reduced Abel equation $\dot{x}=A(t)x^{3}+B(t)x^{2}$ on an interval $[0,T]$. The Smale-Pugh problem asks whether this maximum number is bounded in terms of a given class of coefficients. We establish for the first time a Chebyshev criterion, providing a positive answer to the problem when this class spanned by an extended Chebyshev system (ET-system) $\mathcal{F}=\{f_{0},f_{1},f_{2}\}$ on $[0,T)$ with $f_{0}\not=0$. As an application, we prove that the equation has at most three limit cycles (including $x=0$) when the coefficients $A$ and $B$ are both linear trigonometric functions or quadratic polynomials. This reestablishes the result of Yu et al. (J. Differ. Equ., 2024) and improves the work of Bravo et al. (Disc. Cont. Dyn. Syst., 2015 \& J. Differ. Equ., 2024). We also obtain the same maximum number of limit cycles for the equation with trinomial coefficients.

math.CA↗

arXiv2024

On the study of the limit cycles for a class of population models with time-varying factors

In this paper, we study a class of population models with time-varying factors, represented by one-dimensional piecewise smooth autonomous differential equations. We provide several derivative formulas in "discrete" form for the Poincar\'{e} map of such equations, and establish a criterion for the existence of limit cycles. These two tools, together with the known ones, are then combined in a preliminary procedure that can provide a simple and unified way to analyze the equations. As an application, we prove that a general model of single species with seasonal constant-yield harvesting can only possess at most two limit cycles, which improves the work of Xiao in 2016. We also apply our results to a general model described by the Abel equations with periodic step function coefficients, showing that its maximum number of limit cycles, is three. Finally, a population suppression model for mosquitos considered by Yu and Li in 2020 and Zheng et al. in 2021 is studied using our approach.

math.CA↗