In this paper we present a survey on the uniqueness of limit cycles bifurcating from a center, homoclinic loop or a heterclinic loop in planar systems, introducing five bifurcation theorems, and then apply the theorems to the study of codimension two bifurcations, obtaining a complete analysis on the uniqueness of limit cycles.

This work is on a general integration by parts formula generalizing the Ito formula. An application is given to stochastic inclusions which have a different form than usual.

In this paper, we develop a fractional-order model of new psychoactive substance (NPS) transmission. We first prove the non-negativity and boundedness of the model. Then, prevention function and treatment function are introduced into the model to establish the fractional-order optimal control model. The existence of the optimal control pair is proved, and the solution of the model is obtained. Finally, the numerical simulation of the optimal control model is carried out. The results show that the fractional model is helpful for us to analyze the NPS dynamics model more deeply. Through the analysis of three control strategies, the optimal control strategy is the combination of prevention and treatment measures. In addition, it is worth noting that this paper obtains a new insight that prevention is more effective than treatment in the early stage of NPS transmission, and the control strategy at this time is to invest in maximum prevention measures and moderate treatment measures. However, when the spread of NPS shows an epidemic trend, the influence of preventive measures is greatly reduced, and the control strategy at this time is to focus on increasing the investigation of addicts and improving the treatment rate of psychological addicts.

In this paper, we study the $L^2$-critical Hartree equation with harmonic potential which arises in quantum theory of large system of nonrelativistic bosonic atoms and molecules. Firstly, by using the variational characteristic of the nonlinear elliptic equation and the Hamilton conservations, we get the sharp threshold for global existence and blow-up of the Cauchy problem. Then, in terms of a change of variables, we first find the relation between the Hartree equation with and without harmonic potential. Furthermore, we prove the upper bound of blow-up rate in $\mathbb{R}^3$ as well as the mass concentration of blow-up solution for the Hartree equation with harmonic potential in $\mathbb{R}^N.$

Consensus tracking control problem of the multi-agent systems with both noise and disturbances is studied in this paper. A robust adaptive control scheme is designed according to Lyapunov stability theory. The main contribution of this paper is that the consensus is realized in multi-agent systems with both noise and external disturbances. Numerical simulations are carried out for the proposed scheme to demonstrate the effectiveness of the control strategy proposed in this paper.

This paper concerns a stationary prey-predator model with density-dependent diffusion and hunting cooperation under homogeneous Dirichlet boundary conditions. Based on the spectral analysis, the asymptotic stability of trivial and semi-trivial solutions is obtained. Moreover, the sufficient conditions for the existence of positive solutions are established by using degree theory in cones. Our analytical results suggest that density-dependent diffusion and hunting cooperation obviously influence on the positive solutions.

In this paper, we consider the following Schrödinger-Poisson system with a cubic term $$\begin{align*}\tag{0.1}\label{0.1} \begin{cases} -\Delta u+V(|x|)u+\lambda\phi u=|u|^2u \ \  {\rm in} \ \  \mathbb{R}^3,\\ -\Delta\phi=u^2 \ \ {\rm in} \  \ \mathbb{R}^3, \end{cases} \end{align*}$$ where $λ > 0$ and the radial function $V (x)$ is an external potential. By taking advantage of the Gersgorin disc theorem and Miranda theorem, via the variational method and blow up analysis, we prove that for each positive integer $k,$ problem (0.1) admits a radial nodal solution $U^λ_{k,4}$ that changes sign exactly $k$ times. Furthermore, the energy of $U^λ_{k,4}$ is strictly increasing in $k$ and the asymptotic behavior of $U^λ_{k,4}$ as $λ → 0_+$ is established. These results extend the existing ones from the super-cubic case in [17] to the cubic case.

During the conversion from difference to differential in the grey polynomial model, the “misplaced replacement” problem will occur. A novel unbiased grey polynomial model, i.e., HUGMP(1, 1, $N$) is presented to overcome the above drawback. Meanwhile, the parameter estimation of HUGMP (1, 1, $N$) is directly constructed by the equivalent relation between the parameter estimation and the recurrence relation of the time response function for GMP(1, 1, $N$) model. The recurrence relation is deduced from the solution of the homogenized differential equations, converted from the whitenization equation of GMP(1, 1, $N$) model by introducing new variables. The simulated values are directly calculated by the precise direct integration method in order to reduce round-off error and improve fitting accuracy. Moreover, it is proved that the proposed unbiased grey polynomial model possesses not only complete coincidence of simulation to non-homogeneous exponential sequence with polynomial time terms, but also multiple transformation consistency. At last, the results of applications verify the effectiveness of the proposed model by comparing with other conventional models.

During this century, an increasing interest appeared in studying the planar piecewise differential systems. This is due to their numerous applications for modelling many natural phenomena. For understanding the dynamics of the planar differential systems we must control the existence or non-existence of periodic orbits and limit cycles. So many papers have been published studying the existence or non-existence of periodic orbits and limit cycles for continuous or discontinuous piecewise differential systems. But until now very few papers have studied the periodic orbits and limit cycles of piecewise differential systems where two differential systems of the piecewise differential system are continuous and discontinuous respectively. We study the periodic orbits and limit cycles of the planar continuous–discontinuous piecewise differential systems separated by two parallel straight lines, such that either in one of these straight lines the piecewise differential system is continuous and in the other one discontinuous. In two pieces of these piecewise differential systems there are arbitrary Hamiltonian systems of degree two and in the third piece there is an arbitrary Hamiltonian system of degree one forming the continuous-discontinuous piecewise differential systems. We determine the limit cycles of these piecewise differential systems by considering two cases. In the first the Hamiltonian system of degree one can be in the middle of the three zones, and in the second it is on one side of the three zones.

In this paper, a linear perturbation up to any order in $\epsilon$ for a cubic center with a multiple line of critical points is considered. By the algorithm of any order Melnikov function, the sharp upper bound of the number of limit cycles is 2.

Dengue fever is a vector-borne viral disease that has become a worrisome health issue in tropical and subtropical countries. The seasonal trend of dengue incidence encourages outbreaks with a high risk of infection at particular periods annually that potentially resulted in a significant economic burden. The epidemiological mathematical model, the SIR-SI model, is modified by considering the time-dependent and periodic-forced infection rate parameter through sinusoidal functions to obtain well data fitting. We show the existence and the stability of the disease-free and endemic equilibria for the system and their relation to the basic reproduction number of the disease. Next, we adapt the insurance concept to develop an insurance mathematical model that accommodates the proposed dengue transmission model in calculating nominal premiums. An increase in the basic reproduction number as an important indicator of the level of disease transmission risk resulted in an increase in the nominal premium. We also introduce a reserve function that guarantees sufficient premium payments collected by insurer to cover up future expenditure due to dengue outbreaks. Through this reserve function, we obtain an adjusted premium as a minimum value of premium which ensures that the reserve function is always positive. Mathematical models combined with insurance features have the potential to become important tools for relevant authorities to gain insight into disease transmission dynamics as well as assess the economic burden induced by the occurrence of disease outbreaks.

Stability analysis and bifurcation of the nutrient-microorganism model are presented in this paper. It is found that the model could experience the changes of the equilibrium points and the saddle-node, the Hopf and the codimension-2 Bogdanov-Takens bifurcations. The induced complex dynamics are also illustrated, by virtue of theory like the Sotomayor’s theorem, the normal form and the universal unfolding. From the obtained results, some insights into interaction between the nutrient and the microorganism can be given. Further, numerical simulation is carried out to verify the theoretical results.

This paper focuses on analyzing the properties of equilibrium points and limit cycles in two types of planar piecewise smooth slow-fast systems. The right-half system is a classical van der Pol equation, while the left-half system is either linear or quadratic. Additionally, we provide a detailed description of the characteristics of limit cycles in these systems.

This paper discusses a new coupled system of Riemann-Liouville fractional differential equations, in which the nonlinear terms include the Riemann-Liouville fractional integrals and the boundary value problems involve three-points. We seek also for the existence and uniqueness of solutions for this new system. We first get some useful properties of the Green’s function generated by the system, and then we apply a fixed point theorem of increasing $φ$-$(h, e)$-concave operators to this new coupled system. Finally, we gain the existence and uniqueness results of the solution for this problem. In the end, a concrete example is structured to illustrate the main result.

In this paper, we present an example of piecewise linear systems with infinitely many crossing limit cycles defined in two zones separated by a piecewise linear curve with countable corners. Then we prove that under piecewise linear perturbations, the perturbed system can have infinitely many limit cycles, or exactly $ℓ$ limit cycles for any given nonnegative integer $ℓ.$

In this paper, we study discontinuous Sturm-Liouville problem with fractional Hilfer derivatives. By defining an operator $A$ in the Hilbert space $L_2[−1, 1],$ this research shows that the eigenvalues and corresponding eigenfunctions of the main problem coincide with the eigenvalues and corresponding eigenfunctions of the constructed operator. Moreover, the characteristic function is also constructed such that the eigenvalues of the problem are coincide with the zeros of this function.

Tuberculosis is the second biggest infectious disease killer after coronavirus. In this paper, we analyze a mathematical model of in-host tuberculous granuloma, obtaining the basic reproduction number, as well as the existence and stability of equilibrium points. The sensitivity analysis provides parameters that have a significant effect on model dynamics. Finally, changes in the number of immune cells, infected macrophages and Mycobacterium tuberculosis are analyzed by numerical simulation of three disease states: clearance, latent infection and active tuberculosis. The results suggest that the immune mechanism determing whether an infected individual will suffer from active or latent tuberculosis is the ability of activated infected macrophages to kill Mycobacterium tuberculosis.

Since the aquatic organisms are very sensitive to the increasing copper contamination in water, this paper focuses on investigating the effects of copper contamination on the interaction between phytoplankton and zooplankton species within both deterministic and stochastic environments. We first construct a deterministic phytoplankton-zooplankton interaction model coupled with the copper concentration and analyze its dynamics, including existence and stability of equilibria, as well as the existence of Hopf bifurcation. A new stochastic model is derived in the form of continuous-time Markov chain (CTMC), and branching process theory is applied to the CTMC model to estimate the extinction probability of zooplankton species. Analytical and numerical findings show that the destiny of species is closely related to the copper concentration, and the predictions of the deterministic and stochastic models may be different in some cases.

In this article, we introduce the notion of cyclic $α$-admissible mapping with respect to $θ$ with its special cases, which are cyclic $α$-admissible mapping with respect to $θ^∗$ and cyclic $α^∗$-admissible mapping with respect to $θ.$ We present the notion of orthogonal $(αθ−βF)$-rational contraction and establish new fixed point results over orthogonal $\mathcal{F}$-metric space. The study includes illustrative examples to support our results. We apply our results to prove the existence and uniqueness of solutions for second-order differential equations.

Consider the Cauchy problem for the Benjamin-Ono-Burgers equation. There exists a unique global weak solution under appropriate conditions on the initial function and the external force. Here are many very important and interesting questions.
• Can we accomplish the exact limits for all order derivatives of the global smooth solution of the Benjamin-Ono-Burgers equation, in terms of some given information, representing certain physical mechanisms?
• What are the influences of various physical mechanisms (represented by the initial function, the external force, the order of the derivatives and the diffusion coefficient) on the exact limits?
• Can we establish improved decay estimates with sharp rates for all order derivatives of the solution, so that the most important constants $\mathcal{A}$ and $\mathcal{C}$ are independent of any norm of any order derivatives of the initial function, the external force and the solution, for all sufficiently large $t > 0?$ Other positive constants $\mathcal{B}$ and $\mathcal{D}$ in the estimates are much less important because $Bt^{−1}$ and $Dt^{−1}$ becomes arbitrarily small as $t → ∞.$ This kind of decay estimates may play a substantial role in long time, accurate numerical simulations.
• Can we use the solution of the corresponding linear equation to approximate the solution of the Benjamin-Ono-Burgers equation?
• Can we couple together classical ideas (such as the Fourier transformation, the Parseval’s identity, Lebesgue’s dominated convergence theorem, squeeze theorem, etc) in an appropriate way to establish important and interesting results for the Benjamin-Ono-Burgers equation?
• For very similar dissipative dispersive wave equations, such as the Kortewegde Vries-Burgers equation and the Benjamin-Bona-Mahony-Burgers equation, can we apply the same ideas developed in this paper to establish the same or very similar results?
We will couple together a few novel ideas, several existing ideas and existing results and use rigorous mathematical analysis to provide positive solutions to these important and interesting questions.