Two criteria are constructed to guarantee the existence of periodic solutions for a second-order $n$-dimensional differential system by using continuation theorem. It is noticed that the criteria established are found to be associated with the system’s damping coefficient, natural frequency, parametrical excitation, and the coefficient of the nonlinear term. Based on the criteria obtained, we investigate the periodic motions of the simply supported at the four-edge rectangular thin plate system subjected to the parametrical excitation. The effectiveness of the criteria is validated by corresponding numerical simulation. It is found that the existent range of periodic solutions for the thin plate system increases along with the increase of the ratio of the modulus of nonlinear term’s coefficient and parametric excitation term, which generalize and improve the corresponding achievements given in the known literature.

A new pest management mathematical model with saturated growth is proposed. The integrated pest management (IPM) strategy by introducing two state dependent pulses into the model is considered. Firstly, we analyze singular points of the model qualitatively and get the condition for focus point. Secondly, by using geometry theory of impulsive differential equation, the existence and stability of periodic solution of the system are discussed. Lastly, some examples and numerical simulations are given to illustrate our results.

Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problems $u^{″}+a(t)f(u)=0$, $t\in (0,1)$, $u(0)=0$, and $u(1)=0$, where $f\in C(ℝ,ℝ)$ satisfies $f(0)=0$ and the limits $f_{\infty }={\text{lim}}_{|s|\to \infty }(f(s)/s)$, $f_0={\text{lim}}_{|s|\to 0}(f(s)/s)\in \{0,\infty \}$. Weight function $a(t)\in C^1[0,1]$ satisfies $a(t)>0$ on $[0,1]$.

We study the problem of bifurcation of critical periods of a time-reversible polynomial system of degree $n$. We first present a new method to find the number of zeros of the period function. Then applying our results, we study the number of critical periods for some polynomial systems and obtain new results.

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restricted $N+1$-body problems with a zero mass on the vertical axis of the plane for $N$ equal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a nonconstant periodic solution for any $2\le N\le 472$; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. This result contradicts with our intuition that the small mass should always be at the origin.

The twisting bifurcations of double homoclinic loops with resonant eigenvalues are investigated in four-dimensional systems. The coexistence or noncoexistence of large 1-homoclinic orbit and large 1-periodic orbit near double homoclinic loops is given. The existence or nonexistence of saddle-node bifurcation surfaces is obtained. Finally, the complete bifurcation diagrams and bifurcation curves are also given under different cases. Moreover, the methods adopted in this paper can be extended to a higher dimensional system.

We consider a nonautonomous functional differential equation of the third order with multiple deviating arguments. Using the Lyapunov-Krasovskiì functional approach, we give certain sufficient conditions to guarantee the asymptotic stability and uniform boundedness of the solutions.