This paper investigates the spatial dynamics of an epidemic system consisting of a single reaction-diffusion equation for susceptible populations and an ordinary differential equation for infected populations. First, we analyze the stability of constant steady-state solutions and establish sufficient conditions under which Turing-like spatial heterogeneity may arise. Subsequently, we construct bifurcation solutions near the homogeneous equilibrium and prove that these spatially heterogeneous solutions are inherently unstable - a striking contrast to classical reaction-diffusion epidemic models in which both populations diffuse, where stable Turing-like patterns are commonly observed. Furthermore, we demonstrate that under specific initial conditions, the system may develop blow-up solutions within a finite time, which reflects a loss of boundedness of the mathematical solution. Numerical simulations further illustrate the transient spatial heterogeneity associated with diffusion-driven instability. The theoretical analysis is verified by numerical simulations.
This paper proposes a fifth-order semi-discrete complex mKdV equation, which may be relevant to the study of ultrashort pulse propagation in optical fibers. We systematically investigate its integrable properties, establishing its Lax pair, Hamiltonian structure, and infinite conservation laws. A discrete generalized (p,N-p) -fold Darboux transformation is formulated from the associated Lax pair. This transformation enables the systematic derivation of breather, rogue wave, periodic wave, and mixed interaction solutions, whose dynamical features and generation mechanisms are analyzed in detail. To further analyze the rogue wave solutions, we combine large parameter asymptotic techniques with the fast Fourier transformation, deriving asymptotic expressions for arbitrary order rogue waves. This analysis reveals a profound connection between the spectral characteristics of rogue waves and their separation dynamics. Finally, to bridge the semi-discrete and continuous models, we systematically demonstrate how the Lax pair, conservation laws, Darboux transformation, and rogue wave solutions of the semi-discrete system converge to their continuous counterparts in a suitable continuum limit. Consequently, we conclude that the proposed fifth-order semi-discrete complex mKdV equation serves as a highly effective discrete analogue of its continuous counterpart.
We study a factor map φ :(X,f)→ (Y,g) and its induced map on the hyperspace of nonempty compact subsets. For continuous, possibly noninvertible dynamics and a fully supported base, we prove that φ has relative uniformly positive entropy if and only if its full-hyperspace extension does. The proof uses natural extensions and preservation of relative open-cover entropy. For homeomorphic dynamics on a fixed finite hyperspace, we obtain an equivalence under an explicit matching condition, prove the cases k=1,2 , and give a counterexample for k=3 . We also prove that openness is equivalent for the original map, its full-hyperspace extension and its two-point-hyperspace extension, whereas the corresponding assertion for every finite hyperspace fails. For continuous dynamics, all four relative sensitivity properties considered here descend from the hyperspace. Relative strong sensitivity also ascends; the upward syndetic result is proved under a simultaneous finite-intersection hypothesis.
This paper is concerned with the bifurcation of critical periods from the isochronous center of the cubic Kukles system under sufficiently small polynomial perturbations of degree n that preserve the center at the origin. For a suitable parameterization ξ of the periodic orbits surrounding the center, their periods can be written as T(ξ ,ε )=T_0+T_1(ξ )ε +T_2(ξ )ε ^2+⋯ . For n=1 , the origin remains an isochronous center. For n=2 , the origin is either an isochronous center or the period function is strictly monotonic on every compact subinterval of the period annulus. For n⩾ 3 , if T_1'(ξ )≢0 , then up to the first order in ε , the maximum number N_max(n) of critical periods bifurcating from the periodic orbits of the unperturbed system satisfies [ n/2] -1 ⩽ N_max(n)⩽ n-2 . Furthermore, for cubic symmetric perturbations, if T_1≢0 , at most one critical period can bifurcate up to the first order in ε and this upper bound is sharp. If T_1≡ 0 , we compute T_2 explicitly and prove that, for all sufficiently small nonzero ε , either the period function is strictly increasing on every compact subinterval of the period annulus or the origin remains an isochronous center.
In this paper, we study the number of limit cycles of a near-Hamiltonian system ẋ=y+ε P(x,y,a), ẏ=-x+x^3+ε Q(x,y,a) . When P, Q are n-degree polynomials with n odd and 1≤ n≤ 35 , we can find n-1 limit cycles bifurcating from the center and the heteroclinic loop.
