Journal of Dynamics and Differential Equations

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Online ISSN: 1572-9222
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We introduce the notion of conditional Lipschitz shadowing, which does not aim to shadow every pseudo-orbit, but only those which belong to a certain prescribed set. We establish two types of sufficient conditions under which certain nonautonomous ordinary differential equations have such a property. The first criterion applies to a semilinear differential equation provided that its linear part is hyperbolic and the nonlinearity is small in a neighborhood of the prescribed set. The second criterion requires that the logarithmic norm of the derivative of the right-hand side with respect to the state variable is uniformly negative in a neighborhood of the prescribed set. The results are applicable to important classes of model equations including the logistic equation, whose conditional shadowing has recently been studied. Several examples are constructed showing that the obtained conditions are optimal.
 
In this paper, we study the well-posedness properties of a stochastic rotating shallow water system. An inviscid version of this model has first been derived in Holm (Proc R Soc A 471:20140963, 2015) and the noise is chosen according to the Stochastic Advection by Lie Transport theory presented in Holm (Proc R Soc A 471:20140963, 2015). The system is perturbed by noise modulated by a function that is not Lipschitz in the norm where the well-posedness is sought. We show that the system admits a unique maximal solution which depends continuously on the initial condition. We also show that the interval of existence is strictly positive and the solution is global with positive probability.
 
We consider a derivative nonlinear Schrödinger equation with general nonlinearlity: $$\begin{aligned} i\partial _tu+\partial _x^2u+i|u|^{2\sigma }\partial _xu=0, \end{aligned}$$In Tang and Xu (J Differ Equ 264(6):4094–4135, 2018), the authors prove the stability of two solitary waves in energy space for \(\sigma \in (1,2)\). As a consequence, there exists a solution of the above equation which is close arbitrary to sum of two solitons in energy space when \(\sigma \in (1,2)\). Our goal in this paper is proving the existence of multi-solitons in energy space for \(\sigma \geqslant \frac{3}{2}\). Our proofs proceed by fixed point arguments around the desired profile, using Strichartz estimates.
 
In this paper, we study the shift on the space of uniformly bounded continuous functions band-limited in a given compact interval with the standard topology of tempered distributions. We give a constructive proof of the existence of minimal subsystems with any given mean dimension strictly less than twice its band-width. A version of real-valued function spaces is considered as well.
 
We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value. Stable TW solutions for such systems have previously been observed numerically as well as in biological experiments on the growth of cellulolytic biofilms. In this work, we provide an analytical justification for these observations and prove existence and stability results for TW solutions of such models. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase–plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. Finally, numerical simulations are presented that affirm the theoretical predictions on the existence, stability, and parametric dependence of the travelling waves.
 
Example 2.3
Several notions of topological entropy for non necesarily continuous semiflows were introduced because the classical notion of topological entropy does not work cause the discontinuity of these systems. In particular, in Jaque and San Martin (J Differ Equ 266:3580–3600, 2019) was studied the definition of topological entropy by using separated and spanned sets, respectively, associated to a certain pseudosemimetric. Besides, it was proved that for regular impulsive semiflows the notion obtained by using separated sets can be computed trough a suitable conjugacy with a continuous system. In this paper, we will show that the same result holds for the entropy defined by spanning sets by adding a mild condition on the semiflow. This result extends, for this kind of semiflows, those obtained by Bowen (Trans Am Math Soc 153:401–414, 1971) and Dinaburg (Dokl Akad Nauk SSSR 190:19–22, 1970) respectively for the continuous case.
 
The asymptotic behaviour of stochastic three-dimensional Lagrangian-averaged Navier-Stokes equations with infinite delay and nonlinear hereditary noise is analysed. First, using Galerkin’s approximations and the monotonicity method, we prove the existence and uniqueness of solutions when the non-delayed external force is locally integrable and the delay terms are globally Lipschitz continuous with an additional assumption. Next, we show the existence and uniqueness of stationary solutions to the corresponding deterministic equation via the Lax-Milgram and the Schauder theorems. Later, we focus on the stability properties of stationary solutions. To begin with, we discuss the local stability of stationary solutions for general delay terms by using a direct method and then apply the abstract results to two kinds of infinite delays. Besides, the exponential stability of stationary solutions is also established in the case of unbounded distributed delay. Moreover, we investigate the asymptotic stability of stationary solutions in the case of unbounded variable delay by constructing appropriate Lyapunov functionals. Eventually, we establish criteria on the polynomial asymptotic stability of stationary solutions for the special case of proportional delay.
 
We correct a mistake on the Melnikov function given in (Battelli and Fečkan in J Dyn Differ Equ, 2022. https://doi.org/10.1007/s10884-022-10155-0) for the persistence of periodic solutions in perturbed slowly varying discontinuous differential equations.
 
If the semigroup is slowly non-dissipative, i.e., its black solutions can diverge to infinity as time tends to infinity, one still can study its dynamics via the approach by the unbounded attractors—the counterpart of the classical notion of global attractors. We continue the development of this theory started by Chepyzhov and Goritskii (Unbounded attractors of evolution equations. Advances in Soviet mathematics, American Mathematical Society, Providence, 1992). We provide the abstract results on the unbounded attractor existence, and we study the properties of these attractors, as well as of unbounded ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-limit sets in slowly non-dissipative setting. We also develop the pullback non-autonomous counterpart of the unbounded attractor theory. The abstract theory that we develop is illustrated by the analysis of the autonomous problem governed by the equation ut=Au+f(u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_t = Au + f(u)$$\end{document}. In particular, using the inertial manifold approach, we provide the criteria under which the unbounded attractor coincides with the graph of the Lipschitz function, or becomes close to the graph of the Lipschitz function for large argument.
 
In this paper, we study the instability of standing waves for the nonlinear Schrödinger equation with energy critical growth, also called as Sobolev critical growth. Firstly, we prove the existence of the ground states by solving a variational problem under certain conditions. Secondly, by establishing the approximation relation between the rescaling elliptic equation and the critical Lane–Emden equation, we prove the instability of the standing waves for sufficiently large frequency in \({L^2}\)-subcritical case. Moreover, we prove that all standing waves with positive frequencies are strongly unstable in \({L^2}\)-critical and \({L^2}\)-supercritical cases.
 
A Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting is formulated. The cyclicity of diverse limit periodic sets, including a generic contact point, canard slow-fast cycles, transitory canards, slow-fast cycles with two canard mechanisms, singular slow-fast cycle, etc. are analyzed. The cyclicity of the contact point by the normal form theory in planar slow-fast setting and of all types of canard slow-fast cycles has been obtained. Especially , new techniques for finding the maximum number of limit cycles produced by slow-fast cycles containing both the generic and degenerate contact point away from the origin (such slow-fast cycles are the transitory canards and cycles with two canard mechanisms) are developed. The main tool is geometric singular perturbation theory including cylindrical blow-up and the notion of slow divergence integral. Meanwhile, the dynamic behaviors near the origin using non-standard techniques (constructing generalized normal sectors) are studied. Furthermore , it is found that the parameter regions, which is corresponding canard slow-fast cycle, has cyclicity at most one or two by the slow divergence integral theory and blow-up techniques and the uniqueness and stability of a relaxation oscillation is shown using the notion of entry-exit function. By numerical simulations, some interesting dynamic phenomena, such as relaxation oscillation and canard explosion, are shown to illustrate the theoretical results.
 
This paper concerns the behavior of time-periodic solutions to 1D dissipative autonomous semilinear hyperbolic PDEs under the influence of small time-periodic forcing. We show that the phenomenon of forced frequency locking happens similarly to the analogous phenomena known for ODEs or parabolic PDEs. However, the proofs are essentially more difficult than for ODEs or parabolic PDEs. In particular, non-resonance conditions are needed, which do not have counterparts in the cases of ODEs or parabolic PDEs. We derive a scalar equation which answers the main question of forced frequency locking: Which time shifts $$u(t+\varphi )$$ u ( t + φ ) of the solution u ( t ) to the unforced equation do survive under which forcing?
 
The phase portrait on the carrying simplex of Zeeman’s class 30
The phase portrait on the carrying simplex of Zeeman’s class 28
The phase portrait on the carrying simplex of Zeeman’s class 26
The phase portrait on the carrying simplex of Zeeman’s class 31
In the existing literatures, there exist at least four limit cycles for 3D Lotka-Volterra competitive systems in Zeeman’s classes 26 and 27 with no explicit critical parameter values, and three limit cycles in Zeeman’s classes 27 and 29 with explicit critical parameter values. Giving explicit critical parameter values in Zeeman’s classes \(26-31\) and accurate focus values at them except class 30, we rigorously prove that Zeeman’s classes \(27-30\) have at least three limit cycles, and Zeeman’s classes 26 and 31 have at least two small amplitude limit cycles.
 
Spreading of the fast prey for given parameters {d1=1.3,d2=0.4,d3=0.3,s=1.8}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d_1=1.3, d_2=0.4, d_3=0.3, s=1.8\}$$\end{document}. The conditions in Theorem 1.3 are satisfied, since s1∗≈2.28,s2∗≈1.26,s3∗≈1.78\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1^*\approx 2.28, s_2^*\approx 1.26, s_3^*\approx 1.78$$\end{document}
Spreading of the predator and the fast prey for given parameters {d1=1.3,d2=0.1,d3=0.6,s=1.0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d_1=1.3, d_2=0.1, d_3=0.6, s=1.0\}$$\end{document}. The conditions in Theorem 1.4 are satisfied, since s1∗≈2.28,s2∗≈0.63,s3∗∗∗≈1.31\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1^*\approx 2.28,\; s_2^*\approx 0.63,\; {s_3^{***}\approx 1.31}$$\end{document}
Spreading of two preys for given parameters {d1=1.3,d2=1.8,d3=0.1,s=1.1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d_1=1.3, d_2=1.8, d_3=0.1, s=1.1\}$$\end{document}. The conditions in Theorem 1.5 are satisfied, since s1∗∗≈2.04,s2∗∗=1.2,s3∗≈1.03\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1^{**}\approx 2.04,\; s_2^{**}=1.2,\; s_3^*\approx 1.03$$\end{document}
Spreading of all species for given parameters {d1=1.0,d2=0.8,d3=1.2,s=0.5}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{d_1=1.0, d_2=0.8, d_3=1.2, s=0.5\}$$\end{document}. The conditions in Theorem 1.6 are satisfied, since s1∗∗∗≈1.63,s2∗∗∗≈0.73,s3∗∗∗≈1.86\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s_1^{***}\approx 1.63,\; s_2^{***}\approx 0.73,\; s_3^{***}\approx 1.86$$\end{document}
This paper deals with the long time behavior of a reaction–diffusion system modeling the spatio-temporal interaction of two preys and one predator in a shifting environment. Here the prey populations weakly compete, and the environment becomes hostile for the three species with a constant positive speed. We investigate the survival of the species and describe the spreading speeds of different species. The dynamical behavior exhibits various regions composed of different combinations of species, which propagate with different wave speeds.
 
In this paper, we revisit the Cauchy problem for the three dimensional nonlinear Schrödinger equation with a constant magnetic field. We first establish sufficient conditions that ensure the existence of global in time and finite time blow-up solutions. In particular, we derive sharp thresholds for global existence versus blow-up for the equation with mass-critical and mass-supercritical nonlinearities. We next prove the existence and orbital stability of normalized standing waves which extend the previous known results to the mass-critical and mass-supercritical cases. To show the existence of normalized solitary waves, we present a new approach that avoids the celebrated concentration-compactness principle. Finally, we study the existence and strong instability of ground state standing waves which greatly improve the previous literature.
 
The aim of this paper is to present a new, analytical, method for computing the exact number of relative equilibria in the planar, circular, restricted 4-body problem of celestial mechanics. The new approach allows for a very efficient computer-aided proof, and opens a potential pathway to proving harder instances of the n -body problem.
 
A general time-delay reaction–diffusion–advection system with the Dirichlet boundary condition and spatial heterogeneity is investigated in this paper. By using the implicit function theorem, we obtain the existence and asymptotic expression of the spatially non-homogeneous positive steady-state solution. This is the steady-state bifurcation from zero equilibrium. Via analyzing the corresponding characteristic equation, the stability of the spatially non-homogeneous positive steady-state solution and the occurrence of Hopf bifurcation at the positive steady-state solution are obtained, and the spatially non-homogeneous periodic solution is derived from Hopf bifurcation, this is the secondary bifurcation behavior of the system. Utilizing the normal form method and center manifold theory, we prove that the direction of Hopf bifurcation is supercritical and the bifurcating spatially non-homogeneous periodic solution is stable. Furthermore, We show that there exist two sequences Hopf bifurcation values and the orders of two sequences Hopf bifurcation values are given. Moreover, theoretical and numerical results are applied to competition and cooperation systems, respectively. Finally, the effect of the advection rate and spatial heterogeneity are discussed.
 
Suppose M is a closed Riemannian manifold. For a \(C^2\) generic (in the sense of Mañé) Tonelli Hamiltonian \(H: T^*M\rightarrow \mathbb {R}\), the minimal viscosity solution \(u_\lambda ^-:M\rightarrow \mathbb {R}\) of the negative discounted equation $$\begin{aligned} -\lambda u+H(x,d_xu)=c(H),\quad x\in M,\ \lambda >0 \end{aligned}$$with the Mañé’s critical value c(H) converges to a uniquely established viscosity solution \(u_0^-\) of the critical Hamilton–Jacobi equation $$\begin{aligned} H(x,d_x u)=c(H),\quad x\in M \end{aligned}$$as \(\lambda \rightarrow 0_+\). We also propose a dynamical interpretation of \(u_0^-\).
 
With metamorphosis or not, creatures have varying ability in their different life stages to compete for resource, space or mating. Interaction of species with environment and competition between species are key factors in the evolution of ecological population. Taking these concerns into account, we study a model with two life stages, immature and mature, and incorporate both intra- and inter-specific competitions between two species in a two-patch environment. The structure of monotone dynamics in such a model leads us to explore its local and global dynamics. The investigation starts with the single-species model on which we establish the threshold dynamics that either the species eventually goes extinction or exists on both patches, which is determined by the parameters. Then we study the two-species model and formulate the threshold competition strength which monotonously but oppositely depends on the maturation times of two species, and indicates how the competitor invade an environment. Moreover, we demonstrate two mechanisms which give rise to dominance dynamics, under competition-dependent and -independent criteria respectively. Finally, we conduct numerical simulations to show that the proposed model admits multiple positive equilibria due to the consideration of two life stages.
 
Characterization of the dynamics for the differential equation (4.11) for s∈[-40,40]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in [-40,40]$$\end{document}. In the upper panel the attractor a~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{a}$$\end{document} (in red) and repeller r~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{r}$$\end{document} (in blue) of x′=-x2+p(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x'=-x^2+p(t)$$\end{document}. In the lower panel, the curves λ∞(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^\infty (s)$$\end{document} (solid green curve in the lower panel) and d∞(s)=2-a~(s)+r~(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^\infty (s)=2-\widetilde{a}(s)+\widetilde{r}(s)$$\end{document} (magenta in the lower panel). The (common) points s on which they are strictly positive (i.e., the points s for which a~(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{a}(s)$$\end{document} and r~(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{r}(s)$$\end{document} are close enough) are highlighted in thick red on the axis y=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y=0$$\end{document} in both panels. These are the points for which (4.11) has no bounded solutions, The complementary of the closure of this set, given by the points for which λ∞(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^\infty (s)$$\end{document} and d∞(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d^\infty (s)$$\end{document} are strictly negative, is composed by the points for which (4.11) has an attractor–repeller pair (Color figure online)
Numerical simulation of the bifurcation map λ∗(c,s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _*(c,s)$$\end{document} of (4.10)c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_c$$\end{document} (surface with gradient color). The red grid identifies the plane λ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda =0$$\end{document}. Consequently, the points of the surface below it correspond to Case A, the points above to Case C, and the points of intersection to Case B. The curve in green is the graph of the bifurcation curve λ∞(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^\infty (s)$$\end{document} of (4.11) (see also Fig. 1). Theorem 5.3 guarantees the convergence of λ∗(c,s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _*(c,s)$$\end{document} to λ∞(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^\infty (s)$$\end{document} as c increases. The figure indicates how fast this convergence is
Total tipping on the hull for y′=-(y-2Γc(t))2+ps(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y'=-\big (y-2\,\Gamma _c(t)\big )^2+p_s(t)$$\end{document} for large enough c
Numerical simulation of the bifurcation map λ∗(c,h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _*(c,h)$$\end{document} of (5.1)c,h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{c,h}$$\end{document} for Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document} and p as in (4.9) and c,h∈[0,5]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c,h\in [0,5]$$\end{document}. On the left: the gradient surface represents the graph of λ∗(c,h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _*(c,h)$$\end{document}; the red grid identifies the plane λ∗=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _*=0$$\end{document}: the points of the surface below this plane correspond to Case A, the points above to Case C, and the points of intersection to Case B. The red dashed line is the graph of the bifurcation curve λ∗(c,0)=λ∗(c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _*(c,0)=\lambda _*(c)$$\end{document} of the family (4.1), whereas the solid green line is the graph of the bifurcation curve λ∗(∞,h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _*(\infty ,h)$$\end{document} of (5.1)∞,h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{\infty ,h}$$\end{document}, represented at c=6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=6$$\end{document} for convenience. On the right: a projection of the same picture on the plane c=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0$$\end{document}
A critical transition for a system modelled by a concave quadratic scalar ordinary differential equation occurs when a small variation of the coefficients changes dramatically the dynamics, from the existence of an attractor–repeller pair of hyperbolic solutions to the lack of bounded solutions. In this paper, a tool to analyze this phenomenon for asymptotically nonautonomous ODEs with bounded uniformly continuous or bounded piecewise uniformly continuous coefficients is described, and used to determine the occurrence of critical transitions for certain parametric equations. Some numerical experiments contribute to clarify the applicability of this tool.
 
The weak Minkowski billiard reflection rule: qj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_j$$\end{document} minimizes (2) over all q¯j∈Hj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{q}_j\in H_j$$\end{document}, where Hj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_j$$\end{document} is a K-supporting hyperplane through qj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_j$$\end{document}
T∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^\circ $$\end{document} is a smooth and strictly convex body in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document} and its boundary plays the role of the indicatrix, i.e., the set of vectors of unit Finsler (with respect to T∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^\circ $$\end{document}) length, which therefore is an 1-level set of μT∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{T^\circ }$$\end{document}. Note that the two T∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^\circ $$\end{document}-supporting hyperplanes intersect on Hj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_j$$\end{document} due to the condition ∇μT∘(qj-qj-1)-∇μT∘(qj+1-qj)=μjnHj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla \mu _{T^\circ }(q_j-q_{j-1})-\nabla \mu _{T^\circ }(q_{j+1}-q_j)= \mu _j n_{H_j}$$\end{document}
The pair (q, p) satisfies (3), namely: qj-qj-1∈NT(pj-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_j-q_{j-1}\in N_T(p_{j-1})$$\end{document}, qj+1-qj∈NT(pj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{j+1}-q_j\in N_T(p_j)$$\end{document}, pj-pj-1∈-NK(qj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_j-p_{j-1}\in -N_K(q_j)$$\end{document}, and pj+1-pj∈-NK(qj+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_{j+1}-p_j\in - N_K(q_{j+1})$$\end{document}
Illustration of K×T⊂Rn×Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\times T\subset \mathbb {R}^n\times \mathbb {R}^n$$\end{document} and ∂HK×T\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial H_{K\times T}$$\end{document}
Illustration of the idea behind the proof of Proposition 4 when xq([a′,b′])\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_q([a',b'])$$\end{document} is a subset of the interior of a j-face of K, 1≤j≤n-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le j \le n-2$$\end{document}. We have t0∈[a′,b′]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0\in [a',b']$$\end{document} and clearly see NT(xp(t0))∩(Tn-j)⊥,Rpn={0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_T(x_p(t_0))\cap (T_{n-j})^{\perp ,\mathbb {R}^n_p}=\{0\}$$\end{document}
We rigorously state the connection between the EHZ-capacity of convex Lagrangian products K×T⊂R^n×R^n and the minimal length of closed (K , T )-Minkowski billiard trajectories. This connection was made explicit for the first time by Artstein-Avidan and Ostrover under the assumption of smoothness and strict convexity of both K and T . We prove this connection in its full generality, i.e., without requiring any conditions on the convex bodies K and T. This prepares the computation of the EHZ-capacity of convex Lagrangian products of two convex polytopes by using discrete computational methods.
 
In this paper, we consider a kind of second-order delay differential system. By taking some transforms, the property of delay is reflected in the boundary condition. The wonder is that the corrseponding first-order system is exactly the so-called P-boundary value problem of Hamiltonian system which has been studied deeply by many mathematicians, including the authors of this paper. Firstly, we define the relative Morse index \(\mu _Q(A,B)\) for the delay system and give the relationship with the P-index \(i_{P}(\gamma _{R})\) of Hamiltonian system. Secondly, by this index, topology degree and saddle point reduction, the existence of periodic solutions is established for this kind of delay differential system.
 
This paper is concerned with the spreading behavior of a two-species strong-weak competition system with two free boundaries. The model may describe how a strong competing species invades into the habitat of a native weak competing species. The asymptotic spreading speed of invading fronts has been determined by making use of semi-wave systems in Du et al. (J Math Pures Appl 107:253–287, 2017). Here we give a sharp estimate for the asymptotic spreading speed of invading fronts. Moreover, we prove that the solution of the free boundary problem evolves eventually into a semi-wave solution when the spreading happens, while the solution of the free boundary problem exponentially converges to a semi-trivial solution of such system when the vanishing happens.
 
In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.
 
This paper concerns a discrete diffusive predator–prey system involving two competing predators and one prey in a shifting habitat induced by the climate change. By assuming that both predators can increase when the prey is at the maximal capacity and the prey can still survive under optimal climatic conditions when these two predators have their maximal densities, we investigate the existence and non-existence for different types of forced traveling waves which describe the conversion from the state of a saturated aboriginal prey with two invading alien predators, an aboriginal co-existent predator–prey with an invading alien predator, and the coexistence of three species to the extinction state, respectively. The existence of supercritical and critical forced waves is showed by applying Schauder’s fixed point theorem on various invariant cones via constructing different types of generalized super- and sub-solutions while the non-existence of subcritical forced waves is obtained by contradiction.
 
The theory of compact global attractors for dynamical systems relies on the existence of a bounded absorbing set. In this paper, suppressing this condition, we present sufficient conditions to ensure the existence and uniqueness of maximal attractors for dynamical systems (both autonomous and nonautonomous). Such attractors are, in general, unbounded. For semigroups satisfying these conditions, when a Lyapunov function exists, we also present the characterization of the unique maximal attractor as the unstable set of the critical elements (not necessarily fixed points). As an example we present a semilinear parabolic equation to illustrate the theory in the autonomous case. For the nonautonomous setting, applying the general theory, we present an abstract evolution equation and a concrete parabolic equation as examples.
 
a The boundness of τ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau (t)$$\end{document}. b The monotonicity of t-τ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t-\tau (t)$$\end{document}
The periodic solution of Eq. (1.2) with infinite oscillatory frequency
A sketch of sequences bk,b~k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_k,\,\,{\tilde{b}}_k$$\end{document} and ck\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_k$$\end{document}
A sketch of the sequence (b~kjl)j∈Z,1≤l≤lj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\tilde{b}}_{kj}^l)_{j\in \mathbb {Z},\,1\le l\le l_j}$$\end{document}
This paper mainly investigates the oscillatory behaviors in a class of discontinuous equations with a variable delay. Under the restrictions of boundedness and monotonicity on the time-delay function, the type and distribution of zero points of the solution are firstly determined. And then, the non-existence of oscillatory solutions with infinite frequency is proven for the systems with two classes of discontinuous vector fields. In detail, by applying the inequality techniques on a backward shift operator, an exact lower bound of the time is provided to ensure the vanishing of infinite frequency oscillations after this time. The obtained results improve some previous works in the cases of the constant delay and constant vector fields.
 
This paper presents the dynamics that one microorganism spatially expands with two nutrients in space-time periodic environment. The generalized principal eigenvalue defined by the linear periodic parabolic operator is applied as a threshold to discuss the associated initial value problem. When the generalized principal eigenvalue is nonnegative, the solutions approach to the microorganism-extinction equilibrium uniformly. When the generalized principal eigenvalue is negative, the spreading speeds of the model are established when the initial distribution of microorganism has nonempty compact support, which are determined by a family of periodic parabolic eigenvalue problems. In the homogeneous environment, we show that the solutions locally uniformly converge to the microorganism-existent steady state by constructing upper and lower solutions. Finally, numerical simulations are performed to illustrate that the leftward or rightward spreading speed is approximately a constant under various circumstances.
 
In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our coupling method only requires that the expanding map satisfies three assumptions: (H1) Chernov’s one-step expansion at q-scale; (H2) dynamically Hölder continuity of log Jacobian (on each branch); (H3) eventually covering over a magnet interval, which are much weaker than the standard assumptions for uniformly expanding maps. We further conclude the existence of an absolutely continuous invariant probability measure and establish the regularity of its density function. Moreover, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem) with respect to a large class of unbounded observables that we call “dynamically Hölder series". Our approach is particularly powerful for the piecewise expanding maps which do not satisfy the “big image property" and the maps that have inverse Jacobian of low regularity. As few is known on the statistical properties of such maps in the literature, we demonstrate our results for a specific class of piecewise expanding maps of this kind.
 
We investigate the global dynamics from a measure-theoretic perspective for smooth flows with invariant cones of rank k. For such systems, it is shown that prevalent (or equivalently, almost all) orbits will be pseudo-ordered or convergent to equilibria. This reduces to Hirsch’s prevalent convergence Theorem if the rank \(k=1\); and implies an almost-sure Poincaré–Bendixson Theorem for the case \(k=2\). These results are then applied to obtain an almost sure Poincaré–Bendixson theorem for high-dimensional differential equations.
 
This paper deals with a two-species attraction–repulsion chemotaxis system $$\begin{aligned} \left\{ \begin{aligned}&u_t=\Delta u-\xi _{1}\nabla \cdot (u\nabla v)+\chi _{1}\nabla \cdot (u\nabla z)+f_{1}(u,w),&(x,t)\in \Omega \times (0,\infty ), \\&\tau v_{t}=\Delta v+w-v,&(x,t)\in \Omega \times (0,\infty ),\\&w_t=\Delta w-\xi _{2}\nabla \cdot (w\nabla z)+\chi _{2}\nabla \cdot (w\nabla v)+f_{2}(u,w),&(x,t)\in \Omega \times (0,\infty ), \\&\tau z_{t}=\Delta z+u-z,&(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right. \end{aligned}$$under homogeneous Neumann boundary conditions in a smoothly bounded domain \(\Omega \subseteq {\mathbb {R}}^{n}\), where \(\tau \in \{0,1\},\xi _{i},\chi _{i}>0\) and \(f_{i}(u,w)(i=1,2)\) satisfy $$\begin{aligned} \left\{ \begin{aligned}&f_{1}(u,w)=u\bigg (a_{0}-a_{1}u-a_{2}w+a_{3}\int _{\Omega }udx+a_{4}\int _{\Omega }wdx\bigg ),\\&f_{2}(u,w)=w\bigg (b_{0}-b_{1}u-b_{2}w+b_{3}\int _{\Omega }udx+b_{4}\int _{\Omega }wdx\bigg )\\ \end{aligned} \right. \end{aligned}$$with \(a_{i},b_{i}>0(i=0,1,2),a_{j},b_{j}\in {\mathbb {R}}(j=3,4)\). It is proved that in any space dimension \(n\ge 1\), the above system possesses a unique global and uniformly bounded classical solution regardless of \(\tau =0\) or \(\tau =1\) under some suitable assumptions. Moreover, by constructing Lyapunov functionals, we establish the globally asymptotic stabilization of coexistence and semi-coexistence steady states.
 
We study the dynamics about equilibria of an infinite dimensional system of ordinary differential equations of coagulation–fragmentation–death type that was introduced recently by da Costa et al. (Eur J Appl Math 31(6):950–967, 2020) as a model for the silicosis disease mechanism. For a class of piecewise constant rate coefficients an appropriate change of variables allows for the appearance of a closed finite dimensional subsystem of the infinite-dimensional system and the analysis of the eigenvalues of the linearizations of this finite dimensional subsystem about the equilibria is then used to obtain the results on the stability of the equilibria in the original infinite dimensional model.
 
Holling type functional responses
Limit cycles and relative positions of Wlocs(PA)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^s_{loc}(P_A)$$\end{document} and Wlocu(PK)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^u_{loc}(P_K)$$\end{document}
Codimension 3 Bogdanov–Takens bifurcation diagram on S2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^2$$\end{document} (ξ1≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _1\le 0$$\end{document})
Codimension 3 Hopf bifurcation diagram
This paper is devoted to the high codimension bifurcations of a classical predator–prey system with Allee effects and generalized Holling type III functional response \(p(x)=\frac{mx^2}{ax^2+bx+1}\) where \(b>-2\sqrt{a}\). We show that the maximal orders of nilpotent saddle, cusp singularity and weak focus are all three. The unfoldings of a cusp singularity and of a nilpotent saddle of order 3 with a fixed invariant line are developed. The dependence of codimension of degenerate Hopf bifurcation on b is thoroughly investigated. It is proven that there exist a homoclinic loop of order 2 and a heteroclinic loop of order 2 for \(-2\sqrt{a}<b<0\), and three limit cycles for \(b>0\). Together with existing work for Holling type I, II and IV functional responses, our results complement the analysis of the classical predator–prey systems with Allee effects and four types of Holling functional responses. Furthermore, simple formulas are derived to characterize the order of nilpotent saddle, through which the existence and order of the heteroclinic loop can be easily obtained for a general class of predator–prey systems with any smooth functional response.
 
We study solutions of the equation $$u_t-\Delta u+\lambda u=f$$ u t - Δ u + λ u = f , for initial data that is ‘large at infinity’ as treated in our previous papers on the unforced heat equation. When $$f=0$$ f = 0 we characterise those $$(u_0,\lambda )$$ ( u 0 , λ ) for which solutions converge to 0 as $$t\rightarrow \infty $$ t → ∞ , as not every $$\lambda >0$$ λ > 0 is able to achieve that for all initial data. When $$f\ne 0$$ f ≠ 0 we give conditions to guarantee that the solution is given by the usual ‘variation of constants formula’ $$u(t)=\mathrm{e}^{-\lambda t}S(t)u_0+\int _0^t\mathrm{e}^{-\lambda (t-s)}S(t-s)f(s)\,\mathrm{d}s$$ u ( t ) = e - λ t S ( t ) u 0 + ∫ 0 t e - λ ( t - s ) S ( t - s ) f ( s ) d s , where $$S(\cdot )$$ S ( · ) is the heat semigroup. We use these results to treat the elliptic problem $$-\Delta u+\lambda u=f$$ - Δ u + λ u = f when f is allowed to be ‘large at infinity’, giving conditions under which a solution exists that is given by convolution with the usual Green’s function for the problem. Many of our results are sharp when $$u_0,f\ge 0$$ u 0 , f ≥ 0 .
 
We study the behavior of solutions of the Cauchy problem for a semilinear heat equation with critical nonlinearity in the sense of Joseph and Lundgren. It is known that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. In this paper, we give its optimal and sharp convergence rate of solutions with a critical exponent and two exponentially approaching initial data. This rate contains a logarithmic term which does not contain in the super critical nonlinearity case. Proofs are given by a comparison method based on matched asymptotic expansion.
 
In this article, we consider 3D stochastic primitive equations (PEs) driven by affine-linear multiplicative white noise, with random initial condition. Our main objective is to obtain the global well-posedness of the stochastic equations under the sufficient Malliavin regularity of the initial condition. Apart from the conventional strategy, we adopt the dynamical system approach and techniques from Malliavin calculus to attack the global well-posedness problem for the PEs.
 
The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions.
 
In this paper, we study Chaplain–Lolas model in three dimensional bounded domain, which describes the invasion and diffusion process of solid tumors during the vascular growth stage. Although the model has received extensive attention since it was proposed, the existence of solutions in three-dimensional space is still missing, and only a global small solution is established by Pang and Wang (Math Models Methods Appl Sci 28(11):2211–2235, 2018) with sufficiently small proliferation coefficient. As for long time behavior, there are only corresponding stability result of constant steady-state for the case without ECM remodelling (Hillen et al. in Math Models Methods Appl Sci 25(1):165–198, 2013; Tao and Winkler in SIAM J Math Anal 47:4229–4250, 2016). If the remodeling effect of ECM is considered, the relevant research is still blank. In this paper, we first pay our attention to the study of existence of global strong solution and long time behavior. We prove that when the ratio of cell proliferation coefficient to chemotactic intensity μχ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\mu }{\chi ^2}$$\end{document} is large, there exists a unique global strong solution around the equilibrium state (1, 1, 0), and the global strong solution converges exponentially to the constant equilibrium point. In fact, such a largeness restriction on μχ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\mu }{\chi ^2}$$\end{document} is actually necessary to some extent, since that the constant equilibrium point (1, 1, 0) is actually linearly unstable when such a condition is not satisfied. Subsequently, we turn our attention to the study of dynamic behavior of solutions. We introduce a time periodic external force to this system, and prove that the solution will gradually show the same periodic behavior under the action of periodic external force, and become a time periodic solution. At last, we analysis the stability and instability of equilibrium points, and discuss the influence of spatial diffusion, chemotaxis and haptotaxis effect on the stability of solutions. In particular, we find that only chemotaxis can change the stability of the solution and make the originally stable equilibrium (1, 1, 0) unstable, while the haptotactic term has no effect on the stability.
 
We present two limit theorems for the zero-range process with nonlocal diffusion on inhomogeneous networks. The deterministic model is governed by the reaction–diffusion equation with an integral term in space instead of a Laplacian. By constructing the reproducing kernel Hilbert space to consider the inhomogeneities of the network structure, we prove that the law of large numbers and the central limit theorem hold for our models. Furthermore, under a special case of the kernel, we can show that the fluctuation limit obtained by the central limit theorem has a continuous sample path.
 
In the N-body problem, it is classical that there are conserved quantities of center of mass, linear momentum, angular momentum and energy. The level sets M(c,h)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {M}(c,h)$$\end{document} of these conserved quantities are parameterized by the angular momentum c and the energy h, and are known as the integral manifolds. A long-standing goal has been to identify the bifurcation values, especially the bifurcation values of energy for fixed non-zero angular momentum, and to describe the integral manifolds at the regular values. Alain Albouy identified two categories of singular values of energy: those corresponding to bifurcations at relative equilibria; and those corresponding to “bifurcations at infinity”, and demonstrated that these are the only possible bifurcation values. This work examines the bifurcations for the four body problem with equal masses. There are four singular values corresponding to bifurcations at infinity. To establish that the topology of the integral manifolds changes at each of these values, and to describe the manifolds at the regular values of energy, the homology groups of the integral manifolds are computed for the five energy regions on either side of the singular values. The homology group calculations establish that all four energy levels are indeed bifurcation values, and allows some of the global properties of the integral manifolds to be explored. A companion paper will provide the corresponding analysis of the bifurcations at relative equilibria for the four-body problem with equal masses.
 
We prove that smoothness of nonautonomous linearization is of class C2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^2.$$\end{document} Our approach admits the existence of stable and unstable manifolds determined by a family of nonautonomous hyperbolicities, including the non uniform exponential case, while for the classic exponential dichotomy we obtain the same class of differentiability except for a zero Lebesgue measure set. Moreover, our goal is reached without spectral conditions.
 
We consider some families of three-dimensional quadratic vector fields having a fixed zero-Hopf equilibrium. We are interested in the bifurcation of periodic ν-orbits from the singularity, that is, those small amplitude orbits that make a fixed arbitrary number ν of revolutions about a rotation axis and then returns to the initial point closing the orbit. When the parameters of the family are restricted to certain explicitly computable open semi-algebraic sets Λ, we characterize those parameters that give rise to the appearance of local two-dimensional periodic invariant manifolds through the singularity. Also we use a Bautin-type analysis to study the maximum number of small-amplitude ν-limit cycles that can be made to bifurcate from the equilibrium when the parameters of the family are restricted to Λ. We obtain global upper bounds on the number of bifurcated ν-limit cycles.
 
Various inequivalent notions of attraction for autonomous dynamical systems have been proposed, each of them useful to understand specific aspects of attraction. Milnor’s notion of a measure attractor considers invariant sets with positive measure basin of attraction, while Ilyashenko’s weaker notion of a statistical attractor considers positive measure points that approach the invariant set in terms of averages. In this paper we propose generalisations of these notions to nonautonomous evolution processes in continuous time. We demonstrate that pullback/forward measure/statistical attractors can be defined in an analogous manner and relate these to the respective autonomous notions when an autonomous system is considered as nonautonomous. There are some subtleties even in this special case–we illustrate an example of a two-dimensional flow with a one-dimensional measure attractor containing a single point statistical attractor. We show that the single point can be a pullback measure attractor for this system. Finally, for the particular case of an asymptotically autonomous system (where there are autonomous future and past limit systems) we relate pullback (respectively, forward) attractors to the past (respectively, future) limit systems.
 
In the present paper, we establish an infinite dimensional Kolmogorov–Arnold–Moser (KAM) theorem for reversible systems with double normal frequencies. Applying it, we prove the existence of quasi-periodic solutions for one dimensional coupled nonlinear quantum harmonic oscillators (QHO) with a natural reversible structure. To compensate the lack of smoothing effect of perturbation, we introduce a class of vector fields with polynomial decay which extends the works of Grébert and Thomann (Commun Math Phys 307(2):383–427, 2011) for Hamiltonian QHO. To deal with the reversible, coupled perturbations in the equations, we also introduce a new class of generating vector fields during the KAM iteration. Moreover, the quasi-periodic solutions we obtain may not be linearly stable. This is obviously different from the result in Grébert and Thomann (2011) for Hamiltonian QHO.
 
In this paper, we are interested in the positive periodic solutions of the periodic-parabolic problem $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\Delta u+\lambda u-a(x,t)u^p &{}\text { in } \Omega \times (0,T],\\ Bu=0 &{}\text { on } \partial \Omega \times (0,T],\\ u(x,0)=u(x,T) &{}\text { in } \Omega , \end{array}\right. } \end{aligned}$$where \(\Omega \) is a \(C^{2+\mu }\) bounded domain in \({\mathbb {R}}^N\) (\(N\ge 1\)), \(\lambda >0\) is a real parameter, \(p>1\) is constant, \(a\in C^{\mu ,\mu /2}({{\bar{\Omega }}}\times [0,T])\) is positive and T-periodic in t. We establish that the positive solution has a “blow-up" phenomenon due to large \(\lambda \) or small a(x, t). By analyzing the sharp profiles, we find that the linear part \(\lambda u\) and nonlinear part \(a(x,t)u^p\) make quite different effects on the limiting behavior of positive periodic solutions. The second aim is then to investigate the sharp connections between linear and nonlinear parts on the asymptotic behavior of positive periodic solutions. Our study exhibits that the linear part plays a determined role. We also study the asymptotic profiles of periodic-parabolic problem with nonlocal dispersal. We find that the asymptotic profiles are different between two kinds of diffusion problems.
 
The weak mean equicontinuity for a countable discrete amenable group G acting continuously on a compact metrizable space X is studied. It is shown that weak mean equicontinuity, continuously pointwise ergodicity and uniformity are coincided. Moreover, we prove that (X, G) is mean equicontinuous if and only if the product system (X×X,G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X \times X,G)$$\end{document} is weak mean equicontinuous.
 
This paper is devoted to the temporal decay of solutions of a coupled parabolic-elliptic equations in \({\mathbb {R}}^2\). It is proved that weak solutions of the equations decay to zero in \(L^{2,\infty }\times L^2\) without a uniform rate, and these decay estimates are optimal. Furthermore, the uniform logarithmic decay estimates of weak solutions are established when initial data are in \(L^1\cap L^2\). In addition, the temporal decay estimates of weak solutions at the sharp rate are also shown. The proofs are based on the Fourier splitting method and the generalized Ladyzhenskaya inequality for weak type spaces.
 
We prove the existence in the sense of sequences of solutions for some system of integro-differential type equations in two dimensions containing the normal diffusion in one direction and the anomalous diffusion in the other direction in \(H^{2}({\mathbb R}^{2}, {{\mathbb {R}}}^{N})\) using the fixed point technique. The system of elliptic equations contains second order differential operators without the Fredholm property. It is established that, under the reasonable technical assumptions, the convergence in \(L^{1}({{\mathbb {R}}}^{2})\) of the integral kernels yields the existence and convergence in \(H^{2}({{\mathbb {R}}}^{2}, {\mathbb R}^{N})\) of the solutions. We emphasize that the study of the systems is more difficult than of the scalar case and requires to overcome more cumbersome technicalities.
 
We examine the convergence in the Krylov–Bogolyubov averaging for nonlinear stochastic perturbations of linear PDEs with pure imaginary spectrum and show that if the involved effective equation is mixing, then the convergence is uniform in time.
 
Top-cited authors
Bixiang Wang
  • New Mexico Institute of Mining and Technology
Mingxin Wang
  • Harbin Institute of Technology
Ramon Plaza
  • Universidad Nacional Autónoma de México
Pablo Padilla
  • Universidad Nacional Autónoma de México
Philip Maini
  • University of Oxford