Quorum sensing governs bacterial communication, playing a crucial role in regulating population behaviour. We propose a mathematical model that uncovers chaotic dynamics within quorum sensing networks, highlighting challenges to predictability. The model explores interactions between autoinducers and two bacterial subtypes, revealing oscillatory dynamics in both a constant autoinducer submodel and the full three-component model. In the latter case, we find that the complicated dynamics can be explained by the presence of homoclinic Shilnikov bifurcations. We employ a combination of normal-form analysis and numerical continuation methods to analyse the system.

We study the planar FitzHugh–Nagumo system with an attracting periodic orbit that surrounds a repelling focus equilibrium. When the associated oscillation of the system is perturbed, in a given direction and with a given amplitude, there will generally be a change in phase of the perturbed oscillation with respect to the unperturbed one. This is recorded by the phase transition curve (PTC), which relates the old phase (along the periodic orbit) to the new phase (after perturbation). We take a geometric point of view and consider the phase-resetting surface comprising all PTCs as a function of the perturbation amplitude. This surface has a singularity when the perturbation maps a point on the periodic orbit exactly onto the repelling focus, which is the only point that does not return to stable oscillation. We also consider the PTC as a function of the direction of the perturbation and present how the corresponding phase-resetting surface changes with increasing perturbation amplitude. In this way, we provide a complete geometric interpretation of how the PTC changes for any perturbation direction. Unlike other examples discussed in the literature so far, the FitzHugh–Nagumo system is a generic example and, hence, representative for planar vector fields.

In this paper, we derive simple analytical bounds for solutions of $x - \ln x = y -\ln y$, and use them for estimating trajectories following Lotka–Volterra-type integrals. We show how our results give estimates for the Lambert W function as well as for trajectories of general predator–prey systems, including, for example, Rosenzweig–MacArthur equations.

In this paper, we classify all global dynamics of the three-dimensional type-K monotone Lotka–Volterra system with the identical intrinsic growth rate inside the compactification of the positive octant of $\mathbb {R}^{3}$. By means of the replicator equations, it is proved that this system can have exactly $35$ topologically different phase portraits. As a consequence, we obtain the necessary and sufficient condition for the system to be bounded in the positive octant and verify that the limit set of any orbit of the compactified vector field associated with the system is an equilibrium.

The article considers systems of interacting particles on networks with adaptively coupled dynamics. Such processes appear frequently in natural processes and applications. Relying on the notion of graph convergence, we prove that for large systems the dynamics can be approximated by the corresponding continuum limit. Well-posedness of the latter is also established.

In Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90, the authors investigate about the integrability of the family QSH (the whole class of non-degenerate planar quadratic systems possessing at least one invariant hyperbola). However, some very difficult cases are left open in Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90, and the main aim of this article is to study the Liouvillian integrability some of the systems that were left behind in Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90.

In this article, we explore the bifurcation problem of limit cycles near the double eight figure loop (compound cycle with a 2-polycycle connecting two homoclinic loops). A general theory is established to find the lower bound of the maximal number of limit cycles (isolated periodic orbits) near the double eight figure loop. The Liénard system, a well-known nonlinear dynamical model, appears in a natural way in physics, chemistry, engineering, and so on, where periodic phenomena play a relevant role. As an application, we investigate an $(n+1)$th-order generalized Liénard system and prove the system has at least $7[\frac{n}{6}]+2[\frac{r}{2}]-[\frac{r}{4}]$ limit cycles near the double eight figure loop for any $n\geq5$ and $r=\rm mod(n,6)$, and their distribution is also gained.

This paper studies the spatio-temporal dynamics of a diffusive plant-sulphide model with toxicity delay. More specifically, the effects of discrete delay and distributed delay on the dynamics are explored, respectively. The deep analysis of eigenvalues indicates that both diffusion and delay can induce Hopf bifurcations. The normal form theory is used to set up an exact formula that determines the properties of Hopf bifurcation in a diffusive plant-sulphide model. A sufficiently small discrete delay does not affect the stability and a sufficiently large discrete delay destabilizes the system. Nonetheless, a sufficiently small or large distributed delay does not affect the stability. Both delays cause instability by inducing Hopf bifurcation rather than Turing bifurcation.

Given a set of standard binary patterns and a defective pattern, the pattern retrieval task is to find the closest pattern to the defective one among these standard patterns. The Hebbian network of Kuramoto oscillators with second-order coupling provides a dynamical model for this task, and the mutual orthogonality in memorised patterns enables us to distinguish these memorised patterns from most others in terms of stability. For the sake of error-free retrieval for general problems lacking orthogonality, a unified approach was proposed which transforms the problem into a series of subproblems with orthogonality using the orthogonal lift for two patterns. In this work, we propose the least orthogonal lift for three patterns, which evidently reduces the time of solving subproblems and even the dimensions of subproblems. Furthermore, we provide an estimate for the critical strength for stability/instability of binary patterns, which is convenient in practical use. Simulation results are presented to illustrate the effectiveness of the proposed approach.

We introduce a new non-abelian quantum synchronisation model over the unitary group, represented as a gradient flow, where state matrices asymptotically converge to a common one up to phase translation. We provide a sufficient framework leading to quantum synchronisation based on Riccati-type differential inequalities. In addition, uniform time-delayed interaction is considered for modelling realistic communication, and we demonstrate that quantum synchronisation is persistent when a small time delay is allowed. Finally, numerical simulation is performed to visualise qualitative behaviours and support theoretical results.

A centre of a differential system in the plane $ {\mathbb {R}}^2$ is an equilibrium point $p$ having a neighbourhood $U$ such that $U\setminus \{p\}$ is filled with periodic orbits. A centre $p$ is global when $ {\mathbb {R}}^2\setminus \{p\}$ is filled with periodic orbits. In general, it is a difficult problem to distinguish the centres from the foci for a given class of differential systems, and also it is difficult to distinguish the global centres inside the centres. The goal of this paper is to classify the centres and the global centres of the following class of quintic polynomial differential systems

\begin{align*} \dot{x}= y,\quad \dot{y}={-}x+a_{05}\,y^5+a_{14}\,x\,y^4+a_{23}\,x^2\,y^3+a_{32}\,x^3\,y^2+a_{41}\,x^4\,y+a_{50}\,x^5, \end{align*}

$ {\mathbb {R}}^2$

Analytic rotated vector fields have four significant properties: as the rotated parameter $\alpha$ changes, the amplitude of each stable (or unstable) limit cycle varies monotonically, each semi-stable limit cycle bifurcates at most two limit cycles, the isolated homoclinic loop (if exists) disappears while a unique limit cycle with the same stability arises or no closed orbits arise oppositely, and a unique limit cycle arises near the weak focus (if exists). In this paper, we prove that the four properties remain true for a rotated family of generalized Liénard systems having finitely many switching lines. Furthermore, we discuss variational exponent and use it to formulate multiplicity of limit cycles. Then we apply our results to give exact number of limit cycles to a continuous piecewise linear system with three zones and answer to a question on the maximum number of limit cycles in an SD oscillator.

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a (8,0)-mode Turing–Hopf bifurcation and unstable for a (3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a (0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.

This research studies the robustness of permanence and the continuous dependence of the stationary distribution on the parameters for a stochastic predator–prey model with Beddington–DeAngelis functional response. We show that if the model is extinct (resp. permanent) for a parameter, it is still extinct (resp. permanent) in a neighbourhood of this parameter. In the case of extinction, the Lyapunov exponent of predator quantity is negative and the prey quantity converges almost to the saturated situation, where the predator is absent at an exponential rate. Under the condition of permanence, the unique stationary distribution converges weakly to the degenerate measure concentrated at the unique limit cycle or at the globally asymptotic equilibrium when the diffusion term tends to 0.

Many reaction networks arising in applications are multistationary, that is, they have the capacity for more than one steady state, while some networks exhibit absolute concentration robustness (ACR), which means that some species concentration is the same at all steady states. Both multistationarity and ACR are significant in biological settings, but only recently has attention focused on the possibility for these properties to coexist. Our main result states that such coexistence in at-most-bimolecular networks (which encompass most networks arising in biology) requires at least three species, five complexes and three reactions. We prove additional bounds on the number of reactions for general networks based on the number of linear conservation laws. Finally, we prove that, outside of a few exceptional cases, ACR is equivalent to non-multistationarity for bimolecular networks that are small (more precisely, one-dimensional or up to two species). Our proofs involve analyses of systems of sparse polynomials, and we also use classical results from chemical reaction network theory.

For a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus–Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form $x^{\prime}=f_1(x,y)$, $y^{\prime}=f_2(x,y)$, where f1 has degree one, f2 has degree $n\ge 1$ and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus–Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form $x^{\prime}=y$, $y^{\prime}=g_1(x) +y g_2(x)$ with g1 and g2 polynomials of degrees n and m, respectively.

In this paper, we study existence of rotating periodic solutions for p-Laplacian differential systems. We first build a new continuation theorem by topological degree, and then obtain the existence of rotating periodic solutions for two kinds of p-Laplacian differential systems via this continuation theorem, extend some existing relevant results.

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem.

We work with polynomial three-dimensional rigid differential systems. Using the Lyapunov constants, we obtain lower bounds for the cyclicity of the known rigid centres on their centre manifolds. Moreover, we obtain an example of a quadratic rigid centre from which is possible to bifurcate 13 limit cycles, which is a new lower bound for three-dimensional quadratic systems.