Stability and Neimark-Sacker bifurcation analysis for a discrete single genetic negative feedback autoregulatory system with delay

In this paper, we deal with a discrete single genetic negative feedback autoregulatory system with delay by using Euler method. Choosing the delay $\tau $ as the bifurcation parameter and analyzing the associated characteristic equation corresponding to the unique positive fixed point, it is found that the stability of the positive equilibrium and Neimark-Sacker bifurcation may occur when $\tau $ crosses some critical values. Then the explicit formula which determines the stability, direction, and other properties of bifurcating periodic solution is derived by using the center manifold theorem and normal form theory. Finally, in order to illustrate our theoretical analysis, numerical simulations are also included in the end.     

Bifurcation and chaotic behavior of a discrete-time Ricardo–Malthus model

The dynamics of a discrete-time Ricardo–Malthus model obtained by numerical discretization is investigated, where the step size δ is regarded as a bifurcation parameter. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of $R^{2}_{+}$ by using the theory of center manifold and normal form. Numerical simulations are presented not only to illustrate our theoretical results, but also to exhibit the system’s complex dynamical behavior, such as the cascade of period-doubling bifurcation in orbits of period 2, 4, 8 16, period-11, 22, 28 orbits, quasiperiodic orbits, and the chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.     

Spatiotemporal complexity of a three-species ratio-dependent food chain model

In this paper, we investigate the complex dynamics of a ratio-dependent spatially extended food chain model. Through a detailed analytical study of the reaction–diffusion model, we obtain some conditions for global stability. On the basis of bifurcation analysis, we present the evolutionary process of pattern formation near the coexistence equilibrium point $(N^*,P^*,Z^*)$ via numerical simulation. And the sequence cold spots $\rightarrow $ stripe–spots mixtures $\rightarrow $ stripes $\rightarrow $ hot stripe–spots mixtures $\rightarrow $ hot spots $\rightarrow $ chaotic wave patterns controlled by parameters $a_1$ or $c_1$ in the model are presented. These results indicate that the reaction–diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics.     

On the effect of non-smooth Coulomb damping on flutter-type self-excitation in a non-gyroscopic circulatory 2-DoF-system

This article deals with self-excited vibrations, attractivity of stationary solutions, and the corresponding bifurcation behavior of two-dimensional differential inclusions of the type $\mathbf{M}\mathbf{q}'' + \mathbf{D}\mathbf{q}' + (\mathbf{K} + \bar{\mu}\mathbf{N})\mathbf{q} \in-\mathbf{R}\operatorname{Sign}(\mathbf{q}')$ . For the smooth case R=0, the equilibrium may become unstable due to non-conservative positional forces stemming from the circulatory matrix N. This type of instability is usually referred to as flutter instability and the loss of stability is related to a Hopf bifurcation of the steady state, which occurs for a critical parameter $\bar{\mu}= \bar{\mu}_{\mathrm{crit}}$ . For R≠0, the steady state is a set of equilibria, which turns out to be attractive for all values of the bifurcation parameter $\bar{\mu}$ . Depending on $\bar{\mu}$ , the basin of attraction of the equilibrium set can be infinite or finite. The transition from an infinite to a finite basin of attraction occurs at the stability threshold $\bar{\mu}_{\mathrm{crit}}$ of the underlying smooth problem. For the finite basin of attraction, its size is proportional to the Coulomb friction and inverse-proportional to $(\bar{\mu}- \bar{\mu}_{\mathrm{crit}})$ . By adding Coulomb damping the notion of steady state stability for the smooth problem is replaced by the question whether the basin of attraction of the steady state is infinite or finite. Simultaneously, the local Hopf-bifurcation is replaced by a global bifurcation. This implies that in the presence of Coulomb damping the occurrence of self-excited vibrations can only be investigated with regard to the perturbation level.     

Boundary Value Problems of Robin Type for the Brinkman and Darcy–Forchheimer–Brinkman Systems in Lipschitz Domains

The purpose of this paper is to study boundary value problems of Robin type for the Brinkman system and a semilinear elliptic system, called the Darcy–Forchheimer–Brinkman system, on Lipschitz domains in Euclidean setting. In the first part of the paper, we exploit a layer potential analysis and a fixed point theorem to show the existence and uniqueness of the solution to the nonlinear Robin problem for the Darcy–Forchheimer–Brinkman system on a bounded Lipschitz domain in \({\mathbb{R}^n}\) \({(n \in \{2,3\})}\) with small data in L ²-based Sobolev spaces. In the second part, we show an existence result for the mixed Dirichlet–Robin problem for the same semilinear Darcy–Forchheimer-Brinkman system on a bounded creased Lipschitz domain in \({\mathbb{R}^3}\) with small L ²-boundary data. We also study mixed Dirichlet–Robin problems and boundary value problems of mixed Dirichlet–Robin and transmission type for Brinkman systems on bounded creased Lipschitz domains in \({\mathbb{R}^n}\) (n ≥ 3). Finally, we show the well-posedness of the Navier problem for the Brinkman system with boundary data in some L ²-based Sobolev spaces on a bounded Lipschitz domain in \({\mathbb{R}^3}\) .     

Periodic solution and heteroclinic bifurcation in a predator–prey system with Allee effect and impulsive harvesting

In this article, we investigate a prey– predator model with Allee effect and state-dependent impulsive harvesting. We obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system (1.2) by means of the geometry theory of semicontinuous dynamic system and the method of successor function. We also obtain that system (1.2) exhibits the phenomenon of heteroclinic bifurcation about parameter $\alpha $ . The methods used in this article are novel and prove the existence of order-1 periodic solution and heteroclinic bifurcation.     

Local Dynamics at Focal Points Coupled with Elliptic Directions

We investigate local dynamics of a 4-dimensional system with small and slowly varying time periodic forcing. By assuming the unperturbed system is autonomous and has a fixed point with eigenvalues $(0,0,i,-i)$ ( 0 , 0 , i , ? i ) , we study homoclinic, subharmonic solutions and Hopf bifurcation in a $O(\epsilon )$ O ( ? ) neighborhood of the fixed point, where $\epsilon $ ? is the perturbation parameter.     

Switching stabilization and { H}_{\varvec{\infty }} performance of a class of discrete switched LPV system with unstable subsystems

Stability and $H_\infty $ performance are analyzed in this paper for a class of discrete switched linear parameter-varying (LPV) systems in which all subsystems’ state-space matrices are parametrically affine, and any subsystem is not stable for parameters varying in a convex set. A switching law is designed to stabilize and satisfy the $H_\infty $ performance of the switched LPV system. By means of the multiple Lyapunov functions method, linear matrix inequality (LMI) conditions for the existence of parameter-dependent Lyapunov functions are proposed. An example shows the effectiveness of the proposed methods.     

Pore-Scale Simulations of Gas Displacing Liquid in a Homogeneous Pore Network Using the Lattice Boltzmann Method

A lattice Boltzmann high-density-ratio model, which uses diffuse interface theory to describe the interfacial dynamics and was proposed originally by Lee and Liu (J Comput Phys 229:8045–8063, 2010), is extended to simulate immiscible multiphase flows in porous media. A wetting boundary treatment is proposed for concave and convex corners. The capability and accuracy of this model is first validated by simulations of equilibrium contact angle, injection of a non-wetting gas into two parallel capillary tubes, and dynamic capillary intrusion. The model is then used to simulate gas displacement of liquid in a homogenous two-dimensional pore network consisting of uniformly spaced square obstructions. The influence of capillary number (Ca), viscosity ratio ( $M$ M ), surface wettability, and Bond number (Bo) is studied systematically. In the drainage displacement, we have identified three different regimes, namely stable displacement, capillary fingering, and viscous fingering, all of which are strongly dependent upon the capillary number, viscosity ratio, and Bond number. Gas saturation generally increases with an increase in capillary number at breakthrough, whereas a slight decrease occurs when Ca is increased from $8.66\times 10^{-4}$ 8.66 × 10 - 4 to $4.33\times 10^{-3}$ 4.33 × 10 - 3 , which is associated with the viscous instability at high Ca. Increasing the viscosity ratio can enhance stability during displacement, leading to an increase in gas saturation. In the two-dimensional phase diagram, our results show that the viscous fingering regime occupies a zone markedly different from those obtained in previous numerical and experimental studies. When the surface wettability is taken into account, the residual liquid blob decreases in size with the affinity of the displacing gas to the solid surface. Increasing Bo can increase the gas saturation, and stable displacement is observed for $Bo>1$ B o > 1 because the applied gravity has a stabilizing influence on the drainage process.     

Diversity evolution and parameter matching of periodic-impact motions of a periodically forced system with a clearance

A two-degree-of-freedom periodically forced system with a clearance is considered. The correlative relationship and matching law between dynamics and system parameters are analyzed by the co-simulation analysis of multi-parameter and multi-performance. Key parameters of the system, such as the exciting frequency, clearance value and constraint stiffness, are emphasized to analyze the influence of the main factors on its soft impact characteristics and reveal diversity, evolution and distribution regions of periodic-impact motions. The quantity of the fundamental group of \(p/1\) impact motions, which have the excitation period and differ by the number \(p\) of impacts, is basically determined by the constraint stiffness. A series of grazing bifurcations occur with decreasing the exciting frequency so that the number \(p\) of impacts of the fundamental group of motions correspondingly increases one by one. As the constraint stiffness is very large, the impact number \(p\) of the fundamental group of motions becomes also big enough in low exciting frequency range. Consequently, the system possibly exhibits chattering-impact characteristics and the relative impact velocities successively attenuate in an excitation period. There exist a series of singular points between any two adjacent ones of the fundamental group of motions as the damping ratio or the damping constant of the viscous dashpot connecting two masses is very small, i.e., real-grazing and bare-grazing bifurcation boundaries of one of them, saddle-node and period-doubling bifurcation boundaries of the other mutually cross themselves at the point of intersection and create two types of transition regions: hysteresis and tongue-shaped regions. A series of zones of regular and complex subharmonic impact motions are found to dominate in the tongue-shaped regions. The dimensionless parameters have been designed technically, under which large mass ratio or small supporting stiffness ratio leads to diversity and complexity of periodic-impact motions of the system. Based on the sampling ranges of parameters, the influence of system parameters on relative impact velocities, existence regions and correlative distribution of different types of periodic-impact motions of the system is emphatically studied.     

Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain ${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$ , and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$ .     

The Bifurcation Study of 1:2 Resonance in a Delayed System of Two Coupled Neurons

In this paper, we consider a delayed system of differential equations modeling two neurons: one is excitatory, the other is inhibitory. We study the stability and bifurcations of the trivial equilibrium. Using center manifold theory for delay differential equations, we develop the universal unfolding of the system when the trivial equilibrium point has a double zero eigenvalue. In particular, we show a universal unfolding may be obtained by perturbing any two of the parameters in the system. Our study shows that the dynamics on the center manifold are characterized by a planar system whose vector field has the property of 1:2 resonance, also frequently referred as the Bogdanov–Takens bifurcation with $Z_2$ symmetry. We show that the unfolding of the singularity exhibits Hopf bifurcation, pitchfork bifurcation, homoclinic bifurcation, and fold bifurcation of limit cycles. The symmetry gives rise to a “figure-eight” homoclinic orbit.     

Combined Effect of Stress, Pore Pressure and Temperature on Methane Permeability in Anthracite Coal: An Experimental Study

The effects of particle-size distribution on the longitudinal dispersion coefficient ( $D_{\mathrm{L}})$ D L ) in packed beds of spherical particles are studied by simulating a tracer column experiment. The packed-bed models consist of uniform and different-sized spherical particles with a ratio of maximum to minimum particle diameter in the range of 1–4. The modified version of Euclidian Voronoi diagrams is used to discretize the system of particles into cells that each contains one sphere. The local flow distribution is derived with the use of Laurent series. The flow pattern at low particle Reynolds number is then obtained by minimization of dissipation rate of energy for the dual stream function. The value of $D_{\mathrm{L}}$ D L is obtained by comparing the effluent curve from large discrete systems of spherical particles to the solution of the one-dimensional advection–dispersion equation. Main results are that at Peclet numbers above 1, increasing the width of the particle-size distribution increases the values of $D_{\mathrm{L}}$ D L in the packed bed. At Peclet numbers below 1, increasing the width of the particle-size distribution slightly lowers $D_{\mathrm{L}}$ D L .     

Approximation of the p-Stokes Equations with Equal-Order Finite Elements

Non-Newtonian fluid motions are often modeled by the p-Stokes equations with power-law exponent ${p\in(1,\infty)}$ . In the present paper we study the discretization of the p-Stokes equations with equal-order finite elements. We propose a stabilization scheme for the pressure-gradient based on local projections. For ${p\in(1,\infty)}$ the well-posedness of the discrete problems is shown and a priori error estimates are proven. For ${p\in(1,2]}$ the derived a priori error estimates provide optimal rates of convergence with respect to the supposed regularity of the solution. The achieved results are illustrated by numerical experiments.     

Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings

Nonlinear damping suspension is a promising method to be used in a rotor-bearing system for vibration isolation between the bearing and environment. However, the nonlinearity of the suspension may influence the stability of the rotor-bearing system. In this paper, the motions of a flexible rotor in short journal bearings with nonlinear damping suspension are studied. A computational method is used to solve the equations of motion, and the bifurcation diagrams, orbits, Poincaré maps, and amplitude spectra are used to display the motions. The results show that the effect of the nonlinear damping suspension on the motions of the rotor-bearing system depends on the speed of rotor: (a) For low speeds, the rotor- bearing system presents the same motion pattern under the nonlinear damping ( \(p=0.5, 2, 3\) ) suspension as for the linear damping ( \(p=1\) ) suspension; (b) For high speeds, the effect of nonlinear damping depends on a combination of the damping exponent and damping coefficient. The square root damping model ( \(p=0.5\) ) shows a wider stable speed range than the linear damping for large damping coefficients. The quadratic damping ( \(p=2\) ) shows similar results to linear damping with some special damping coefficients. The cubic damping ( \(p=3\) ) shows more stable response than the linear damping in general.     

The Rot-Div System in Exterior Domains

The goal of this paper is to reconsider the classical elliptic system rot v =  f, div v =  g in simply connected domains with bounded connected boundaries (bounded and exterior sets). The main result shows solvability of the problem in the maximal regularity regime in the L _p -framework taking into account the optimal/minimal requirements on the smoothness of the boundary. A generalization for the Besov spaces is studied, too, for \({{\bf f} \in \dot B^s_{p,q}(\Omega)}\) for \({-1+\frac 1p < s < \frac 1p}\) . As a limit case we prove the result for \({{\bf f} \in \dot B^0_{3,1}(\Omega)}\) , provided the boundary is merely in \({B^{2-1/3}_{3,1}}\) . The dimension three is distinguished due to the physical interpretation of the system. In other words we revised and extended the classical results of Friedrichs (Commun Pure Appl Math 8;551–590, 1955) and Solonnikov (Zap Nauch Sem LOMI 21:112–158, 1971).     

Geometric Approach to Nonvariational Singular Elliptic Equations

In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms, as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter (γ - 1), for 0 < γ < 1, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases.We establish existence as well as sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and we obtain fine geometric-measure properties of the free boundary ${\mathfrak{F} = \partial{u > 0}}$ . In particular, we show sharp Hausdorff estimates which imply local finiteness of the perimeter of the region {u > 0} and the ${\mathcal{H}^{n-1}}$ almost-everywhere weak differentiability property of ${\mathfrak{F}}$ .     

Output feedback \varvec{\mathcal {H}}_{{\varvec{\infty }}} control of stochastic nonlinear time-delay systems with state and disturbance-dependent noises

This paper is concerned with the output feedback \(\mathcal {H}_\infty \) control problem for a class of stochastic nonlinear systems with time-varying state delays; the system dynamics is governed by the stochastic time-delay It \(\hat{o}\) -type differential equation with state and disturbance contaminated by white noises. The design of the output feedback \(\mathcal {H}_\infty \) control is based on the stochastic dissipative theory. By establishing the stochastic dissipation of the closed-loop system, the delay-dependent and delay-independent approaches are proposed for designing the output feedback \(\mathcal {H}_\infty \) controller. It is shown that the output feedback \(\mathcal {H}_\infty \) control problem for the stochastic nonlinear time-delay systems can be solved by two delay-involved Hamilton–Jacobi inequalities. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating Obstacle

We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the C _b -regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L ³ spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L ³ spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.     

Dynamics of stochastic Lorenz–Stenflo system

This paper discusses the Lorenz–Stenflo system under the influence of L \(\acute{\hbox {e}}\) vy noise. We find conditions under which the solution to stochastic Lorenz–Stenflo system is exponentially stable. We then investigate the estimation of the global attractive set and stochastic bifurcation behavior of the stochastic Lorenz–Stenflo system. Results show that the jump noise can make the solution stable, the bounds and bifurcation to undergo change under some conditions. Numerical results show the effectiveness and advantage of our methods.