On almost periodic solutions of chemotaxis-fluid systems

In this article, we investigate the existence, uniqueness, and exponential stability of periodic mild solutions to the chemotaxis–Navier–Stokes equations on the real hyperbolic space H d ( R ) , with d ≥ 2 . First, we establish the well-posedness of mild solutions for the corresponding linear systems by employing dispersive and smoothing estimates for the scalar and vectorial heat semigroups. Then, by combining a Messera-type principle with the ergodic method, we derive the existence of periodic mild solutions to the linear systems. Utilizing these linear results and fixed-point arguments, we further prove the well-posedness of periodic mild solutions for the chemotaxis–Navier–Stokes system. Finally, the exponential stability of such solutions is obtained via the Gronwall inequality.

Periodic solutions for Boussinesq systems in weak-Morrey spaces

In this paper, we investigate the well-posedness and polynomial stability of almost periodic mild solutions for Boussinesq systems on the whole space R n (for dimension n ⩾ 3 ) and whole line time-axis R t in framework of weak-Morrey spaces. By using linear and bilinear estimates on Morrey–Lorentz spaces, we prove the existence of bounded mild solutions for the corresponding linear systems on the whole line time-axis R t . Then we obtain the well-posedness of almost periodic solutions to the linear systems. By using fixed point arguments, we establish the well-posedness and polynomial stability of such solutions for Boussinesq systems.

In this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space and real hyperbolic space . We work in framework of critical spaces such as on weak‐Lorentz space to obtain the results for the Keller–Segel system on and on for to obtain those on . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in and the one in .

We investigate the global existence and exponential decay of mild solutions for the Boussinesq systems in L p -phase spaces on the framework of a real hyperbolic manifold H d ( R ) , where d ⩾ 2 and 1 < p ⩽ d . We consider a couple of Ebin–Marsden’s Laplace and Laplace–Beltrami operators associated with the corresponding linear system which provides a vectorial matrix semigroup. First, we show the existence and the uniqueness of the bounded mild solution for the linear system by using dispersive and smoothing estimates of the vectorial matrix semigroup. Next, using the fixed point arguments, we can pass from the linear system to the semilinear system to establish the existence of the bounded mild solutions. By using Gronwall’s inequality, we establish the exponential stability of such solutions. Finally, we give an application of stability to the existence of periodic mild solutions for the Boussinesq systems.

In this work, we study the existence, uniqueness and exponential stability of periodic mild solution of the Boussinesq system on the real hyperbolic manifold H d ( R ) ( d ⩾ 2 ). We consider a couple of Ebin-Marsden's Laplace and Laplace–Beltrami operators associated with the corresponding linear system. Our method is based on the dispersive and smoothing estimates of semigroups generated by Ebin-Marsden's Laplace and Laplace–Beltrami operators. First, we prove the existence and uniqueness of the bounded periodic mild solution for the linear system. Next, using the fixed point arguments, we pass from the linear system framework to the semilinear system framework to establish the existence of the periodic mild solution. Additionally, we prove the unconditional uniqueness of periodic mild solution for the Boussinesq system on hyperbolic manifolds. Finally, we establish the exponential stability of periodic mild solution.

We study the global existence, uniqueness and exponential stability of mild solutions to the Boussinesq systems equipped with generalized gravitational field on the non-compact Riemannian manifold (M,g) with dimension d ⩾ 2. The curvature tensors of manifold (M,g) satisfy some bounded and negative conditions. We consider a couple of Stokes (or vectorial heat) semigroup and scalar heat semigroup associated with the corresponding linear system which provides a vectorial matrix semigoup. By using dispersive and smoothing estimates of the vectorial matrix semigroup we establish the global well-posedness of mild solutions for linear systems in framework of Lp-spaces for p > d. Then, we use results of linear systems and fixed point arguments to establish the well-posedness of solutions for the semilinear systems. Moreover, we establish an exponential stability of such solutions by using Gronwall’s inequality. Besides, in the case of a manifold with dimension d ⩾ 3 and small Ricci curvature, we also establish the well-posedness and stability of mild solutions to the Boussinesq system in the framework of weak-Ld space, i.e., Ld,∞-space.

ABSTRACTIn this paper, we investigate the existence and stability of almost periodic solutions of impulsive fractional-order differential systems with uncertain parameters. The impulses are realised at fixed moments of time. For the first time, we determine the impact of the uncertainties on the qualitative behaviour of such systems. The main criteria for the existence of almost periodic solutions are proved by employing the fractional Lyapunov method. The global perfect robust uniform-asymptotic stability of such solutions is also considered. We apply our results to uncertain impulsive neural network systems of fractional order.

Fractional-order autonomous systems do not possess any non-constant periodic solutions, and to the best of our knowledge, there are no existing results regarding the existence of the periodic solution for fractional-order non-autonomous systems. The main objective of this work is to fill the above gap by studying the existence of a periodic solution of the Caputo-Fabrizio fractional-order system and also to find ways to stabilize a non-periodic solution. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo-Fabrizio system cannot admit a non-constant periodic solution. Under a similar assumption as the one for an integer-order differential system, and by using the properties of the Caputo-Fabrizio derivative, the existence of a periodic solution of a non-autonomous Caputo-Fabrizio fractional-order differential system is established. The main result is utilized in constructing and finding the periodic solution of the linear non-homogeneous Caputo-Fabrizio system. By using the result on linear systems, we derive a periodic solution of a fractional-order Gunn diode oscillator under a periodic input voltage, and observe that the diameter of the periodic orbit keeps reducing as the fractional-order continuously increases. In the end, by using the result on a linear non-homogeneous system, and by constructing a suitable linear feedback control, the solution of the linear non-homogeneous fractional-order system is stabilized to a periodic solution. An example is presented to support the obtained result. The main advantage of the proposed method over others is the simple considerations like the concept of equilibrium point and the utilization of the property of the Caputo-Fabrizio derivatives instead of other types of fractional derivatives.

Competition of two microbial species in a turbidostat

We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDEs of KdV type, including generalized KdV and Benjamin--Ono equations. In this investigation, we consider the spectral stability of such solutions that arise as small perturbations of an equilibrium state. A key feature of our analysis is the development of a nonlocal Floquet-like theory that is suitable to analyze the $L^2(\mathbb{R})$ spectrum of the associated linearized operators. Using spectral perturbation theory then, we derive a relationship between the power of the nonlinearity and the symbol of the fractional dispersive operator that determines the spectral stability and instability to arbitrary small localized perturbations.

The role of perturbation methods and bifurcation theory in predicting the stability and complicated dynamics of machining is discussed using a nonlinear single-degree-of-freedom model that accounts for the regenerative effect, linear structural damping, quadratic and cubic nonlinear stiffness of the machine tool, and linear, quadratic, and cubic regenerative terms. Using the width of cut w as a bifurcation parameter, we find, using linear theory, that disturbances decay with time and hence chatter does not occur if w < wc and disturbances grow exponentially with time and hence chatter occurs if w > wc. In other words, as w increases past wc, a Hopf bifurcation occurs leading to the birth of a limit cycle. Using the method of multiple scales, we obtained the normal form of the Hopf bifurcation by including the effects of the quadratic and cubic nonlinearities. This normal form indicates that the bifurcation is supercritical; that is, local disturbances decay for w < wc and result in small limit cycles (periodic motions) for w > wc. Using a six-term harmonic-balance solution, we generated a bifurcation diagram describing the variation of the amplitude of the fundamental harmonic with the width of cut. Using a combination of Floquet theory and Hill’s determinant, we ascertained the stability of the periodic solutions. There are two cyclic-fold bifurcations, resulting in large-amplitude periodic solutions, hysteresis, jumps, and subcritical instability. As the width of cut w increases, the periodic solutions undergo a secondary Hopf bifurcation, leading to a two-period quasiperiodic motion (a two-torus). The periodic and quasiperiodic solutions are verified using numerical simulation. As w increases further, the torus doubles. Then, the doubled torus breaks down, resulting in a chaotic motion. The different attractors are identified by using phase portraits, Poincare´ sections, and power spectra. The results indicate the importance of including the nonlinear stiffness terms.

In this paper, we study the existence, uniqueness and stability of the time periodic mild solutions to the incompressible Navier–Stokes equations on the non-compact manifolds with negative Ricci curvature tensor. In our strategy, we combine the dispersive and smoothing estimates for Stokes semigroups and Massera-type theorem to establish the existence and uniqueness of the time periodic mild solution to Stokes equation on Riemannian manifolds. Then using fixed point arguments, we can pass to semilinear equations to obtain the existence and uniqueness of the periodic solution to the imcompressible Navier–Stokes equations under the action of a periodic external force. The stability of the solution is also proved by using the cone inequality.

On the periodic Schrödinger–Boussinesq system

Dynamic analysis of a reverse-idler gear pair with concurrent clearances