Vol. 10, No. 2 (2021): Regular Issue

When deterministically extended structures are taken into consideration, it is admissible that fractals are dense both in the nature and in the dynamics. In particular, this is true because fractal structures are closely related to chaos. To make advances in the direction, first of all, one should consider fractals as states of dynamics. If one realizes this approach, fractals will be proved to be dense in the universe, since modeling the real world is based on differential equations and their developments.

This paper studies a new class of boundary value problems of Caputo-Hadamard fractional differential equations of order $\varrho\in (2, 3]$ supplemented with nonlocal multi-point (discrete) boundary conditions. Existence and uniqueness results for the given problem have obtained by applying standard fixed-point theorems. Finally, two examples are given to illustrate the validity of our main results.

A characteristic feature of seismic waves is the presence of dominant frequency/wave number in the spectrum. A well-known model for such waves is the Nikolaevskiy equation, which is also applicable to some reaction-diffusion systems and Rayleigh-Benard convection. For the critical case when there is one neutral mode, we describe the dynamics of the Fourier modes (elastic waves) under the Nikolaevskiy equation using the centre manifold technique. After quickly attracted to the surface (manifold), the modes then evolve slow algebraic decay. An inverse square-root law for the decaying regime is obtained. The result is confirmed by direct computations of the dynamical system for the modes.

This paper studies a nonlinear fractional implicit differential equation (FIDE) with boundary conditions involving a Hilfer-Hadamard type fractional derivative. We establish the equivalence between the Cauchy-type problem (FIDE) and its mixed type integral equation through a variety of tools of some properties of fractional calculus and weighted spaces of continuous functions. The existence and uniqueness of solutions are obtained. Further, the Ulam-Hyers and Ulam-Hyers-Rassias stability are discussed. The arguments in the analysis rely on Schaefer fixed point theorem, Banach contraction principle and generalized Gronwall inequality. At the end, an illustrative example will be introduced to justify our results.

In this paper, we investigate the approximate controllability for time-dependent impulsive neutral stochastic partial differential equations with fractional Brownian motion and Poisson jumps in Hilbert space. The results are obtained by using semigroup theory, stochastic analysis, and fixed point approach, we derive a new set of sufficient conditions for the approximate controllability of nonlinear stochastic system under the assumption that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate our results.

A canonical modified two population neuronal network model of Laplace convolution type is considered via the 'linear chain trick'. Linear stability analysis of this system and conditions for Hopf bifurcation initiating spatiotemporal oscillations are investigated, including deriving the normal form at bifurcation, and deducing the stability of the resulting limit cycle attractor. For more steeply negative firing-rate functions, the Hopf bifurcations occur at larger values of both the delay and the inhibitory time constant. Other bifurcations such as double Hopf or generalized Hopf modes occurring from the homogeneous background state are also shown to be impossible for our model. In this first model, the Hopf-generated limit cycles turn out to be remarkably stable under very large variations of all four system parameters beyond the Hopf bifurcation point, and do not undergo further symmetry breaking, cyclic-fold, flip, transcritical or Neimark-Sacker bifurcations. Numerical simulations reveal strong distortion of the limit cycle shapes in phase space as the parameters are pushed far into the post-Hopf regime, and also reveal other features, such as the increase of the oscillation amplitudes of the physical variables on the limit cycle attractor, as well as decrease of their time periods, as both the delay and the inhibitory time constant are increased. The final section considers alternative Fourier convolution models with general functional forms for the synaptic connectivity functions. In particular, we develop an approach to derive the large variable or asymptotic behaviors in both space and time for arbitrary functional forms of the connectivity functions.

This paper investigates the synchronization of Takagi-Sugeno (T-S) fuzzy sampled-data-controller for Hindmarsh-Rose (H-R) neuron model with constant {communication time delay} in the from the master-slave framework. The utilization of the state information of $e(t_k),~e(t),~e(t_{k+1}),~e(t_k-\lambda),~e(t-\lambda),~e(t_{k+1}-\lambda)$, is done adequately {be} the information of a novel looped-functional in the construction of a Lyapunov functional (LF). To establish that the slave system is synchronized with the master system, some satisfactory conditions with less conservativeness are derived by using the above mentioned functional and utilizing wirtingers inequality, jensen's inequalities, free matrix-based integral inequality method. The linear matrix inequality (LMI) techniques the fuzzy sampled-data control can be designed. Finally, a numerical example is given to illustrate the effectiveness of our theoretical results.

In this paper we study the existence and uniqueness results of }$\psi ${ -Hilfer nonlocal fractional differential equation with constant coefficient by using some properties of Mittag-Leffler function and fixed point theorems such as Banach and Schaefer's fixed point theorems. The generalized Gronwall inequality lemma is used in analyze E}$_{\alpha }${ -Ulam-Hyers stability. Finally, an example is provided to illustrate the obtained results.

In this paper, periodic motions in a discontinuous dynamical system with two circular boundaries are studied analytically by generic mappings. A bifurcation tree of stable and unstable periodic motions varying with excitation frequency is predicted analytically. On the bifurcation tree, there are three main bifurcations: the grazing bifurcation for the motions switching, the period-doubling bifurcations for period-doubled periodic motion, and saddle-node bifurcations for onset and vanishing of periodic motions. Periodic motions are numerically illustrated, and the $G$-functions are presented for illustrations of the analytical conditions of motions switchability, such as, the passable motion and grazing motion at the boundaries, and the formation and vanishing of sliding motions on the discontinuous boundaries. In this study discussed are the parameter effects on periodic motions in discontinuous dynamical systems. Such discussion is very helpful for one to design a discontinuous system for specific motions under specific system parameters.

In this manuscript, we establish the controllability of neutral impulsive stochastic integrodifferential equations driven by a Rosenblatt process with infinite delay in separable Hilbert space. The controllability results is obtained by using fixed-point technique and via resolvent operator.

The main object of this paper is to introduce a new class of Bazilevi\'{c} functions $\Omega^n_\alpha(m, \delta)$ in the open unit disk $\mathbb{D}$ associated with linear differential operator. In addition to, we obtained the coefficient estimates as well as best possible upper bound to the third Hankel determinant for the functions belong to this class.

In this paper we introduce a class of second order neutral stochastic integro differential equations with impulses that are governed by fractional Brownian motion in Hilbert space. First, we establish the existence of mild solution using Banach fixed point theorem. Further approximate controllability for this system is formulated by assuming that the corresponding linear system is approximately controllable. The results are illustrated with example.