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pp. 173-184 | DOI: 10.5890/DNC.2021.06.001

Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily

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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.

- On Caputo-Hadamard Type Fractional Differential Equations with Nonlocal Discrete
Boundary Conditions

pp. 185-194 | DOI: 10.5890/DNC.2021.06.002

Murugesan Manigandan,Muthaiah Subramanian, Palanisamy Duraisamy, Thangaraj Nandha Gopal

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This paper studies a new class of boundary value problems of Caputo-
Hadamard fractional differential equations of orderr ∈ (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.

- Decay in Systems with Neutral Short-Wavelength Stability: The Presence of a Zero
Mode

pp. 195-205 | DOI: 10.5890/DNC.2021.06.003

Adham A. Ali, Fatima Z. Ahmed

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A characteristic feature of seismic waves is the presence of dominant frequency/
wave number in the spectrum. A well-knownmodel for such waves
is the Nikolaevskiy equation, which is also applicable to some reactiondiffusion
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.

- Some Existence and Stability Results of Hilfer-Hadmard Fractional Implicit
Differential Equation in a Weighted Space

pp. 207-225 | DOI: 10.5890/DNC.2021.06.004

Laxman A. Palve, Mohammed S. Abdo, Satish K. Panchal

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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.

- Approximate Controllability for Time-dependent Impulsive Neutral Stochastic Partial
Differential Equations with Fractional Brownian Motion and Poisson Jumps

pp. 227-235 | DOI: 10.5890/DNC.2021.06.005

K. Ramkumar, K. Ravikumar, A. Anguraj

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In this paper, we investigate the approximate controllability for timedependent
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.

- Bifurcations and Dynamics in Modified Two Population Neuronal Network Models

pp. 237-257 | DOI: 10.5890/DNC.2021.06.006

S. Roy Choudhury, Gizem S. Oztepe

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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.

- Synchronization of T-S Fuzzy Sampled-data Controller for H-R Neuron Model With
Delay using a New Looped-Functional

pp. 259-273 | DOI: 10.5890/DNC.2021.06.007

P. Nirvin1, R. Rakkiyappan

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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.

- $E_\alpha$-Ulam-Hyers Stability Result for $\psi$-Hilfer Nonlocal Fractional Differential Equation

pp. 275-288 | DOI: 10.5890/DNC.2021.06.008

Mohammed A. Almalahi, Satish K. Panchal

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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.

- A Parameter Study on Periodic Motions in a Discontinuous Dynamical System with Two Circular Boundaries

pp. 289-309 | DOI: 10.5890/DNC.2021.06.009

Siyu Guo, Albert C. J. Luo†

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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.

- Controllability of Neutral Impulsive Stochastic Integrodifferential Equations Driven
by a Rosenblatt Process and Unbounded Delay

pp. 311-321 | DOI: 10.5890/DNC.2021.06.010

K. Ramkumar, K. Ravikumar

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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.

- Third Hankel Determinant for Certain Class of Bazilevi´c Functions Associated with
Linear Differential Operator

pp. 323-331 | DOI: 10.5890/DNC.2021.06.011

Saba N. Al-khafaji, Ahmed Hadi Hussain, Sameer Annon Abbas

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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 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.

- Approximate Controllability of Second Order Neutral Stochastic Integro Differential
Equations with Impulses Driven By Fractional Brownian Motion

pp. 333-345 | DOI: 10.5890/DNC.2021.06.012

S. Madhuri, Deekshitulu G.V.S.R.

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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.