163 research outputs found
Sampling expansions associated with quaternion difference equations
Starting with a quaternion difference equation with boundary conditions, a
parameterized sequence which is complete in finite dimensional quaternion
Hilbert space is derived. By employing the parameterized sequence as the kernel
of discrete transform, we form a quaternion function space whose elements have
sampling expansions. Moreover, through formulating boundary-value problems, we
make a connection between a class of tridiagonal quaternion matrices and
polynomials with quaternion coefficients. We show that for a tridiagonal
symmetric quaternion matrix, one can always associate a quaternion
characteristic polynomial whose roots are eigenvalues of the matrix. Several
examples are given to illustrate the results
Turing instability in a diffusive predator-prey model with multiple Allee effect and herd behavior
Diffusion-driven instability and bifurcation analysis are studied in a
predator-prey model with herd behavior and quadratic mortality by incorporating
multiple Allee effect into prey species. The existence and stability of the
equilibria of the system are studied. And bifurcation behaviors of the system
without diffusion are shown. The sufficient and necessary conditions for Turing
instability occurring are obtained. And the stability and the direction of Hopf
and steady state bifurcations are explored by using the normal form method.
Furthermore, some numerical simulations are presented to support our
theoretical analysis. We found that too large diffusion rate of prey prevents
Turing instability from emerging. Finally, we summarize our findings in the
conclusion
Second-order differential equation with indefinite and repulsive singularities
This paper concerns a second-order differential equation with indefinite and
repulsive singularities. It is the first time to study differential equation
containing both indefinite and repulsive singularities simultaneously. A set of
sufficient conditions are obtained for the existence of positive periodic
solutions. The theoretical underpinnings of this paper are the positivity of
Green's function and fixed point theorem in cones. Our results improve and
extend the results in previous literatures. Finally, three examples and their
numerical simulations (phase diagrams and time diagrams of periodic solutions)
are given to show the effectiveness of our conclusions
Steady periodic irrotational blood flow with time-dependent body force
In this paper, we study steady 2D periodic blood flow propagating along blood
vessels with free boundary conditions. In particular, we focus on the dynamic
behavior of irrotational flows with time dependent body force. An equivalent
formulation with fix boundary is developed by utilizing flow force functions.
The local bifurcation result is obtained by Crandall-Rabinowitz theorem. The
existence of a local -curve of small-amplitude solution is strictly
proved
Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization
Floquet multipliers (characteristic multipliers) play significant role in the
stability of the periodic equations. Based on the iterative method, we provide
a unified algorithm to compute the Floquet multipliers (characteristic
multipliers) and determine the stability of the periodic linear differential
equations on time scales unifying discrete, continuous, and hybrid dynamics.
Our approach is based on calculating the value of A and B (see Theorem 3.1),
which are the sum and product of all Floquet multipliers (characteristic
multipliers) of the system, respectively. We obtain an explicit expression of A
(see Theorem 4.1) by the method of variation and approximation theory and an
explicit expression of B by Liouville's formula. Furthermore, a computer
program is designed to realize our algorithm. Specifically, you can determine
the stability of a second order periodic linear system, whether they are
discrete, continuous or hybrid, as long as you enter the program codes
associated with the parameters of the equation. In fact, few literatures have
dealt with the algorithm to compute the Floquet multipliers, not mention to
design the program for its computer realization. Our algorithm gives the
explicit expressions of all Floquet multipliers and our computer program is
based on the approximations of these explicit expressions. In particular, on an
arbitrary discrete periodic time scale, we can do a finite number of
calculations to get the explicit value of Floquet multipliers (see Theorem
4.2). Therefore, for any discrete periodic system, we can accurately determine
the stability of the system even without computer! Finally, in Section 6,
several examples are presented to illustrate the effectiveness of our
algorithm
Topological Conjugacy between Two Kinds of Nonlinear Differential Equations via Generalized Exponential Dichotomy
Based on the notion of generalized exponential dichotomy, this paper considers the topological decoupling problem between two kinds of nonlinear differential equations. The topological equivalent function is given
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