Rinko Miyazaki

Boundedness and almost periodicity in dynamical systems

We consider spectral criteria for the existence of bounded solutions to difference equations of the form with specific spectral properties. The results will be then applied to find periodic, almost periodic solutions to and with (in general, unbounded) T—periodic A(·)T—periodic F(t), f(·). This provides a new and simple approach to find spectral criteria for the existence of periodic, almost periodic solutions to differential equations (*), (**)

Dynamics in a tumor immune system with time delays

In this paper, we study the dynamical behavior of a tumor-immune system (T-IS) interaction model with two discrete delays, namely the immune activation delay for effector cells (ECs) and activation delay for helper T cells (HTCs). By analyzing the characteristic equations, we establish the stability of two equilibria (tumor-free equilibrium and immune-control equilibrium) and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Our results exhibit that both delays do not affect the stability of tumor-free equilibrium. However, they are able to destabilize the immune-control equilibrium and cause periodic solutions. We numerically illustrate how these two delays can change the stability region of the immune-control equilibrium and display the different impacts to the control of tumors. The numerical simulation results show that the immune activation delay for HTCs can induce heteroclinic cycles to connect the tumor-free equilibrium and immune-control equilibrium. Furthermore, we observe that the immune activation delay for HTCs can even stabilize the unstable immune-control equilibrium.

Delayed Feedback Control by Commutative Gain Matrices

The delayed feedback control (DFC) is a control method for stabilizing unstable periodic orbits in nonlinear autonomous differential equations. We give an important relationship between the characteristic multipliers of the linear variational equation around an unstable periodic solution of the equation and those of its delayed feedback equation. The key of our proof is a result about the spectrum of a matrix which is a difference of commutative matrices. The relationship, moreover, allows us to design control gains of the DFC such that the unstable periodic solution is stabilized. In other words, the validity of the DFC is proved mathematically. As an application for the Rossler equation, we determine the best range of k such that the unstable periodic orbit is stabilized by taking a feedback gain K = kE.

Asymptotic constancy for a linear differential system with multiple delays

In this letter we consider a linear differential system with multiple delays which has nonisolated equilibria. In order to study the asymptotic behavior of linear delay differential equations, characteristic equations are generally used. But it is hard to establish the properties of zeros of the characteristic equations, especially if there are multiple time delays. So we use the invariance principle combined with two functionals to show whether any solutions converge. One of the functionals plays the role of a Lyapunov functional, and the other is a conserved quantity. Furthermore we give explicit expressions for the limits of the solutions by using the conserved quantity.