Hopf Bifurcation Control in a FAST TCP and RED Model via Multiple Control Schemes

We focus on the Hopf bifurcation control problem of a FAST TCP model with RED gateway. The system gain parameter is chosen as the bifurcation parameter, and the stable region and stability condition of the congestion control model are given by use of the linear stability analysis. When the system gain passes through a critical value, the system loses the stability and Hopf bifurcation occurs. Considering the negative influence caused by Hopf bifurcation, we apply state feedback controller, hybrid controller, and time-delay feedback controller to postpone the onset of undesirable Hopf bifurcation. Numerical simulations show that the hybrid controller is the most sensitive method to delay the Hopf bifurcation with identical parameter conditions. However, nonlinear state feedback control and time-delay feedback control schemes have larger control parameter range in the Internet congestion control system with FAST TCP and RED gateway. Therefore, we can choose proper control method based on practical situation including unknown conditions or parameter requirements. This paper plays an important role in setting guiding system parameters for controlling the FAST TCP and RED model

Control bifurcations

A parametrized nonlinear differential equation can have multiple equilibria as the parameter is varied. A local bifurcation of a parametrized differential equation occurs at an equilibrium where there is a change in the topological character of the nearby solution curves. This typically happens because some eigenvalues of the parametrized linear approximating differential equation cross the imaginary axis and there is a change in stability of the equilibrium. The topological nature of the solutions is unchanged by smooth changes of state coordinates so these may be used to bring the differential equation into Poincare/spl acute/ normal form. From this normal form, the type of the bifurcation can be determined. For differential equations depending on a single parameter, the typical ways that the system can bifurcate are fully understood, e.g., the fold (or saddle node), the transcritical and the Hopf bifurcation. A nonlinear control system has multiple equilibria typically parametrized by the set value of the control. A control bifurcation of a nonlinear system typically occurs when its linear approximation loses stabilizability. The ways in which this can happen are understood through the appropriate normal forms. We present the quadratic and cubic normal forms of a scalar input nonlinear control system around an equilibrium point. These are the normal forms under quadratic and cubic change of state coordinates and invertible state feedback. The system need not be linearly controllable. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations

Normal Forms and Bifurcations of Control Systems

Research supported in part by AFOSR-49620-95-1-0409 and by NSF 9970998. To be presented at the IEEE CDC 2000, Sydney.We present the quadratic and cubic normal forms of a nonlinear control system around an equilibrium point. These are the normal forms under change of state coordinates and invertible state feedback. The system need not be linearly controllable. A control bifurcation of a nonlinear system occurs when its linear approximation loses stabilizability. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations

Nonlinear dynamics and control in a tumor-immune system

Advances in modeling tumor-immune dynamics and therapies offer deeper understandings of the mechanism of tumor evolution in the interdisciplinary field of mathematics and immune-oncology. The main mathematical models are constructed in terms of ordinary differential equations (ODEs) or partial differential equations (PDEs) and analyzed through tools such as Poincaré map, simulation, or numerical bifurcation analysis to understand the system properties. These models succeed in characterizing essential features of tumor behaviors including periodic bursts and the existence of latency. In relationship to practice, these models are also applied to estimate the feasibility and efficacy of treatments ranging from traditional chemotherapy to immunotherapy (ACI). In recent literature, there have been applications of control methods such as optimal control, hybrid automata, and feedback linearization-based tracking control with almost disturbance decoupling in the studies of tumor-immune systems. This thesis presents an attempt to apply the bifurcation control method with washout filters in tumor treatments. This thesis research investigates the dynamics and controlling of the tumor-immune response of immunotherapies, mainly the Adoptive Cell Immunotherapy (ACI) and Interleukin-2 (IL-2). The first part of the thesis presents the nonlinear dynamics of the classic nonlinear ODE tumor-immune model given by Denise Kirschner and John Carl Panetta in 1998. This model concentrates on the nonlinear phenomena of the tumor-immune system under immunotherapies, primarily the bifurcation phenomenon along with the antigenicity of effector cells. Bifurcation phenomena refer to the qualitative changes in system dynamics due to quasi-static changes in system parameters. Antigenicity refers to a capability to distinguish tumor cells from healthy cells. The Kirschner-Panetta model captures a saddle-node bifurcation and a Hopf bifurcation of the tumor-immune response, which separates the tumor evolution into three stages, the “dangerous equilibrium”, the periodic recurrence, and the “safe equilibrium”. The second part applies and analyzes several control strategies on the immunotherapies based on the KP model in order to eradicate tumors or inhibit tumor growths. The first section studies the combination immunotherapy of ACI and IL-2 as an open-loop control system based on Kirschner’s work, which generates a locally asymptotically stable equilibrium. In the second section, this thesis provides a new idea of treatment in the tumor-immune system, that is a closed-loop control strategy taking advantage of its bifurcation structure by applying dynamic feedback control with a washout filter of ACI or IL-2. Bifurcation control moves the Hopf bifurcation point without changing the equilibrium structure as the bifurcation parameter varies. In this tumor-immune case, the linear dynamic feedback control with a washout filter of ACI could either extend the “safe equilibrium” region or reduce the amplitude of the tumor population at the stage of tumor recurrence. In addition, other bifurcation amplitude controls of either ACI or IL-2 are attempted to reduce the amplitudes of periodic orbits of the tumor immune system but without obvious effects

Feedback-induced self-oscillations in large interacting systems subjected to phase transitions

In this article it is shown that large systems with many interacting units endowing multiple phases display self-oscillations in the presence of linear feedback between the control and order parameters, where an Andronov–Hopf bifurcation takes over the phase transition. This is simply illustrated through the mean field Landau theory whose feedback dynamics turn out to be described by the Van der Pol equation and it is then validated for the fully connected Ising model following heat bath dynamics. Despite its simplicity, this theory accounts potentially for a rich range of phenomena: here it is applied to describe in a stylized way (i) excess demand-price cycles due to strong herding in a simple agent-based market model; (ii) congestion waves in queuing networks triggered by user feedback to delays in overloaded conditions; and (iii) metabolic network oscillations resulting from cell growth control in a bistable phenotypic landscape

Time-delayed feedback control of unstable periodic orbits near a subcritical Hopf bifurcation

We show that Pyragas delayed feedback control can stabilize an unstable periodic orbit (UPO) that arises from a generic subcritical Hopf bifurcation of a stable equilibrium in an n-dimensional dynamical system. This extends results of Fiedler et al. [PRL 98, 114101 (2007)], who demonstrated that such feedback control can stabilize the UPO associated with a two-dimensional subcritical Hopf normal form. Pyragas feedback requires an appropriate choice of a feedback gain matrix for stabilization, as well as knowledge of the period of the targeted UPO. We apply feedback in the directions tangent to the two-dimensional center manifold. We parameterize the feedback gain by a modulus and a phase angle, and give explicit formulae for choosing these two parameters given the period of the UPO in a neighborhood of the bifurcation point. We show, first heuristically, and then rigorously by a center manifold reduction for delay differential equations, that the stabilization mechanism involves a highly degenerate Hopf bifurcation problem that is induced by the time-delayed feedback. When the feedback gain modulus reaches a threshold for stabilization, both of the genericity assumptions associated with a two-dimensional Hopf bifurcation are violated: the eigenvalues of the linearized problem do not cross the imaginary axis as the bifurcation parameter is varied, and the real part of the cubic coefficient of the normal form vanishes. Our analysis of this degenerate bifurcation problem reveals two qualitatively distinct cases when unfolded in a two-parameter plane. In each case, Pyragas-type feedback successfully stabilizes the branch of small-amplitude UPOs in a neighborhood of the original bifurcation point, provided that the phase angle satisfies a certain restriction.Comment: 35 pages, 19 figure