Local stability and Hopf bifurcation of two-dimensional nonlinear descriptor system

Descriptor systems, which are also called differential-algebraic systems, singular systems, degenerate systems, constrained systems and so on, have been one of the major research topics in engineering field. However, problems related to the local stability and the Hopf bifurcation of nonlinear descriptor systems with time delay have not been thoroughly investigated. In this paper, we consider the dynamical behavior of two-dimensional nonlinear descriptor systems with time delay. First, local stability of the system is analyzed using the location of the roots of the characteristic equation of the corresponding linearized system. It is well known that the equilibrium of the linearized system is locally asymptotically stable if all roots of the corresponding characteristic equation locate in the left half of the complex plane, and they are uniformly bounded away from the imaginary axis. Otherwise, the equilibrium is unstable. The conditions for the existence of Hopf bifurcation are investigated in detail by using the time delay as a bifurcation parameter. Furthermore, based on the descriptor and the “neutral-type” model transformation, some special neutral differential equations can be equivalently transformed into the nonlinear descriptor systems. Finally, the correctness and the effectiveness of the theoretical analysis are justified by numerical examples.

[1]  C. Lien.  Guaranteed Cost Observer–Based Controls for a Class of Uncertain Neutral Time-Delay Systems , 2005 .

[2]  Hassan A. El-Morshedy,et al.  Nonoscillation, oscillation and convergence of a class of neutral equations , 2000 .

[3]  R. Agarwal,et al.  Asymptotic stability of certain neutral differential equations , 2000 .

[4]  Aydin Huseynov,et al.  On the sign of Green's function for an impulsive differential equation with periodic boundary conditions , 2009, Appl. Math. Comput..

[5]  Emilia Fridman,et al.  A new Lyapunov technique for robust control of systems with uncertain non-small delays , 2006, IMA J. Math. Control. Inf..

[6]  Yeong-Jeu Sun.  Exponential stability for continuous-time singular systems with multiple time delays , 2003 .

[7]  Guo-Ping Jiang,et al.  State feedback control at Hopf bifurcation in an exponential RED algorithm model , 2014 .

[8]  Hartmut Logemann,et al.  Destabilizing effects of small time delays on feedback-controlled descriptor systems☆ , 1998 .

[9]  R. Datko.  Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks , 1988 .

[10]  John L. Casti,et al.  Introduction to the theory and application of differential equations with deviating arguments , 1973 .

[11]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[12]  Jinde Cao,et al.  Hopf bifurcation and stability of periodic solutions for van der Pol equation with time delay , 2005 .

[13]  Hartmut Logemann,et al.  Conditions for Robustness and Nonrobustness of theStability of Feedback Systems with Respect to Small Delays inthe Feedback Loop , 1996 .

[14]  R. Datko,et al.  Two examples of ill-posedness with respect to small time delays in stabilized elastic systems , 1993, IEEE Trans. Autom. Control..

[15]  Hongyong Zhao,et al.  Hopf bifurcation for a small-world network model with parameters delay feedback control , 2011 .

[16]  Xiaofeng Liao,et al.  Bogdanov–Takens bifurcation in a tri-neuron BAM neural network model with multiple delays , 2013 .

[17]  Yeong-Jeu Sun,et al.  Global stabilizability of uncertain systems with time-varying delays via dynamic observer-based output feedback , 2002 .

[18]  Defang Liu,et al.  Hopf and resonant double Hopf bifurcation in congestion control algorithm with heterogeneous delays , 2010 .

[19]  Emilia Fridman.  Effects of small delays on stability of singularly perturbed systems , 2002, Autom..

[20]  Wen-Jye Shyr,et al.  Robust d-stability for linear uncertain discrete time-delay systems , 2003, IEEE Trans. Autom. Control..

[21]  Emilia Fridman,et al.  A descriptor system approach to H∞ control of linear time-delay systems , 2002, IEEE Trans. Autom. Control..

[22]  Michael P. Polis,et al.  An example on the effect of time delays in boundary feedback stabilization of wave equations , 1986 .

[23]  Kwok-Wo Wong,et al.  Hopf Bifurcation on a Two-Neuron System with Distributed Delays: A Frequency Domain Approach , 2003 .

[24]  Stephen L. Campbell,et al.  Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions , 2009, Appl. Math. Comput..

[25]  Xiaofeng Liao,et al.  Asymptotic stability analysis of certain neutral differential equations: A descriptor system approach , 2009, Math. Comput. Simul..

[26]  Stuart Townley,et al.  The effect of small delays in the feedback loop on the stability of neutral systems , 1996 .

[27]  Chuandong Li,et al.  Exponential Convergence Estimates for a Single Neuron System of Neutral-Type , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[28]  Jack K. Hale,et al.  Effects of Small Delays on Stability and Control , 2001 .

[29]  Chang-Hua Lien,et al.  Robust observer-based control of systems with state perturbations via LMI approach , 2004, IEEE Transactions on Automatic Control.

[30]  Shengyuan Xu,et al.  Robust stability and stabilization for singular systems with state delay and parameter uncertainty , 2002, IEEE Trans. Autom. Control..

[31]  J. Lagnese,et al.  An example of the effect of time delays in boundary feedback stabilization of wave equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[32]  Shing-Tai Pan,et al.  D-stability for a class of discrete descriptor systems with multiple time delays , 2002 .

[33]  X. Liao,et al.  Dynamics of an inertial two-neuron system with time delay , 2009 .

[34]  Chun-Liang Lin.  On the stability of uncertain linear descriptor systems , 1999 .

[35]  W. Desch,et al.  Destabilization due to delay in one dimensional feedback , 1989 .

[36]  J. F. Barman,et al.  L 2 -stability and L 2 -instability of linear time-invariant distributed feedback systems perturbed by a small delay in the loop , 1973 .

[37]  Kwok-Wo Wong,et al.  Hopf Bifurcation and Stability of Periodic Solutions for van der Pol Equation with Distributed Delay , 2001 .

[38]  Emilia Fridman,et al.  A descriptor system approach to nonlinear singularly perturbed optimal control problem , 2001, Autom..