A model of competition between two species in a turbidostat with delayed feedback control is inve... more A model of competition between two species in a turbidostat with delayed feedback control is investigated. By choosing the delay in the measurement of the optical sensor to the turbidity of the fluid as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Computer simulations illustrate the results.
The properties of Hopf bifurcations induced by the coupling time delay in a pair of identical tri... more The properties of Hopf bifurcations induced by the coupling time delay in a pair of identical tri-neuron network loops are investigated. Using center manifold reduction and normal-form theorem of retarded differential equations, explicit conditions ensuring the stability and direction of the Hopf bifurcation are given. Numerical simulations are used to illustrate our theoretical results.
A model of competition between two species in a turbidostat with delayed feedback control is inve... more A model of competition between two species in a turbidostat with delayed feedback control is investigated. By choosing the delay in the measurement of the optical sensor to the turbidity of the fluid as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Computer simulations illustrate the results.
The properties of Hopf bifurcations induced by the coupling time delay in a pair of identical tri... more The properties of Hopf bifurcations induced by the coupling time delay in a pair of identical tri-neuron network loops are investigated. Using center manifold reduction and normal-form theorem of retarded differential equations, explicit conditions ensuring the stability and direction of the Hopf bifurcation are given. Numerical simulations are used to illustrate our theoretical results.
Journal of Mathematical Analysis and Applications, 2009
... j = 2 1 r 2 cos( j ) + sin( j ) r 2 2 + 2 + r 2 1 cos( j ) + r 2 sin( j ) r ... j (x, y, ), d... more ... j = 2 1 r 2 cos( j ) + sin( j ) r 2 2 + 2 + r 2 1 cos( j ) + r 2 sin( j ) r ... j (x, y, ), d dt y = AQ 1 y + summationdisplay 1 j! f 2 j (x, y, ), jgreaterorequalslant2 88 S. Yuan, Y. Song / J. Math. ... 1 j! f 2 j (x, y, ) are homogeneous polynomials in (x, y, ) of degree j with coe cients in C 2 ,ker , respectively ...
Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-typ... more Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-type model with two distributed delays. For this model, He et al. [SIAM J. Math. Anal. 29 (1998) 681] showed that the positive equilibrium can be globally asymptotically stable if the mean delays are sufficiently small. In this paper, using the average time delay as a
Journal of Mathematical Analysis and Applications, 2009
... j = 2 1 r 2 cos( j ) + sin( j ) r 2 2 + 2 + r 2 1 cos( j ) + r 2 sin( j ) r ... j (x, y, ), d... more ... j = 2 1 r 2 cos( j ) + sin( j ) r 2 2 + 2 + r 2 1 cos( j ) + r 2 sin( j ) r ... j (x, y, ), d dt y = AQ 1 y + summationdisplay 1 j! f 2 j (x, y, ), jgreaterorequalslant2 88 S. Yuan, Y. Song / J. Math. ... 1 j! f 2 j (x, y, ) are homogeneous polynomials in (x, y, ) of degree j with coe cients in C 2 ,ker , respectively ...
Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-typ... more Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-type model with two distributed delays. For this model, He et al. [SIAM J. Math. Anal. 29 (1998) 681] showed that the positive equilibrium can be globally asymptotically stable if the mean delays are sufficiently small. In this paper, using the average time delay as a
A model of competition between two species in a turbidostat with delayed feedback control is inve... more A model of competition between two species in a turbidostat with delayed feedback control is investigated. By choosing the delay in the measurement of the optical sensor to the turbidity of the fluid as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Computer simulations illustrate the results.
The properties of Hopf bifurcations induced by the coupling time delay in a pair of identical tri... more The properties of Hopf bifurcations induced by the coupling time delay in a pair of identical tri-neuron network loops are investigated. Using center manifold reduction and normal-form theorem of retarded differential equations, explicit conditions ensuring the stability and direction of the Hopf bifurcation are given. Numerical simulations are used to illustrate our theoretical results.
A model of competition between two species in a turbidostat with delayed feedback control is inve... more A model of competition between two species in a turbidostat with delayed feedback control is investigated. By choosing the delay in the measurement of the optical sensor to the turbidity of the fluid as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. The direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Computer simulations illustrate the results.
The properties of Hopf bifurcations induced by the coupling time delay in a pair of identical tri... more The properties of Hopf bifurcations induced by the coupling time delay in a pair of identical tri-neuron network loops are investigated. Using center manifold reduction and normal-form theorem of retarded differential equations, explicit conditions ensuring the stability and direction of the Hopf bifurcation are given. Numerical simulations are used to illustrate our theoretical results.
Journal of Mathematical Analysis and Applications, 2009
... j = 2 1 r 2 cos( j ) + sin( j ) r 2 2 + 2 + r 2 1 cos( j ) + r 2 sin( j ) r ... j (x, y, ), d... more ... j = 2 1 r 2 cos( j ) + sin( j ) r 2 2 + 2 + r 2 1 cos( j ) + r 2 sin( j ) r ... j (x, y, ), d dt y = AQ 1 y + summationdisplay 1 j! f 2 j (x, y, ), jgreaterorequalslant2 88 S. Yuan, Y. Song / J. Math. ... 1 j! f 2 j (x, y, ) are homogeneous polynomials in (x, y, ) of degree j with coe cients in C 2 ,ker , respectively ...
Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-typ... more Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-type model with two distributed delays. For this model, He et al. [SIAM J. Math. Anal. 29 (1998) 681] showed that the positive equilibrium can be globally asymptotically stable if the mean delays are sufficiently small. In this paper, using the average time delay as a
Journal of Mathematical Analysis and Applications, 2009
... j = 2 1 r 2 cos( j ) + sin( j ) r 2 2 + 2 + r 2 1 cos( j ) + r 2 sin( j ) r ... j (x, y, ), d... more ... j = 2 1 r 2 cos( j ) + sin( j ) r 2 2 + 2 + r 2 1 cos( j ) + r 2 sin( j ) r ... j (x, y, ), d dt y = AQ 1 y + summationdisplay 1 j! f 2 j (x, y, ), jgreaterorequalslant2 88 S. Yuan, Y. Song / J. Math. ... 1 j! f 2 j (x, y, ) are homogeneous polynomials in (x, y, ) of degree j with coe cients in C 2 ,ker , respectively ...
Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-typ... more Beretta and Takeuchi [Differ. Equat. Dyn. Syst. 2 (1994) 19] proposed and studied a chemostat-type model with two distributed delays. For this model, He et al. [SIAM J. Math. Anal. 29 (1998) 681] showed that the positive equilibrium can be globally asymptotically stable if the mean delays are sufficiently small. In this paper, using the average time delay as a
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