We consider a system of two nonlinear differential equations with slowly varying parameter μ = εt. For frozen const the has focus type equilibrium state stability which changes when passing through value 0, i.e., we deal an Andronov–Hopf bifurcation. Using normal form method combined averaging method, study asymptotics respect to small ε → 0 for solutions having narrow transient layer near bifu...

the present article deals with the inter specific competition and intra-specific competition among predator populations of a prey-dependent three component food chain model consisting of two competitive predator sharing one prey species as their food. the behaviour of the system near the biologically feasible equilibria is thoroughly analyzed. boundedness and dissipativeness of the system are e...

We perform a bifurcation analysis of the mathematical model of Jones and Kompala [K.D. Jones, D.S. Kompala, Cybernetic model of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures, J. Biotechnol. 71 (1999) 105-131]. Stable oscillations arise via Andronov-Hopf bifurcations and exist for intermediate values of the dilution rate as has been noted from experiments previ...

We study Hopf-Andronov bifurcations in a class of random differential equations (RDEs) with bounded noise. We observe that when an ordinary differential equation that undergoes a Hopf bifurcation is subjected to bounded noise then the bifurcation that occurs involves a discontinuous change in the Minimal Forward Invariant set.

In this paper we extend the results of Frankel and Kiemel [SIAM J. Appl. Math, 53 (1993), pp. 1436–1446] to a network of slowly coupled oscillators. First, we use Malkin’s theorem to derive a canonical phase model that describes synchronization properties of a slowly coupled network. Then, we illustrate the result using slowly coupled oscillators (1) near Andronov–Hopf bifurcations, (2) near sa...

In this paper, we apply a rigorous quasi-steady state approximation method on a family of models describing a gene regulated by a polymer of its own protein. We study the absence of oscillations for this family of models and prove that Poincaré-Andronov-Hopf bifurcations arise if and only if the number of polymerizations is greater than 8. A result presented in a former paper at Algebraic Biolo...

Cardiac arrhythmias have been closely linked to a variety of bifurcations and chaos. In this paper control of bifurcations and chaos for nonlinear models of cardiac electro-physiologic activity is investigated. Both the Andronov-Hopf bifurcation and period doubling bifur-cation are treated. Washout lter aided feedback controllers are employed to control the location and stability of the bifurca...

A circuit model is proposed for studying the global behavior of the normal form describing the Bogdanov-Takens bifurcation, which is encountered in the study of autonomous dynamical systems arising in different branches of science and engineering. The circuit is easy-to-implement and one can experimentally study the rich dynamics and bifurcations simply by altering the values of some linear cir...

A codimension-three bifurcation, characterized by a pair of purely imaginary eigenvalues and a nonsemisimple double zero eigenvalue, arises in the study of a pair of weakly coupled nonlinear oscillators with Z2 Z2 symmetry. The methodology is based on Arnold's ideas of versal deformations of matrices for the linear analysis, and Poincar e normal forms for the nonlinear analysis of the system. B...