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 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, dynamical analysis is presented for a group of unicycles in leader-follower formation. The equilibrium formations were characterized along with the local stability analysis. It was demonstrated that with the variation in control gain, the collective dynamics might undergo Andronov-Hopf and Fold-Hopf bifurcations. An increase in the number of unicycles increase the vigor of quasi-...

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...

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...

The main objective of this addendum to the mentioned article [49] by Park is to provide some remarks on bifurcation theories for nonlinear partial differential equations (PDE) and their applications to fluid dynamics problems. We only wish to comment and list some related literatures, without any intention to provide a complete survey. For steady state PDE bifurcation problems, the often used c...

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...

Abstract We address the question of which small, bimolecular, mass action chemical reaction networks (CRNs) are capable Andronov–Hopf bifurcation (from here on abbreviated to ‘Hopf bifurcation’). It is easily shown that any such network must have at least three species and four irreversible reactions, one example a with exactly reactions was previously known due Wilhelm. In this paper, we devel...