We prove Wigner-Eckart theorem for the irreducible tensor operators for arbitrary Hopf algebras, provided that tensor product of their irreducible representation is completely reducible. The proof is based on the properties of the irreducible representations of Hopf algebras, in particular on Schur lemma. Two classes of tensor operators for the Hopf algebra Ut(su(2)) are considered. The reduced...

A reaction-diffusion population model with a general time-delayed growth rate per capita is considered. From a careful analysis of the characteristic equation, the stability of the positive steady state solution and the existence of supercritical Hopf bifurcation from the positive steady state solution are obtained via the implicit function theorem, where the time delay is used as the bifurcati...

Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spati...

In this paper, a system of three globally coupled limit cycle oscillators with a linear time-delayed coupling are investigated. Considering the delay as a parameter, we also study the effect of time delay on the dynamics. Next, Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction are obtained. Based on the symmetric Hopf bifurcation theorem, we inv...

A time-delayed model of speculative asset markets is investigated to discuss the effect of time delay and market fraction of the fundamentalists on the dynamics of asset prices. It proves that a sequence of Hopf bifurcations occurs at the positive equilibrium v, the fundamental price of the asset, as the parameters vary. The direction of the Hopf bifurcations and the stability of the bifurcatin...

In this note, we consider Hopf bifurcation control for an Internet congestion model with a single route accessed by a single source. It has been shown that the system without control cannot guarantee a stationary sending rate. As the positive gain parameter of the system passes a critical point, Hopf bifurcation occurs. To control the Hopf bifurcation, a time-delayed feedback controller using p...

In this paper, a system of Lorenz-type ordinary differential equations is considered and, under some assumptions about the parameter space, presence supercritical non-degenerate Hopf bifurcation demonstrated. The technical tool used consists Central Manifold theorem, well-known formula to calculate Lyapunov coefficient and Hopf’s Theorem. For particular values parameters in space established ma...

In this paper, we consider a two-dimensional delay differential system with two delays. By analyzing the distribution of eigenvalues, linear stability of the equilibria and existence of Hopf, Bautin, and Hopf–Hopf bifurcations are obtained in which the time delays are used as the bifurcation parameter. General formula for the direction, period, and stability of the bifurcated periodic solutions...

In the field of economic growth, a huge interest has been devoted to explain the mechanisms through which a nonlinear system looses its stability and starts to oscillate around the steady state. Since the seminal Kalekian analysis, limit cycles represent the theoretical way of characterizing the emergence of persistent oscillations of macro-economic variables, namely the rise of indeterminacy p...