In this paper, we study the cubic three-parameter autonomous planar system ẋ1 = k1 + k2x1 − x1 − x2, ẋ2 = k3x1 − x2, where k2, k3 > 0. Our goal is to obtain a bifurcation diagram; i.e., to divide the parameter space into regions within which the system has topologically equivalent phase portraits and to describe how these portraits are transformed at the bifurcation boundaries. Results may be a...

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-...

Chaos generated by the existence of Smale horseshoe is the well-known phenomenon in the theory of dynamical systems. The Poincaré-Andronov-Melnikov periodic and subharmonic bifurcations are also classical results in this theory. The purpose of this note is to extend those results to ordinary differential equations with multivalued perturbations. We present several examples based on our recent a...

We analyze a frequency decrease as well as a frequency transition with a temperature increase in the Hodgkin-Huxley (HH) oscillator undergoing saddle homoclinic bifurcations. A gradient of frequency for temperature is derived by perturbation analysis of the stable HH oscillators, in combination with the other gradient of frequency for input current and a so-called phase response curve (PRC) mul...

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...

Abstract This is a study of dynamical system depending on parameter . Under the assumption that has family equilibrium positions or periodic trajectories smoothly , focus details stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations problem are considered. In first, constant subject analysis phenomenon soft h...

In this paper, the dynamics of a 3D autonomous dissipative nonlinear system ODEs-Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain new analytical formula for first Lyapunov’s (focal) value at boundary stability corresponding equilibrium state. On other hand, global analysis reveals that may exhibit phenomena Shilnikov chaos. Further, it is shown via calcula...

In the following chapters we present the theory of bifurcations of dynamical systems with simple dynamics. It is difficult to over-emphasize the role of bifurcation theory in nonlinear dynamics the reason is quite simple: the methods of the theory of bifurcations comprise a working tool kit for the study of dynamical models. Besides, bifurcation theory provides a universal language to communica...

<p style='text-indent:20px;'>In this paper we study generalized Poincaré-Andronov-Hopf bifurcations of discrete dynamical systems. We prove a general result for attractors in <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-dimensional manifolds satisfying some suitable conditions. This allows us to obtain sharper Ho...

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...