Non-resonant Double Hopf Bifurcation Research Articles

The nonresonant double Hopf bifurcation in delayed neural network

A two-neuron network with self/neighbour-delayed-connections has been investigated. The self-connections are proposed by excitatory synapses and the neighbour ones by excitatory and inhibitory synapses. It is found that the excitatory self-connections can lead to non-resonant double Hopf bifurcations, for example, 1:√2 double Hopf bifurcation, since the double Hopf bifurcation disappears without self-delayed connections. Various dynamic behaviours are classified in the neighbourhood of the double Hopf bifurcation point, and the bifurcating solutions are computed in an approximate form by the centre manifold reduction. It follows from the approximate solutions that the network of two identical neurons with neighbour-connection cannot be synchronized completely. However, generalized synchronization can occur in the network under consideration. The time delay can also lead to the disappearance of periodic motion. These results may have some potential applications, such as controlling associative memory and designing neural networks.

Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control

The stability and bifurcation of a van der Pol-Duffing oscillator with the delay feedback are investigated, in which the strength of feedback control is a nonlinear function of delay. A geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and Hopf bifurcations. The Hopf bifurcation curves and multi-stable regions are obtained as two parameters vary. Some weak resonant and non-resonant double Hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the “death island” regions simultaneously. By applying the center manifold theory, the normal forms near the double Hopf bifurcation points, as well as classifications of local dynamics are analyzed. Furthermore, some quasi-periodic and chaotic motions are verified in both theoretical and numerical ways.

Double Hopf Bifurcation Analysis Using Frequency Domain Methods

The dynamic behavior close to a non-resonant double Hopf bifurcation is analyzed via a frequency-domain technique. Approximate expressions of the periodic solutions are computed using the higher order harmonic balance method while their accuracy and stability have been evaluated through the calculation of the multipliers of the monodromy matrix. Furthermore, the detection of secondary Hopf or torus bifurcations (Neimark–Sacker bifurcation for maps) close to the analyzed singularity has been obtained for a coupled electrical oscillatory circuit. Then, quasi-periodic solutions are likely to exist in certain regions of the parameter space. Extending this analysis to the unfolding of the 1:1 resonant double Hopf bifurcation, cyclic fold and torus bifurcations have also been detected in a controlled oscillatory coupled electrical circuit. The comparison of the results obtained with the suggested technique, and with continuation software packages, has been included.

Classification and Computation of Non-resonant Double Hopf Bifurcations and Solutions in Delayed van der Pol-Duffing System

Classification and Computation of Non-resonant Double Hopf Bifurcations and Solutions in Delayed van der Pol-Duffing System

Restrictions and unfolding of double Hopf bifurcation in functional differential equations

The normal form of a vector field generated by scalar delay-differential equations at nonresonant double Hopf bifurcation points is investigated. Using the methods developed by Faria and Magalhães (J. Differential Equations 122 (1995) 181) we show that (1) there exists linearly independent unfolding parameters of classes of delay-differential equations for a double Hopf point which generically map to linearly independent unfolding parameters of the normal form equations (ordinary differential equations), (2) there are generically no restrictions on the possible flows near a double Hopf point for both general and Z 2 -symmetric first-order scalar equations with two delays in the nonlinearity, and (3) there always are restrictions on the possible flows near a double Hopf point for first-order scalar delay-differential equations with one delay in the nonlinearity, and in nth-order scalar delay-differential equations ( n⩾2) with one delay feedback.

Normal form coefficients for the nonresonant double Hopf bifurcation

Expressions are given for the coefficients in the normal form for the nondegenerate double Hopf bifurcation when the two frequencies are not in strong resonance.

Double Hopf Bifurcation and Quasi-Periodic Flow in a Model for Baroclinic Instability

Abstract The interaction between pairs of dispersive waves is studied with different zonal wavenumbers in the β-plane model for baroclinic instability. We find that degenerate (codimension 2) double Hopf bifurcations occur at isolated points in the (F2/r2, F−1) parameter space. Here F is Froude number and r is an effective viscosity. Using perturbation methods, we derive the phase and amplitude equations for the interacting modes, which have distinct frequencies. These equations take the standard normal form for a nonresonant double Hopf bifurcation and can readily be analyzed using phase plane methods. We compute numerical values of the coefficients that determine stability in several cases and find that both pure and mixed (quasi-periodic) modes exist and that these modes can be stable or unstable, depending upon the zonal wavenumbers and other parameters. We discuss our results in the light of experimental studies. Our main conclusion is that the two-wave interaction and attendant double Hopf bifurcati...