Resonant Double Hopf Bifurcation Research Articles

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

Resonant double Hopf bifurcation in a diffusive Ginzburg–Landau model with delayed feedback

Normal resonances in a double Hopf bifurcation

We introduce a framework to systematically investigate the resonant double Hopf bifurcation. We use the basic invariants of the ensuing T1-action to analyse the approximating normal form truncations in a unified manner. In this way we obtain a global description of the parameter space and thus find the organising resonance droplet, which is the present analogue of the resonant gap. The dynamics of the normal form yields a skeleton for the dynamics of the original system. In the ensuing perturbation theory both normal hyperbolicity (centre manifold theory) and kam theory are being used.

An analytical method for studying double Hopf bifurcations induced by two delays in nonlinear differential systems

An analytical method is introduced to investigate double Hopf bifurcations induced by two delays qualitatively and quantitatively. As an illustrative example, the clear procedure is demonstrated to study delay-induced weak resonant double Hopf bifurcation in a nonlinear system with multiple delays. When two delays are close to double Hopf bifurcation point, all solutions derived from the bifurcation are classified qualitatively and expressed explicitly. Numerical simulations are a good agreement with our theoretical analysis, and also already work in references. The results show that our work in this paper proposes a simple and valid method for investigating delay-induced double Hopf bifurcations. The important feature of our work is that the explicit expression of periodic solutions is easy to be obtained by solving algebraic equations.

Weakly resonant double Hopf bifurcation in coupled nonlinear systems with delayed freedback and application of homotopy analysis method

In this paper, we study the dynamical behaviors of coupled nonlinear systems with delay coupling by the multiple scales method and the homotopy analysis method. Firstly, we analyze the distribution...

DOUBLE HOPF BIFURCATION FOR AN HOPFIELD EURAL NETWORK MODEL WITH TIME DELAYED FEEDBACK

In this paper, we consider a system of delay differential equations which represents the general model of a Hopfield neural networks type. We focus on the case that the corresponding linear system has two pairs of purely imaginary eigenvalues at the trivial equilibrium, giving rise to double Hopf bifurcations. An analytical approach is used to find the explicit expressions for the critical values of the system parameters at which nonresonant or resonant double Hopf bifurcations may occur. We also investigate the occurrence of an double Hopf bifurcation about the trivial equilibrium.

Galloping of internally resonant towers subjected to turbulent wind

Bifurcation analysis of a structure constituted by two towers, linked by a viscous device at the tip and subjected to turbulent wind, is carried out. The towers have geometrical and mechanical parameters so that the steady part of the wind, whose contribution is evaluated in the framework of the steady theory, induces a 1:1 resonant double-Hopf bifurcation. The turbulent part of the wind, assumed as composed by two frequencies that are equal and double to the main frequency of the unlinked towers, respectively, induces parametric and external harmonic forces. These forces interact with the self-excitation due to the steady part of the wind, bringing imperfection in the bifurcation scenario. Transitions from resonant to non-resonant cases are analyzed in terms of behavior charts, and post-critical dynamics is studied in the space of bifurcation parameters.

Bifurcation analysis in a recurrent neural network model with delays

In this paper, we study the dynamical behaviors of a three-node recurrent neural network model with four discrete time delays. We study several types of bifurcation, and use the method of multiple time scales to derive the normal forms associated with Hopf-zero bifurcation, non-resonant and resonant double Hopf bifurcations. Moreover, bifurcations are classified in two-dimensional parameter space near these critical points, and numerical simulations are presented to demonstrate the applicability of the theoretical results.

WEAK RESONANT DOUBLE HOPF BIFURCATIONS IN AN INERTIAL FOUR-NEURON MODEL WITH TIME DELAY

In this paper, a four-neuron delayed bidirectional associative memory (BAM) model with inertia is considered. Weak resonant double Hopf bifurcations are completely analyzed in the parameter space of the coupling weight and the coupling delay by the perturbation-incremental scheme (PIS). Numerical simulations are given for justifying the theoretical results. To the best of our knowledge, the paper is the first one to introduce inertia to a four-neuron delayed system and clarify the relationship between system parameters and dynamical behaviors.

Multiple scales analysis for double Hopf bifurcation with 1:3 resonance

The dynamical behavior of a general n-dimensional delay differential equation (DDE) around a 1:3 resonant double Hopf bifurcation point is analyzed. The method of multiple scales is used to obtain complex bifurcation equations. By expressing complex amplitudes in a mixed polar-Cartesian representation, the complex bifurcation equations are again obtained in real form. As an illustration, a system of two coupled van der Pol oscillators is considered and a set of parameter values for which a 1:3 resonant double Hopf bifurcation occurs is established. The dynamical behavior around the resonant double Hopf bifurcation point is analyzed in terms of three control parameters. The validity of analytical results is shown by their consistency with numerical simulations.

Hopf and resonant double Hopf bifurcation in congestion control algorithm with heterogeneous delays

The congestion control algorithm, which has dynamic adaptations at both user ends and link ends, with heterogeneous delays is considered and analyzed. Some general stability criteria involving the delays and the system parameters are derived by generalized Nyquist criteria. Furthermore, by choosing one of the delays as the bifurcation parameter, and when the delay exceeds a critical value, a limit cycle emerges via a Hopf bifurcation. Resonant double Hopf bifurcation is also found to occur in this model. An efficient perturbation-incremental method is presented to study the delay-induced resonant double Hopf bifurcation. For the bifurcation parameter close to a double Hopf point, the approximate expressions of the periodic solutions are updated iteratively by use of the perturbation-incremental method. Simulation results have verified and demonstrated the correctness of the theoretical results.

Double Hopf Bifurcation with Strong Resonances in Delayed Systems with Nonlinerities

An efficient method is proposed to study delay‐induced strong resonant double Hopf bifurcation for nonlinear systems with time delay. As an illustration, the proposed method is employed to investigate the 1 : 2 double Hopf bifurcation in the van der Pol system with time delay. Dynamics arising from the bifurcation are classified qualitatively and expressed approximately in a closed form for either square or cubic nonlinearity. The results show that 1 : 2 resonance can lead to codimension‐three and codimension‐two bifurcations. The validity of analytical predictions is shown by their consistency with numerical simulations.

Computation of the simplest normal form of a resonant double Hopf bifurcation system with the complex normal form method

The complex normal form method is presented to find the simplest normal form (SNF) of an odd ratio double Hopf bifurcation system. New nonlinear transformations and rules of getting the form of the SNF coefficients are presented to find an efficient approach to get the final explicit expressions of the SNF. During the course of obtaining the key equations, based on the algorithm of complex operations, all the matrix deduction is substituted by complex operation and no former matrix operation is involved compared with the matrix representation method. Explicit iterative formulas have been derived to determine the coefficients of the SNF and associated nonlinear transformations up to an arbitrary order. Meanwhile, a rule of analyzing the general expression of the SNF is also summarized, which enables to predict the components of the final explicit equations before any substantive analyses. As the result of these innovations is deduced, the computation process is basically simplified. An ordinary differential equation (ODE) system shows the feasibility and convenience of the new method.

An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks

An efficient method, called the perturbation‐incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, the delay‐induced weak or high‐order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely, the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay‐induced weak resonant double Hopf bifurcation and dynamics in the van der Pol–Duffing and the Stuart–Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is...

An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks

An efficient method, called the perturbation‐incremental scheme (PIS), is proposed to study, both qualitatively and quantitatively, the delay‐induced weak or high‐order resonant double Hopf bifurcation and the dynamics arising from the bifurcation of nonlinear systems with delayed feedback. The scheme is described in two steps, namely, the perturbation and the incremental steps, when the time delay and the feedback gain are taken as the bifurcation parameters. As for applications, the method is employed to investigate the delay‐induced weak resonant double Hopf bifurcation and dynamics in the van der Pol–Duffing and the Stuart–Landau systems with delayed feedback. For bifurcation parameters close to a double Hopf point, all solutions arising from the resonant bifurcation are classified qualitatively and expressed approximately in a closed form by the perturbation step of the PIS. Although the analytical expression may not be accurate enough for bifurcation parameters away from the double Hopf point, it is used as an initial guess for the incremental step which updates the approximate expression iteratively and performs parametric continuation. The analytical predictions on the two systems show that the delayed feedback can, on the one hand, drive a periodic solution into an amplitude death island where the motion vanishes and, on the other hand, create complex dynamics such as quasi‐periodic and coexisting motions. The approximate expression of periodic solutions with parameter varying far away from the double Hopf point can be calculated to any desired accuracy by the incremental step. The validity of the results is shown by their consistency with numerical simulations. We show that as an analytical tool the PIS is simple but efficient.

COMPLEX DYNAMICS CLOSE TO NON-RESONANT DOUBLE HOPF BIFURCATION

The interactions between two frequencies in a nonlinear model of resonant coupled electrical circuits are analyzed. This situation is related with the existence of one Hopf degeneracy known as non resonant double Hopf bifurcation. The dynamics of its unfolding is evaluated through the determination of the Neimark-Sacker bifurcation curves, which involve the appearance of quasiperiodic solutions. Moreover, fold and flip bifurcations of cycles have also been detected nearby the singularity using a frequency domain approach.

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.

Multiscale analysis of defective multiple-Hopf bifurcations

An adapted version of the Multiple Scale Method is formulated to analyze 1:1 resonant multiple Hopf bifurcations of discrete autonomous dynamical systems, in which, for quasi-static variations of the parameters, an arbitrary number m of critical eigenvalues simultaneously crosses the imaginary axis. The algorithm therefore requires discretizing continuous systems in advance. The method employs fractional power expansion of a perturbation parameter, both in the state variables and in time, as suggested by a formal analogy with the eigenvalue sensitivity analysis of nilpotent (defective) matrices, also illustrated in detail. The procedure leads to an order- m differential bifurcation equation in the complex amplitude of the unique critical eigenvector, which is able to capture the dynamics of the system around the bifurcation point. The procedure is then adapted to the specific case of a double Hopf bifurcation ( m = 2), for which a step-by-step, computationally-oriented version of the method is furnished that is directly applicable to solve practical problems. To illustrate the algorithm, a family of mechanical systems, subjected to aerodynamic forces triggering 1:1 resonant double Hopf bifurcations is considered. By analyzing the relevant bifurcation equation, the whole scenario is described in a three-dimensional parameter space, displaying rich dynamics.

Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of1:2 and 1:3 are investigated. The Multiple Scale Method is employedto derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the criticalstate. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter ε, of the order ofthe amplitude of motion. However, while resonant terms only appear atthe ε3-order in the 1:3 case, they already arise at theε2-order in the 1:2 case. Thus, by truncating theanalysis at the ε3-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of thethree control parameters and the asymptotic results are compared withexact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and theirstability analyzed.

One to one resonant double Hopf bifurcation in aeroelastic oscillators with tuned mass dampers

The effects of a tuned added mass on the aeroelastic stability of a single degree of freedom bluff body exposed to a steady flow are investigated. The model captures the essential aspects of the behaviour of flexible structures equipped with Tuned Mass Dampers undergoing galloping oscillations. The system exhibits simple as well double Hopf bifurcations, of non-resonant and 1:1 resonant type. Postcritical behaviour of the system in the neighbourhood of the 1:1 resonant type bifurcation is investigated. Employing the Multiple Scale Method, a second order bifurcation equation in the complex amplitude of motion is obtained. Analytical solutions are used to describe the bifurcation scenario in the cases of both undercritical and supercritical aerodynamic behaviour of the bluff body. The effectiveness of the Tuned Mass Damper even in the postcritical range is proved.

SYMBOLIC COMPUTATION OF NORMAL FORMS FOR RESONANT DOUBLE HOPF BIFURCATIONS USING A PERTURBATION TECHNIQUE

This paper presents a perturbation method for computing the normal forms of resonant double Hopf bifurcations with the aid of a computer algebraic system. This technique, based on the method of multiple time scales, can be used to deal with general n -dimensional systems without the application of center manifold theory. Explicit iterative formulas have been derived for uniquely determining the coefficients of normal forms and associated non-linear transformations up to an arbitrary order. User-friendly symbolic computer programs written in Maple have been developed, which can be easily applied for computing the normal forms of a given system. Examples are presented to show the applicability of the methodology and the convenience of using the computer software.