Computation of all the coefficients for the global connections in the Z2-symmetric Takens-Bogdanov normal forms

Bursting Dynamics in a Singular Vector Field with Codimension Three Triple Zero Bifurcation

“…Next, based on the NTT (nonlinear time transformation) method, they derived a recursive algorithm for calculating the analytic formula of the global connection curve of the high-order approximation symmetric Bogdanov-Takens normal form. The theoretically derived phase trajectories can be well matched with the results of numerical simulations [22].…”

“…It can be seen from Figure 13c,d that when A = 0.3, with the change of β, the mode of the spiking state of bursting oscillation due to codimension-2 Bautin bifurcation will change qualitatively, where the red curve represents β = 7.5, and the blue curve represents β = 12. Figure 14 shows Poincar é mapping (22) of the oscillation trajectory of one period, in which it can be found that the spiking state behaves as a quasi-periodic characteristic. Therefore, the structure of the bursting oscillation is the Point-Cycle-Cycle type in this case.…”

Analytical approximation of the canard explosion in a van der Pol system with the nonlinear time transformation method

“…There, a parabola of an integrable system (that corresponds to a homoclinic orbit connecting a critical point at infinity) is perturbed. Taking into account the excellent results obtained with the NTT method in the analysis of global connections up to any order of approximation [35][36][37], here we adapt the NTT method to this canard problem. In fact, the NTT method is an efficient alternative to Melnikov method, mainly in the calculation of the coefficients of the Poincaré application near the homoclinic connection.…”

“…On the other hand, the NTT method has been very efficient in calculating, in the case of homoclinic and heteroclinic connections, the coefficients at any order corresponding to the Melnikov method [35][36][37]. In this way, we will show that, using the NTT method, it is easy to compute all the coefficients of the Melnikov method for this homoclinic orbit at infinity.…”

“…al [32]. Recently, it has been applied to approximate the connecting orbits in a 1:3 resonance problem [33], to study a double Takens-Bogdanov normal form in R 4 [34] and to compute the coefficients for the global connections in the Takens-Bogdanov normal forms [35][36][37]. The main contribution of our work is the efficient calculation of the formula (35) together with the approximation to the critical manifold without discontinuities ( 33)- (34).…”