Melnikov method

In next section we shall give a short account of the Melnikov method in a form convenient for our problem. Then we shall prove Theorem 1. 

Short summary of the Melnikov method We shall explain the Melnikov technique following [11],... 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

In order to clarify the underlying classical mechanism of the fringed tunnehng, we developed theoretical analyses in the low-frequency regime based on a complex adiabatic solution [25], together with the Melnikov method extended to the complex domain [25]. 

Some important facts about the critical point and the branch associated with it, which are numerically observed, can be proven with the Melnikov method and the adiabatic solution in the low-frequency regime. They are summarized as follows [25] ... 

As shown in Appendix B, item 1 is proven by using the Melnikov method. It means that the heteroclinic-like entanglement between the complexihed stable manifold and the initial time plane t occurs. 

Integrating the energy gain equation (17) by using the lowest-order approximation, which is essentially the same as the Melnikov method, gives the... 

As mentioned in the previous subsection, the adiabatic solution (34) together with the Melnikov method enables us to prove items 1, 2a, 2b, and 3. Then the significant properties of tunneling trajectories and of the branches consisting of them, which are numerically observed, can be explained in terms of the adiabatic approximation associated with the Melnikov method. 

Substituting the estimation of Imtic by the Melnikov method [Eq. (B.ll)] into Eq. (39), we get the characteristic peiturbation strength... 

Item 1 in Section V.A is proved by using Melnikov method which is extended into the complex domain [25]. Suppose that the Hamiltonian is written by... 

Analyses by the Melnikov Method of Damped Parametrically Excited Cross Waves... 

The Melnikov method fails to predict chaos for this dissipative system when a > O.b ... 

S. Fadel, Apphcation of the generalized Melnikov method to weakly damped para.-metrically excited cross waves with snrfaee tension, PhD dissertation, Oregon State University, USA (1998). 

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