Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions

Dynamics of the Selkov oscillator

“…We next turn to the case α ≤ α 0 . For this case d'Onofrio proved a result (Proposition 5.3 of his paper) as an application of a Theorem of Hwang and Tsai [9]. It was shown in [1] that one of the conditions in his result is satisfied automatically for α ≤ α 0 , thus showing that the result can be generalized.…”

“…The relation of this statement to that of Theorem 2.1 of [9] is that y has been replaced by −y, the function φ has been taken to be equal to one and the function π(y) equal to y. Under these circumstances conditions (A1), (A2) and (A4) of the theorem of [9] are satisfied automatically and the remaining conditions reduce to (i) and (ii) above. In the case of the Selkov system the interval can be chosen to be (−1, ∞) and condition (i) is satisfied by λ = 0.…”

“…We do so in Theorem 7 below. The following theorem is closely related to Theorem 2.1 of [9]. Theorem 6 Suppose that in the system (32)-(33) defined on the region (r 1 , r 2 )× R the following conditions hold: (i) there exists λ ∈ (r 1 , r 2 ) such that g ′ (λ) > 0 and (x − λ)g(x) > 0 for all…”

On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line

“…Several examples are included to illustrate our main theoretical result. It is known that many planar ecological models and chemical reaction models can be transformed into the Liénard systems, and the uniqueness of limit cycle in the original system can be proved via the uniqueness of limit cycle for the Liénard systems [11,16,25,26]. The uniqueness of limit cycle for the continuous Liénard systems has been proved by many authors including [26,28], and our result here can be regarded as partial extension of these results to discontinuous Liénard systems.…”

“…We apply our result for the discontinuous Liénard systems to prove the uniqueness of crossing limit cycle of a discontinuous Schnakenberg type chemical reaction system, which is a prototypical autocatalytic chemical reaction model [18,24]. The uniqueness of continuous Schnakenberg systems has been proved in [11]. A discontinuous Schnakenberg system naturally appears if the reaction is controlled by the concentration of reacting chemicals, and we show that the limit cycle is preserved under the discontinuity perturbation but it is non-smooth at the intersection points of the limit cycle with the discontinuity line (see Section 4 for the result and Matlab numerical simulation).…”