Limit Cycles of Polynomially Integrable Piecewise Differential Systems
Abstract
:1. Introduction
- (i)
- If , then .
- (ii)
- If , , and there are two positive integers, p and q, such that and , then
- (iii)
- If , and and there are two positive integers p and q such that and , then
2. Proof of Theorem 1
- Subcase (I.1): . Then, , and the variables in system (12) can be separated. Thus, the first integral is
- Subcase (I.2): and . Then,
- Case (II): . Changing variables and to system (3) gives
- Subcase (II.1): , which agrees with case (I) previously discussed. Therefore, the PFI exists.
- Subcase (II.1.1): , for which it is known that one PFI is
- Subcase (II.1.2): and . This subcase coincides with conditions (14) in Subcase (I.2). Therefore, in order to have a PFI , there must be a such that ,
3. Proof of Theorem 2
- (i)
- and in this case
- (ii)
- with and . In this case,
- (ii)
- System (27) has a PFI if and only if
- (ii.1)
- , with ; or
- (ii.2)
- where and , with
- (i)
- if p is even, for all , decreases for and increases outside, with a local minimum at and an inflexion point at ; and,
- (ii)
- if p is odd, if and only if , increases for all , with an inflexion point at and, if , another one at .
- (i)
- When p is even, it is obvious that for all and . From (34)Moreover, the sign of changes at . It is negative before and positive after it, so at this point has a local minimum.Since is equivalent to , then (35) can be rewritten as
- (ii)
- When p is odd, is positive if and only if , i.e., . This implies that if and only if .In order to study the monotonicity, (34) is used:This implies that for all . Since we conclude that for all . So is an increasing function in the whole domain.Finally, we use (35) to study the convexity,As far as for all x, for and , or and . In the first case, if and , then if . In the second case, if and . However, , and since and , then there is no other solution for x where . In conclusion, if and only if . In a similar way, we conclude that if, and only if, .
- Case 3: Only system (27) is Hamiltonian. Again, from Proposition 3, in system (27) and in system (26) and with . In this case, every limit cycle of the DPwLS (26) and (27) crosses at two different points, and , satisfying the system
4. Examples
5. Discussion
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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García, B.; Llibre, J.; Pérez del Río, J.S.; Pérez-González, S. Limit Cycles of Polynomially Integrable Piecewise Differential Systems. Axioms 2023, 12, 342. https://doi.org/10.3390/axioms12040342
García B, Llibre J, Pérez del Río JS, Pérez-González S. Limit Cycles of Polynomially Integrable Piecewise Differential Systems. Axioms. 2023; 12(4):342. https://doi.org/10.3390/axioms12040342
Chicago/Turabian StyleGarcía, Belén, Jaume Llibre, Jesús S. Pérez del Río, and Set Pérez-González. 2023. "Limit Cycles of Polynomially Integrable Piecewise Differential Systems" Axioms 12, no. 4: 342. https://doi.org/10.3390/axioms12040342
APA StyleGarcía, B., Llibre, J., Pérez del Río, J. S., & Pérez-González, S. (2023). Limit Cycles of Polynomially Integrable Piecewise Differential Systems. Axioms, 12(4), 342. https://doi.org/10.3390/axioms12040342