Integrability and Limit Cycles via First Integrals
Abstract
:1. Introduction to the Darboux Theory of Integrability
1.1. Polynomial Differential Systems
1.2. First Integrals
1.3. Integrating Factors
1.4. Invariant Algebraic Curves
1.5. Exponential Factors
1.6. The Method of Darboux
- (i)
- The function:
- (ii)
- When , there are , not all zero, satisfying .
- (iii)
- If , all orbits of the differential system (1) are contained in invariant algebraic curves, because the system has a rational first integral.
- (iv)
- (v)
- We defined . Since are invariant algebraic curves with cofactors, and and are exponential factors with cofactors , we have , and . Therefore, the statement (i) follows from the equality:
- (i)
- We obtained that is a first integral of system (9).
- (ii)
- Since the degree of the polynomial cofactors and is at most , we obtained that , the space of all polynomials of of a degree of at most . We observed that the vector space over has a dimension .
- (iii)
- Under the hypotheses of this statement, we applied statement (ii) to the two following subsets of functions formed by the invariant algebraic curves and the exponential factors. In this way, we obtained two linear dependencies between their corresponding cofactors, with which, after some relabeling and linear algebra, we could obtain:
- (iv)
- Since equality is equivalent to the equality:Therefore, statement (iv) was proved.
2. Limit Cycles in Piecewise Differential Systems via First Integrals
2.1. Discontinuous Piecewise Differential Systems
- (a)
- , the Crossing set.
- (b)
- , the Escaping set.
- (c)
- , the Sliding set.
2.2. Limit Cycles of a Piecewise Differential System Formed by a Linear Differential System and a Quadratic Polynomial Differential System Separated by the Straight Line
2.3. Limit Cycles of Piecewise Differential Systems Formed by Three Linear Centers
2.4. Periodic Orbits of a Relay System in
- (a)
- (b)
- For every pair of points and with and , the discontinuous piecewise linear differential system (19) has a periodic orbit γ intersecting in these two points.
- if , each trajectory has one endpoint in ;
- if either or and , each trajectory has one endpoint in ;
2.5. Limit Cycles of a Class of Piecewise Differential Systems Separated by a Parabola
2.6. Piecewise Differential System with a Non-Regular Discontinuity Line
3. Discussion
4. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
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Llibre, J. Integrability and Limit Cycles via First Integrals. Symmetry 2021, 13, 1736. https://doi.org/10.3390/sym13091736
Llibre J. Integrability and Limit Cycles via First Integrals. Symmetry. 2021; 13(9):1736. https://doi.org/10.3390/sym13091736
Chicago/Turabian StyleLlibre, Jaume. 2021. "Integrability and Limit Cycles via First Integrals" Symmetry 13, no. 9: 1736. https://doi.org/10.3390/sym13091736
APA StyleLlibre, J. (2021). Integrability and Limit Cycles via First Integrals. Symmetry, 13(9), 1736. https://doi.org/10.3390/sym13091736