Andrés Mauricio Rivera Acevedo

• 2006-Present. Pontificia Universidad Javeriana, Cali, Colombia
• 2011, Universidad del Zulia, Maracaibo, Venezuela.
• 2006. Universidad del Valle, Cali, Colombia.

Professor Rivera’s research focuses on the use of concepts of mathematical analysis and differential equations in order to understand physical, biological and economic problems. He uses mathematical tools like Perturbation theory, Bifurcation theory, Piecewise linear systems and Global Continuation Methods to predict and explain the complex motion of various dynamical systems, most of them coming from nonlinear oscillators such as satellites (In Celestial Mechanics: Restricted N-body problems), Nonlinear circuits with passive elements (e.g. Memristors, which behave as a resistance with non-linear memory by relate electric charge with magnetic flux). Prof. Rivera’s contributions include the analytic proof of the existence of a family of even and periodic solutions bifurcates from the equilibrium solution in a Generalized Sitnikov (N+1)-body problem (A special case of the restricted (N+1) - body problem) that’s includes the Sitnikov problem. The proof of the existence of at least three limit cycles in discontinuous piecewise linear systems with two zones and a straight line of discontinuity of Saddle-Focus type.  In driven nonlinear oscillators with parametric external force, he obtain a quantification of the interval where the linear stability of periodic solutions obtains as bifurcations of the trivial one is guarantee.

Currently Funded Research

• Qualitative theory of differential equations.
• Dynamical Systems. Discrete, piecewise lineal and continuous systems.
• Nonlinear oscillations of physical, biological and economic models.

• Rivera, Andrés et al. Bifurcation of periodic orbits for the N-body problem, from a non-geometrical family of solutions. Celestial Mechanics and Dynamical Astronomy. (2022) Vol 134 No 1. https://doi.org/10.1007/s10569-022-10062-y
• Rivera, Andrés et al. Existence and Stability of Periodic Solutions of a Shifted Comb-Drive Finger Actuator. Journal of Applied Nonlinear Dynamics. (2022) Vol 11. No 1. pp. 247-269. DOI:10.5890/JAND.2022.03.015.
• Rivera, Andrés et al. A generalized within-host model of dengue infection with a non-constant monocyte production rate. Journal of Biological Dynamics. (2020) Vol 14. No 1. pp. 143-161.
• Rivera, Andrés et al. On the stability of periodic solutions with defined sign in MEMS via lower and upper solutions. Nonlinear Analysis Real World Applications. (2019) Vol 46. pp. 195-218.
• Rivera, Andrés et al. Stability and bifurcations of even periodic orbits in the Sitnikov problem. Celestial Mechanics and Dynamical Astronomy. (2018) https://doi.org/10.1007/s10569-018-9875-z
• Rivera, Andrés et al. Periodic solutions of the Nathanson’s and the Comb-drive Models. International Journal of Non-Linear Mechanics. (2018) Vol 104. pp. 109-115.
• Rivera, Andrés et al. Quantitative stability of certain families of periodic solutions in the Sitnikov problem. Siam Journal on Applied Dynamical Systems. SIAM J. Appl. Dyn. Syst. (2018). Vol. 17 No 1. pp. 52-77.
• Rivera, Andrés et al. Effects of voltage change on the dynamics in a Comb-drive finger of an electrostatic actuator. International Journal of Non-Linear Mechanics. Vol 95. 2017. pp. 224-232.
• Rivera, Andrés et al. Stability of odd periodic solutions in a resonant oscillator. Annali di Matematica Pura ed Applicata. (2016) Vol 1. DOI 10.1007/s10231-016-0580-9.
• Rivera, Andrés. Nuñez, Daniel. Twist periodic solutions in the relativistic driven harmonic oscillator. Abstract and Applied Analysis. (2016) Vol 2. Article ID 6084082. http://dx.doi.org/10.1155/2016/6084082.
• Rivera, Andrés. Nuñez, Daniel. Quantitative Poincare’s Continuation Method for Nonlinear Oscillators. Abstract and Applied Analysis. Vol 2015. Article ID 836312. http://dx.doi.org/10.1155/2015/836312.
• Rivera, Andrés et al. Simultaneous local and global bifurcations in planar piecewise linear Filippov systems. Poster. In Conference Advances in Applied Nonlinear Mathematics. September 2014. Bristol, U.K.
• Rivera, Andrés. Periodic solutions in a Generalized Sitnikov (N+1)-body problem. In Siam Journal on Applied Dynamical Systems. SIAM J. Appl. Dyn. Syst. Vol 12. 2013 No 3. pp. 1515-1540.
• Rivera, Andres and Ortega, Rafael. Global bifurcations from the center of mass in the Sitnikov problem. In Discrete and Continuous Dynamical Systems, (DCDS-Series B) Vol. 14 No2 2011, pp. 719-732.
• Rivera, Andrés. Critical points of solutions to semilinear elliptic problems. In Divulgaciones Matemáticas: ISSN 1315-2068 Vol XII No 1 2012. pp. 3-19.
• Rivera, Andrés, Torres Villarroya, Pedro and Escudero, Carlos. Chemical Oscillations out the Chemical Noise. In Siam Journal on Applied Dynamical Systems. SIAM J. Appl. Dyn. Syst. Vol. 10 2011 pp. 960- 986.
• Rivera, Andrés. De lo discreto a lo continuo en el modelamiento de membranas. Matemáticas: Enseñanza Universitaria, ISSN: 0120-6788, XII, 1, p.3-19, 2004.
• Rivera, Andrés and Berón J. Periodic Oscillations in MEMS under Squeeze Film Damping Force. Journal of Applied Mathematics Vol 2022 Article ID 1498981. https://doi.org/10.1155/2022/1498981
• Rivera, Andrés et. al. Periodic oscillations in the restricted Hip-Hip 2N+1 body problem. 

  arXiv:2210.01740  (2022)

  https://doi.org/10.48550/arXiv.2210.01740 Focus to learn more

Pre-prints en Celestial Mechanics

• Periodic oscillations in a 2N-body problem. Sometido en SIAM Journal of Applied Dynamical System (SIADS) Marzo de 2022. https://www.siam.org/publications/journals/siam-journal-on-applied-dynamical-systems-siads
• Periodic Oscillations in the restricted Hip-Hop 2N+1-body problem. Sometido en Mayo de 2022. Archive for Rational Mechanics and Analysis. https://www.springer.com/journal/205

Pre-prints en Economic Dynamics and Control

• Existence of periodic solutions for an extension of the Paloma model. En preparación para ser sometido en Journal of Macroeconomics.
• Optimal policies in favor of population growth of species under human intervention. En preparación para ser sometido en el Journal of Economic Dynamics and Control.

Pre-prints en osciladores no lineales

• Periodic solutions in time-dependent growth models. Sometido y aceptado en el MAPI 2. https://scm.org.co/mapi-2/ marzo de 2022.
• Periodic oscillations in electrostatic actuators under Time delayed feedback controller . To be submitted.