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5. Parameter-Dependent Dynamics

Author

Barreira, Luís, Valls, Claudia, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, Barreira, Luís, and Valls, Claudia

Published
2024
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6. The Case of Differential Equations

Author

Barreira, Luís, Valls, Claudia, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, Barreira, Luís, and Valls, Claudia

Published
2024
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7. Resonances and Normal Forms

Author

Barreira, Luís, Valls, Claudia, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, Barreira, Luís, and Valls, Claudia

Published
2024
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8. Stable and Unstable Foliations

Author

Barreira, Luís, Valls, Claudia, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, Barreira, Luís, and Valls, Claudia

Published
2024
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9. Construction of Smooth Conjugacies

Author

Barreira, Luís, Valls, Claudia, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, Barreira, Luís, and Valls, Claudia

Published
2024
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10. Spectra and Examples

Author

Barreira, Luís, Valls, Claudia, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, Barreira, Luís, and Valls, Claudia

Published
2024
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11. Infinite-Dimensional Dynamics

Author

Barreira, Luís, Valls, Claudia, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, Barreira, Luís, and Valls, Claudia

Published
2024
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12. Asymptotic Behavior

Author

Barreira, Luís, Valls, Claudia, Bellomo, Nicola, Series Editor, Benzi, Michele, Series Editor, Jorgensen, Palle, Series Editor, Li, Tatsien, Series Editor, Melnik, Roderick, Series Editor, Scherzer, Otmar, Series Editor, Steinberg, Benjamin, Series Editor, Reichel, Lothar, Series Editor, Tschinkel, Yuri, Series Editor, Yin, George, Series Editor, Zhang, Ping, Series Editor, Barreira, Luís, and Valls, Claudia

Published
2024
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13. Existence of a cylinder foliated by periodic orbits in the generalized Chazy differential equation

Author

Llibre, Jaume, Novaes, Douglas D., and Valls, Claudia

Subjects
Mathematics - Dynamical Systems, 34C23, 34C25, 34C45, 34A36
Abstract

The generalized Chazy differential equation corresponds to the following two-parameter family of differential equations \begin{equation*}\label{gcdeq} \dddot x+|x|^q \ddot x+\dfrac{k |x|^q}{x}\dot x^2=0, \end{equation*} which has its regularity varying with $q$ , a positive integer. Indeed, for $q=1$ it is discontinuous on the straight line $x=0$, whereas for $q$ a positive even integer it is polynomial, and for $q>1$ a positive odd integer it is continuous but not differentiable on the straight line $x=0$. In 1999, the existence of periodic solutions in the generalized Chazy differential equation was numerically observed for $q=2$ and $k=3$. In this paper, we prove analytically the existence of such periodic solutions. Our strategy allows to establish sufficient conditions ensuring that the generalized Chazy differential equation, for $k=q+1$ and any positive integer $q$ , has actually an invariant topological cylinder foliated by periodic solutions in the $(x,\dot x,\ddot x)$-space. In order to set forth the bases of our approach, we start by considering $q=1,2,3$, which are representatives of the different classes of regularity. For an arbitrary positive integer $q$, an algorithm is provided for checking the sufficient conditions for the existence of such an invariant cylinder, which we conjecture that always exists. The algorithm was successfully applied up to $q=100$.

Published
2023
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34. Zero-Hopf bifurcation in the general Van der Pol-Duffing equation

Author

Cândido, Murilo and Valls, Claudia

Subjects
Mathematics - Dynamical Systems, 34C23, 37G15, 34C45, 34C29
Abstract

We study analytically the coexistence of multiple periodic solutions and invariant tori in the general Van der Pol-Duffing oscillator equations. We use several results related to the averaging method in order to analytically obtain our results. We also provide numerical examples for all the analytical results that we provide.

Published
2019

35. Periodic solutions and invariant torus in the R\'ossler System

Author

Cândido, Murilo R., Novaes, Douglas D., and Valls, Claudia

Subjects
Mathematics - Dynamical Systems, 34C23, 37G15, 34C45, 34C29
Abstract

The R\"ossler System is characterized by a three-parameter family of quadratic 3D vector fields. There exist two one-parameter families of R\"ossler Systems exhibiting a zero-Hopf equilibrium. For R\"ossler Systems near to one of these families, we provide generic conditions ensuring the existence of a torus bifurcation. In this case, the torus surrounds a periodic solution that bifurcates from the zero-Hopf equilibrium. For R\"ossler Systems near to the other family, we provide generic conditions for the existence of a periodic solution bifurcating from the zero-Hopf equilibrium. This improves currently known results regarding periodic solutions for such a family. In addition, the stability properties of the periodic solutions and invariant torus are analysed.

Published
2019
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