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Weekly seminar links: Dynamical Systems and ODEs

##### Dynamical Systems and ODEs

1. Differential Dynamical Systems (WEN Lan, SUN Wenxiang, GAN Shaobo, SHI Yi)

1) Star Flows

Star systems are ones that have robustly no non-hyperbolic critical elements (singularity or periodic orbit). Since structural stability implies the star condition, a main step in studying the stability conjecture is to work on star systems. Hayashi (1992) showed that star diffeomorphisms satisfy Axiom A. However, the Lorenz attractor shows that star flows may not satisfy Axiom A if the system has a singularity. This highlights the significance of the role of singularities for flows.

In [Li-Gan-Wen, DCDS (2005)], Gan, Wen and Li introduced the extended linear Poincaré flow to study flows with singularities. They are able to deal with singularities and regular points simultaneously, for instance for star flows or flows with partial hyperbolicity. Both the final singular-hyperbolicity description of star flows and the solution of Palis weak density conjecture for 3-D flows rely heavily on this technique.

In [Gan-Wen, Invent. Math. (2006)], Gan and Wen proved that every nonsingular star flow satisfies Axiom A and the no-cycle condition. This implies the Omega stability conjecture stating that Omega stability implies Axiom A, and in turn implies the stability conjecture stating that the structural stability implies Axiom A.

They went on to characterize the singular hyperbolicity of star flows [Shi-Gan-Wen, JMD (2014)], which was proven to the best description by the recent example of Bonatti-da Luz.

2) Palis Weak Density Conjecture

A long-standing goal in dynamical systems since S. Smale is to characterize the behavior of most systems. To this end, Palis raised a series of density conjectures, one of which states that every system can be approximated either by Morse-Smale systems or by systems exhibiting Smale’s horseshoe. Gan and Wen proved this density conjecture for 3-D diffeomorphisms in [Bonatti-Gan-Wen, ETDS (2007)]. The method played a key role in the final solution of the conjecture for diffeomorphisms [Crovisier, Ann. of Math. (2) (2010)]. By using the technique of the extended linear Poincaré flow in [Li-Gan-Wen, DCDS (2005)], they proved this density conjecture for 3-D flows in [Gan-Yang, ASENS (to appear)].

3) Entropy and Nonuniformly Hyperbolic Systems

Wenxiang Sun is interested in smooth ergodic theories; in particular, he focuses on the ergodic theory of nonuniformly hyperbolic systems. With his co-authors, he solved several open problems proposed by Bowen, Walters, and Liao.

In [Wang-Sun, Trans. Amer. Math. Soc. (2010)], with his student, he proved that the Lyapunov exponents of hyperbolic measures can be approximated by that of periodic measure. This is a first result on Lyapunov exponents approximation in both uniformly hyperbolic systems and non-uniformly hyperbolic systems.

In [Liang-Liao-Sun, Proc. Amer. Math. Soc. (2014)], with his students, he proved that hyperbolic measures can be approximated by periodic measures.

In [Sun-Todd-Zhou, Trans. Amer. Math. Soc. (2009)], with his co-authors, he constructed equivalent smooth flows such that one has zero entropy and the other has positive entropy. This solves an open problem by Ohno in 1980. The flow constructed in their paper is likely the “black hull” in mathematical version.

In [Sun-Zhang-Zhou, Topology Appl. (2016)], with his students, he constructed equivalent topological flows such that one has zero entropy and infinite growth rate of period and the other has infinite entropy and zero growth rate of period. This shows the extreme degeneracy for entropy and growth rate of period.

In [Liang-Liao-Sun-Tian, Trans. Amer. Math. Soc. (2017)], with his students, he established a variation principle for non-uniformly hyperbolic systems.

2. Differential Equations (LIU Bin, LI Weigu, YANG Jiazhong)

1) Hamiltonian Systems, KAM Theory, and Quasi-periodic Solutions for Differential Equation

In [Liu, J. Differential Equations (2009)], Bin Liu considered a bounded perturbation of a class of forced isochronous oscillators with repulsive singularity. Under a Lazer-Landesman type condition combined with other regular assumptions on the associated potential function, he proved the boundedness of all solutions as well as the existence of infinitely many quasi-periodic solutions.

In [Capietto-Dambrosio-Liu, Z. Angew. Math. Phys. (2009)], with his co-authors, Liu proved all the solutions are bounded for a class Duffing-type equations with some class of singular potentials, which answered a question of Littlewood in the early 1960's for these equations.

In [Jin-Liu-Wang, J. Math. Anal. Appl. (2011)], with his co-authors, Liu proved there exists infinitely many quasiperiodic solutions for a class of coupled Duffing-type equations.

In [Fu-Liu-Mi, Acta Math. Hungar. (2015)], with his co-authors, they showed the exact asymptotic behavior of the unique solution for some singular boundary value problem.

In [Huang-Li-Liu, Nonlinearity (2016)], with his co-authors, they proved there exists infinitely many quasiperiodic solutions for a class of asymmetric oscillators, and all these solutions are bounded.

2) Rotation Numbers, Limit Cycles, Normal Forms and Linearization Theories for Differential Equations

In [Li-Lu, Trans. Amer. Math. Soc. (2008)], Prof. Weigu Li (and K. Lu) introduced a concept of rotation number for lifts of random orientation-preserving homeomorphisms on the circle, not necessarily on [0, 1]. The construction also works for continuous systems generated by random ODE's. They gave conditions under which there is an analytical random conjugacy with a pure rotation given by this rotation number itself.

In [Wu-Li, J. Differential Equations (2008)], he (and H. Wu) proved an analogue of the Poincaré normal form theorem for non-autonomous systems using a homotopy method. In [Li-Llibre-Wu, Ergodic Theory Dynam. Systems (2009)], he (and J. Llibre, H. Wu) studied normal forms for almost periodic differential systems.

In [Li-Llibre-Yang-Zhang, J. Dynam. Differential Equations (2009)], he (and J. Llibre, J. Yang, Z. Zhang) gave a sharp upper bound for the maximum number of limit cycles bifurcating from the period annulus of any homogeneous and quasi-homogeneous center.

In [Li-Lu, Discrete Contin. Dyn. Syst. Ser. B (2016)], he (and K. Lu) studied questions related to smooth linearization of random dynamical systems when a zero Lyapunov exponent exists and proved a version of the Takens Theorem for random diffeomorphisms.

3) Normal Form Theory and Linearization Theory

Jiazhong Yang is interested in normal form theory and linearization theory, esp. for vector fields and diffeomorphisms. His research interests also focus on qualitative analysis of planar polynomial differential systems with emphasis on limit cycles, bifurcations theory, integrabilities and center-focus problems.

In [Li-Liu-Yang, J. Differential Equations (2009)], Li-Liu-Yang studied the number and configuration of limit cycles of cubic system. This is related to the well-known Hilbert’s 16th problem. For more than more century, the topic of limit cycles remains to be one of the most interesting subjects. In this paper, the authors theoretically proved that there exist cubic systems which can have up to 13 limit cycles.

The center-focus problem is one of the most classical problems in o.d.e. To distinguish a focus from a center-like system is extremely difficult. In [Qiu-Yang, J. Differential Equations (2009)], by an explicit construction, Qiu-Yang gave a polynomial system having a linear center of degreen, which can reach as high asn2 of focus order. This result negatively convinced us that integrability problem is far from trivial.

When a polynomial system admit a center region, i.e., its orbits are closed curves in the region, then it is very interesting to consider the critical values of the period function. In [Gasull-Liu-Yang, J. Differential Equations (2010)], Gasull-Liu-Yang showed by giving a concrete example that the period function of polynomial systems can be extremely complicated since the critical values of the system can of be the order ~n2.

In [Liu-Chen-Yang, Nonlinearity (2012)], Liu-Chen-Yang considered the existence of hyperelliptic limit cycles of Lienard systems. The latter has very important applications in various fields and has been widely studied. In this paper, they obtained a complete classification of such systems.

In [Dong-Liu-Yang, Qual. Theory Dyn. Syst. (2015)], Dong-Liu-Yang considered the so-called generalized center focus problem and gave an estimation of highest possible saddle order. This result will have certain positive influence on further study of integrability problem, especially in the category of complex systems.

3. Smooth Ergodic Theories (SUN Wenxiang, LIU Peidong, SHU Lin)

Wenxiang Sun is interested in smooth ergodic theories; in particular, he focuses on the ergodic theory of nonuniformly hyperbolic systems. With his co-authors, he solved several open problems proposed by Bowen, Walters, and Liao.

In [Wang-Sun, Trans. Amer. Math. Soc. (2010)], with his student, they proved that the Lyapunov exponents of hyperbolic measures can be approximated by that of periodic measure. This is a first result on Lyapunov exponents approximation in both uniformly hyperbolic systems and non-uniformly hyperbolic systems.

In [Liang-Liao-Sun, Proc. Amer. Math. Soc. (2014)], with his students, they proved that hyperbolic measures can be approximated by periodic measures.

In [Sun-Todd-Zhou, Trans. Amer. Math. Soc. (2009)], with his co-authors, they constructed equivalent smooth flows such that one has zero entropy and the other has positive entropy. This solves an open problem by Ohno in 1980. The flow constructed in our paper is likely the “black hull” in mathematical version.

In [Sun-Zhang-Zhou, Topology Appl. (2016)], with his students, they constructed equivalent topological flows such that one has zero entropy and infinite growth rate of period and the other has infinite entropy and zero growth rate of period.  This shows the extreme degeneracy for entropy and growth rate of period.

In [Liang-Liao-Sun-Tian, Trans. Amer. Math. Soc. (2017)], with his students, they established a variation principle for non-uniformly hyperbolic systems.

Peidong Liu is interested in entropy formulas and conjectures related to Sinai-Ruelle-Bowen (SRB) measures, which are a class of important measures of physical significance in dynamical systems.

In [Liu, Comm. Math. Phys. (2008)], he gave an equality relating entropy, folding entropy and negative Lyapunov exponents for a non-invertible map of a finite-dimensional manifold and showed that the equality holds if and only if the invariant measure have smooth conditional measures on the stable manifolds. In [Liu-Shu, Nonlinearity (2011)], he and L. Shu investigated the entropy production of a non-invertible dynamical system and showed it is zero if and only if the invariant measure is absolutely continuous with respect to Lebesgue measure.

In [Liu-Lu, Discrete Contin. Dyn. Syst. (2015)], he and K. Lu showed that Shub’s entropy conjecture is true for a partially hyperbolic attractor in a finite dimensional manifold and proved the existence of SRB measures by using random perturbations. In [Lian-Liu-Lu, J. Differential Equations (2016)], he and Z. Lian, K. Lu considered partially hyperbolic attractors of a discrete-time dynamical system in a Hilbert space. They showed the existence of SRB measures and also investigated their ergodic properties.

Dr. Lin Shu is interested in dimension theories and rigidity problems.

Eckamann-Ruelle conjectured that an ergodic measure on a compact Riemannian manifold without boundary, preserved by aC2 endomorphism, is exact dimensional. She confirmed this and gave a new Lyapunov dimension formula for endomorphisms [Shu, Comm. Math. Phys. (2010); Shu, Comm. Math. Phys. (2009)].

In [Ledrappier-Shu, Trans. Amer. Math. Soc. (2014)], she (and F. Ledrappier) used the linear drift and stochastic entropy in the universal cover space to characterize the locally symmetric property of a manifold without focal points. In [Ledrappier-Shu, Ann. Inst. Fourier (Grenoble) (to appear)], they continued to study the differentiability of these quantities under conformal metric changes. In particular, they gave the formula of the differentials and showed the locally symmetric metrics are critical points of the linear drift and entropy.