We consider perturbations of quadratic maps $f_a$ admitting an absolutely continuous invariant probability measure, where $a$ is in a certain positive measure set $\mathcal{A}$ of parameters, and show that in any neighborhood of any such an $f_a$, we find a rich fauna of dynamics. There are maps with periodic attractors as well as non-periodic maps whose critical orbit is absorbed by the continuation of any prescribed hyperbolic repeller of $f_a$. In particular, Misiurewicz maps are dense in $\mathcal{A}$. Almost all maps $f_a$ in the quadratic family is known to possess a unique natural measure, that is, an invariant probability measure $μ_a$ describing the asymptotic distribution of almost all orbits. We discuss weak*-(dis)continuity properties of the map $a\mapsto μ_a$ near the set $\mathcal{A}$, and prove that almost all maps in $\mathcal{A}$ have the property that $μ_a$ can be approximated with measures supported on periodic attractors of certain nearby maps. On the other hand, for any $a \in \mathcal{A}$ and any periodic repeller $Γ_a$ of $f_a$, the singular measure supported on $Γ_a$ can also approximated with measures supported on nearby periodic attractors. It follows that $a\mapsto μ_a$ is not weak*continuous on any full-measure subset of $(0,2]$. Some of these results extend to unimodal families with critical point of higher order, and even to not-too-flat flat topped families.
Chaotic phenomena in generic unfoldings of the Hamilton Hopf bifurcation with emphasis on the restricted planar circular 3-body problem beyond the Gascheau-Routh mass ratio
Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations
研究了中立型微分方程（NFDE）的非共振Hopf-Hopf奇异性.利用形式伴随理论、中心流形定理和RFDE的传统范式方法，建立了一种计算三阶范式的算法。作为一个例子，研究了具有扩展延迟反馈的Van der Pol方程。应用该算法给出了Hopf-Hopf歧点附近的开折结果。通过理论分歧图和数值例子，得到了周期解、拟周期解。Hopf-Hopf分岔图表明混沌吸引子的可能存在，这是由一系列的模拟证实。
Homoclinic tangencies to resonant saddles and discrete Lorenz attractors
研究了一类二次同宿切线到鞍不动点的三参数一般开折中周期轨道的分支。我们应用重新标度技术的第一次返回（庞加莱）的地图，并表明，重新标度的地图可以带来一个地图渐近接近的三维Henon地图$\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$，如已知的，表现出野生双曲洛伦兹吸引子在一些开放的参数域。在此基础上，我们证明了类Lorenz吸引子的无穷级联的存在性。
