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For piecewise-linear maps, the phenomenon that a branch of a one-dimensional unstable manifold of a periodic solution is completely contained in its stable manifold is codimension-two. Unlike codimension-one homoclinic corners, such “subsumed” homoclinic connections can be associated with stable periodic solutions. The purpose of this paper is to determine the dynamics near a generic subsumed homoclinic connection in two dimensions. Assuming the eigenvalues associated with the periodic solution satisfy $0<\left|\lambda \right|<1<\sigma <\frac{1}{\left|\lambda \right|}$, in a two-parameter unfolding there exists an infinite sequence of roughly triangular regions within which the map has a stable single-round periodic solution. The result applies to both discontinuous and continuous maps, although these cases admit different characterizations for the border-collision bifurcations that correspond to boundaries of the regions. The result is illustrated with a discontinuous map of Mira and the two-dimensional border-collision normal form.

A switchable soliton mode-locked and multi-wavelength operation in thulium-doped fiber laser (TDFL) is demonstrated based on nonlinear polarization rotation (NPR) technique. The TDFL produces a soliton pulse operating at 1917.66nm using a 5m long thulium-doped fiber (TDF) as a gain medium as well as nonlinear medium. The solitonic behavior is further identified with two orders of Kelly sidebands in the output spectrum. The mode-locked emission is obtained from threshold pump power of 522–1052mW with consistent pulse repetition rate of 14.7MHz. Maximum pulse energy is calculated as 0.89nJ at pump power of 1052mW, whereas the maximum pulse width is estimated to be approximately 1.36ps corresponding to sech2 pulse profile. By shifting the polarization orientation, the cavity can change to multi-wavelength operation with signal-to-noise ratio (SNR) higher than 53dB. At pump power of 1037mW, three stable peak wavelengths are generated with a power fluctuation and constant spacing of $\pm 1$ dB and 7.4nm, respectively.

Transition metal oxides (TMOs) are distinguished by their thermal and chemical stability as well as excellent optical, electrical, and magnetic properties. Moreover, the 3$d$-orbital electrons of the transition metal ions in TMOs have a strong $d-d$ Coulomb interaction, enabling them to exhibit a variety of unusual physical phenomena, dramatically different from conventional metals and semiconductors. Additionally, TMOs possess excellent nonlinear optical properties, such as third-order nonlinear response and saturation absorption, which make them broadly applicable for ultrafast photonics devices. Herein, a review is presented of the nonlinear optical properties of typical TMOs, as well as their employment as saturable absorbers in near-infrared pulsed fiber lasers in recent years. Finally, some challenges and promising prospects for nonlinear ultrafast photonic devices based on TMOs are discussed.