Stability of Traveling Waves to a Burgers Equation with 2nd-Order Nonlinear Diffusion

We are interested in the study of asymptotic stability for Burgers equation with second-order nonlinear diffusion. We first transform the original equation by the ansatz transformation to establish the existence of traveling wave. We further employ the energy estimate under small perturbation and arbitrary wave amplitude. This energy estimate is then used to establish the stability.

An analytical approach for LQR design for improving damping performance of multi-machine power system

In a multi-machine environment, the inter-area low-frequency oscillations induced due to small perturbation(s) has a significant adverse effect on the maximum limit of power transfer capacity of power system. Conventionally, to address this issue, power systems were equipped with lead-lag power system stabilizers (CPSS) for damping oscillations of low-frequency. In recent years the research was directed towards optimal control theory to design an optimal linear-quadratic-regultor (LQR) for stabilizing power system against the small perturbation(s). The optimal control theory provides a systematic way to design an optimal LQR with sufficient stability margins. Hence, LQR provides an improved level of performance than CPSS over broad-range of operating conditions. The process of designing of optimal LQR involves optimization of associated state (Q) and control (R) weights. This paper presents an analytical approach (AA) to design an optimal LQR by deriving algebraic equations for evaluating optimal elements for weight matrix ‘Q’. The performance of the proposed LQR is studied on an IEEE test system comprising 4-generators and 10-busbars.

Morphing of soft tubes by anisotropic growth

We present a study of smart growth in layered cylindrical structures. We start from the characterization of a compatible growth field in an anisotropic growing tube with the aim to show a small perturbation in the compatible growth field that may produce a controlled deprivation of compatibility and localization of elastic energy storage in a composite structure made up of anisotropic growing tubes.

Fast growth of the number of periodic points arising from heterodimensional connections

We consider $C^{r}$ -diffeomorphisms ( $1 \leq r \leq +\infty$ ) of a compact smooth manifold having two pairs of hyperbolic periodic points of different indices which admit transverse heteroclinic points and are connected through a blender. We prove that, by giving an arbitrarily $C^{r}$ -small perturbation near the periodic points, we can produce a periodic point for which the first return map in the center direction coincides with the identity map up to order $r$ , provided the transverse heteroclinic points satisfy certain natural conditions involving higher derivatives of their transition maps in the center direction. As a consequence, we prove that $C^{r}$ -generic diffeomorphisms in a small neighborhood of the diffeomorphism under consideration exhibit super-exponential growth of number of periodic points. We also give examples which show the necessity of the conditions we assume.

Some Special Types of Orbits around Jupiter

This paper intends to show some special types of orbits around Jupiter based on the mean element theory, including stationary orbits, sun-synchronous orbits, orbits at the critical inclination, and repeating ground track orbits. A gravity model concerning only the perturbations of J2 and J4 terms is used here. Compared with special orbits around the Earth, the orbit dynamics differ greatly: (1) There do not exist longitude drifts on stationary orbits due to non-spherical gravity since only J2 and J4 terms are taken into account in the gravity model. All points on stationary orbits are degenerate equilibrium points. Moreover, the satellite will oscillate in the radial and North-South directions after a sufficiently small perturbation of stationary orbits. (2) The inclinations of sun-synchronous orbits are always bigger than 90 degrees, but smaller than those for satellites around the Earth. (3) The critical inclinations are no-longer independent of the semi-major axis and eccentricity of the orbits. The results show that if the eccentricity is small, the critical inclinations will decrease as the altitudes of orbits increase; if the eccentricity is larger, the critical inclinations will increase as the altitudes of orbits increase. (4) The inclinations of repeating ground track orbits are monotonically increasing rapidly with respect to the altitudes of orbits.

Conservation laws for even order elliptic systems in the critical dimension - a new approach

We consider elliptic systems of order 2m in dimension 2m which are generalizations of extrinsic and intrinsic polyharmonic maps. We show the existence of a conservation law for these systems by using a small perturbation of Uhlenbeck’s gauge fixing matrix.