**Network Model for Credit Risk**

We are developing a reduced-order network model in the Enterprise Risk Architect at Bank of the West, BNP Paribas. The goal is to identify the stages in economic cycle in order to build accurate yet intuitive models for the bank's credit risk. This will enable the institution to precisely manage the risk and, in turn, help the millions of consumers and businesses to achieve a better financial status.

*Work in Progress*

**Hamiltonian Structure of Fish**

The study of a deformable body in fluid has always been a challenging topic. In this work, we investigate the multi-link body submerged in a potential fluid in 3D, which serves as a simplified model of fish swimming. Similar systems have been studied numerically, but rarely analytically. Among those analytical works, Newtonian and Lagrangian formulations are usually employed. However, a Hamiltonian formulation has never been established. The objective of this work is to unveil the Hamiltonian structure of such system.

**Hamiltonian structure and stability of a submerged deformable body**F. Jing, V. Ostrovskyi,

*in preparation*

**Fish Flapping Stability**

Most aquatic vertebrates swim by lateral flapping of their bodies and caudal fins. While much effort has been devoted to understanding the flapping kinematics and its influence on the swimming efficiency, little is known about the stability (or lack of) of periodic swimming. In this paper, we examine the stability of periodic locomotion due to sideways flapping in unbounded potential flow. It is believed that stability limits maneuverability and body designs/flapping motions that are adapted for stable swimming are not suitable for high maneuverability and vice versa. Here, we consider a simplified model where the swimmer is a planar elliptic body undergoing prescribed periodic heaving and pitching. We show that periodic locomotion can be achieved due to the resulting hydrodynamic forces, and its value depends on several parameters including the aspect ratio of the body, the amplitudes and phases of the prescribed flapping. We obtain closed-form solutions for the locomotion and efficiency for small flapping amplitudes, and numerical results for finite flapping amplitudes. We then study the stability of the (finite amplitude flapping) periodic locomotion using Floquet theory. We find that stability depends nonlinearly on all parameters. Interesting trends of switching between stable and unstable motions emerge and evolve as we continuously vary the parameter values. This suggests that, when it comes to live organisms, maneuverability and stability need not be thought of as disjoint properties, rather the organism may manipulate its motion in favor of one or the other depending on the task at hand.

**Stability of periodic locomotion in potential flow**

F. Jing, E. Kanso,* Regul. Chaotic Dyn., *18(**4**), 380-393 (2013), abs, ArXiv

**Snake Optimization**

In this work, we analyze two- and three-link planar snake-like locomotion on a frictional substrate, and optimize the motion for propulsive efficiency. We begin with a mechanical model for snake locomotion that was recently proposed in PNAS, and which uses the fact that snakes have anisotropic friction with the ground due to their scales. In the present work, we identify the frictional coefficients and motions that optimize efficiency in the two-link case, and the motions that optimize efficiency in the three-link case. We find motions which are reminiscent of biological motions such as power and recovery strokes, and the "concertina'' mode (a "contraction-expansion" motion) of snake locomotion.

**Optimization of two- and three-link snake-like locomotion**

F. Jing, S. Alben, *Phys. Rev. E*, **87**, 022711 (2013) abs, ArXiv, pdf

**Vortex Merger**

We revisit the two vortex merger problem (both symmetric and asymmetric) for the Navier-Stokes equations using the core growth model for vorticity evolution coupled with the passive particle field and an appropriately chosen time-dependent rotating reference frame. Using the combined tools of analyzing the topology of the streamline patterns along with careful tracking of passive fields, we highlight the key features of the stages of evolution of vortex merger, pinpointing deficiencies in the low-dimensional model with respect to similar experimental/numerical studies. The model, however, reveals a far richer and delicate sequence of topological bifurcations than has previously been discussed in the literature for this problem, and at the same time points the way towards a method of improving the model.

**Insights into symmetric and asymmetric vortex mergers using the core growth model**

F. Jing, E. Kanso, P.K. Newton, *Phys. Fluids*, **24**, 073101 (2012) abs, ArXiv, pdf

**Fish Gliding Stability**

We examine the stability of the “coast” motion of fish, that is to say, the motion of a neutrally buoyant fish at constant speed in a straight line. The fish motion is said to be unstable if a perturbation in the conditions surrounding the fish results in forces and moments that tend to increase the perturbation and it is stable if these emerging forces tend to reduce the perturbation and return the fish to its original state. Stability may be achieved actively or passively. Passive stabilization requires no energy input by the fish and is dependent upon the fish morphology, i.e. geometry and elastic properties. In this paper, we use a deformable body consisting of an articulated body equipped with torsional springs at its hinge joints and submerged in an unbounded perfect fluid as a simple model to study passive stability as a function of the body geometry and spring stiffness. We show that for given body dimensions, the spring elasticity, when properly chosen, leads to passive stabilization of the (otherwise unstable) coast motion.

**Effects of body elasticity on stability of underwater locomotion**

F. Jing, E. Kanso, *J. Fluid Mech.*, **690**, 461-473 (2012) abs, ArXiv, pdf

**Vortex Model**

We describe the viscous evolution of a collinear three-vortex structure that corresponds initially to an inviscid point vortex fixed equilibrium, with the goal of elucidating some of the main transient dynamical features of the flow. Using a multi-Gaussian “core-growth” type of model, we show that the system immediately begins to rotate unsteadily, a mechanism we attribute to a “viscously induced” instability. We then examine in detail the qualitative and quantitative evolution of the system as it evolves toward the long-time asymptotic Lamb-Oseen state, showing the sequence of topological bifurcations that occur both in a fixed reference frame and in an appropriately chosen rotating reference frame. The evolution of passive particles in this viscously evolving flow is shown and interpreted in relation to these evolving streamline patterns.

**Viscous evolution of point vortex equilibria: The collinear state**

F. Jing, E. Kanso, P.K. Newton, *Phys. Fluids*, **22**, 123102 (2011) abs, ArXiv, pdf

**Hurricane Control**

This paper considers the motion control of a particle and a spinning disk on rotating earth, as model for tracking atmospheric drifters (e.g. hurricane) over certain periods of time. The equations of motion are derived using Lagrangian mechanics. The approximated steady drift motion typically used in geophysical studies on rotating earth is examined, and the conditions for such approximation to be accurate are studied. Trajectory planning is studied as an optimization problem using the method referred to as Discrete Mechanics and Optimal Control (DMOC). Control effort and solvability of the problem are studied for couple of examples.

**Motion control of a spinning disk on rotating earth**

F. Jing, E. Kanso, P.K. Newton, *46th **IEEE Conf. Decision Control Proc.* (2007) abs, ArXiv, pdf

**Satellite**** Control**

In this work, we study the dynamics and feedback control of an orbiting satellite with flexible solar panels. The satellite is modeled in 2D as a circular disc with two flexible panels and two sets of jet propellers attached symmetrically to both sides, and assumed to orbit in a circular trajectory around the Earth. The orientation of the satellite is required to be maintained for the purpose of communication. Dynamics, controllability and observability of the system are studied analytically. A feedback control system of the satellite is designed using LQR control, and the performance is analyzed based on the computational simulation.

**Dynamics and control of satellite with symmetric solar panels**

F. Jing,* B.S. Thesis*, Beijing University (2005) English, Chinese