Iowa State University

MiniSymposium-01: Numerical Methods and Stability of Nonlinear Dynamical Systems

Abstract: Nonlinear differential equations are widely used in science and engineering. They are usually difficult and sometimes impossible to solve analytically. In this minisymposium, the speakers will discuss the several topics regarding nonlinear dynamical systems, including analytical solutions of periodic motions, the method of implicit mapping, stability, bifurcation and chaos, and numerical simulations.


  • Yu Guo, Midwestern State University
  • Bo Yu, University of Wisconsin Platteville

Saturday, October 19th, 2019 at 10:20 - 11:40 (202 Carver Hall)
10:20 - 10:40 Aaron Barrett,
University of Kansas
Investigating Redundancy in Asynchronous Fixed Point Linear Solvers to Reduce the Effects of Delayed Communication

Recent years have seen the proliferation of smart network-connected devices that exist on the ”edge” of large control systems that are capable of distributed calculations. In particular, the power grid has become progressively more complex, especially with the incorporation of distributed energy resources (DER’s). This increase of ”smart” devices results in a new attack surface reinforcing the need to avoid single points of failure that are common in centralized systems. Additionally, these devices also communicate unreliably with the network, meaning that changes in communication should not halt the entire distributed calculation. In order to remove these kinds of vulnerabilities, we need resilient algorithms to implement on decentralized infrastructure networks. This motivates the study of algorithms which can make use of collaborative. In this talk, we present a parallel asynchronous Jacobi iteration where each process is responsible for updating and distributing several components of the solution vector.

10:40 - 11:00 Chuan Guo,
Southern Illinois University Edwardsville
Periodic Motions in an Inverted Pendulum

In this paper, periodic motions in an inversed pendulum system are analytically predicted through a discrete implicit mapping method. The implicit mapping is established via the discretized differential equation. The corresponding stability and bifurcation conditions of the periodic motions are predicted through eigenvalue analysis. Numerical simulation of the periodic motions in the inversed pendulum is completed from analytical predictions.

11:00 - 11:20 Siyu Guo,
Southern Illinois University
Analytical Prediction of Periodic Motions in a Discontinuous Dynamical System

A way to investigate the motions in discontinuous dynamical systems will be introduced. First of all, the G-function criterion will be introduced to determine the flow behavior in the case of hitting on boundaries as background. Then the mapping structure will be discussed. Based on G-function criterion and mapping structures, a set of algebraic equations which presents the periodic motions in the discontinuous system will be given. By implementing Newton-Raphson algorithm to the algebraic equations, analytical prediction can be obtained, which will be illustrated to discuss more details of the dynamical systems characteristics.

11:20 - 11:40 Juergen Kritschgau,
Iowa State University
An Application of Quantum Annealing to the Bike Share Rebalancing Problem

Quantum annealing has been used to find solutions to hard combinatorial optimization problems. In this talk, we consider an application of the D-Wave machine to solve the bike share rebalancing problem. In particular, we compare classical to quantum implementations of clustering heuristics that are used in capacitated vehicle routing problems.

Saturday, October 19th, 2019 at 14:40 - 16:00 (202 Carver Hall)
14:40 - 15:00 Chuanping Liu,
Southern Illinois Edwardsville University
Analytical Dynamics of a Discontinuous System with Hyperbolic Boundary

In this talk, the switchability and periodic motions of a discontinuous dynamic system with hyperbolic boundary are studied. The discontinuous system contains three sub-systems on three different subdomains separated by hyperbolic boundary, and the three sub-systems are three linear systems with under-damping, over-damping and negative damping accordingly. From the theory of discontinuous dynamic systems, the switchability conditions of the discontinuous system at the hyperbolic boundary are developed for motion passability, grazing and slidings on the boundary. The generic mappings from boundary to boundary for specific domains are developed, and the mapping structure of periodic motion is constructed from such generic mappings. Under the switchability conditions, the mapping structure with the Newton-Raphson method generates periodic motions analytically. Based on the analytical prediction, numerical simulations of periodic motions are completed and the corresponding switchability at the boundaries is illustrated. Such studies provide a better understanding of such a discontinuous dynamic system, which helps design and control in engineering applications.

15:00 - 15:20 Weifan Liu,
Syracuse University
Steady-states of Thin Film Droplets on Chemically Heterogeneous Substrates

The spreading and dewetting of thin liquid films subject to van der Waals interactions can be described by a nonlinear fourth-order parabolic PDE. Subject to no-flux boundary condition, the steady-state equation is given by a second order ODE. The bifurcation of steady-state thin films on homogeneous substrates has been previously studied by Bertozzi et al (2001). We extend the previous studies by presenting results on the steady-state thin films on chemically heterogeneous substrates. Specifically, we use phase planes to study the bifurcation of thin films on stepwise-patterned substrates and develop asymptotic approximation for the steady-state solutions. We find a new bifurcation branch of solutions, characterizing droplets pinned at the interface of heterogeneity, which arises as a consequence of wettability contrast of the substrate. In addition, we discuss an effective measure of the fluid leakage for films in presence of an increasing heterogeneity contrast and show that the leakage is inversely proportional to the heterogeneity contrast. Last, we show all of the analysis in 1-D can be easily extended to axisymmetric solutions in 2-D.

15:20 - 15:40 Yeyin Xu,
Southern Illinois University Edwardsville
Analytical Prediction of Period-3 Motion to Chaos in a Nonlinear Rotor System

In this presentation, a bifurcation tree of period-3 motions to chaos in a nonlinear Jeffcott rotor system is predicted analytically through an implicit mapping method. Such bifurcation tree is demonstrated by the route from period-3 motion to period-6 motion. Stable and unstable solutions of periodic motions are achieved from discrete implicit mappings. Stability and bifurcations of periodic motions are determined by the eigenvalue analysis of the resultant Jacobian matrix. Frequency-amplitude characteristics of periodic motions are brought out for a better understanding of the rotor vibration in frequency domain. Numerical simulation is completed for comparison of the numerical and analytical results. Harmonic spectrums of periodic motions are presented. Phase trajectories, displacement orbits and velocity planes are exhibited.

15:40 - 16:00 Bo Yu,
University of Wisconsin Platteville
Periodic Motions in a Single-Degree-of-Freedom System under Both an Aerodynamic Force and a Harmonic Excitation

In this presentation, a semi-analytical approach was used to predict periodic motions in a single-degree-of-freedom system under both aerodynamic force and harmonic excitation. Using the implicit mappings, the predictions of period-1 motions varying with excitation frequency are obtained. Stability of the period-1 motions are discussed, and the corresponding eigenvalues of period-1 motions are presented. Finally, numerical simulations of stable period-1 motions are illustrated.

Coordinated by
Iowa State University