Related papers: A study of the double pendulum using polynomial op…
Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to…
This paper presents an approach to deal with safety of dynamical systems in presence of multiple non-convex unsafe sets. While optimal control and model predictive control strategies can be employed in these scenarios, they suffer from high…
We use Lyapunov-like functions and convex optimization to propagate uncertainty in the initial condition of nonlinear systems governed by ordinary differential equations. We consider the full nonlinear dynamics without approximation,…
In this paper we apply the method of Lagrangian descriptors as an indicator to study the chaotic and regular behavior of trajectories in the phase space of the classical double pendulum system. In order to successfully quantify the degree…
The stationary and highly non-stationary resonant dynamics of the harmonically forced pendulum are described in the framework of a semi-inverse procedure combined with the Limiting Phase Trajectory concept. This procedure, implying only…
This paper considers the problem of designing a continuous-time dynamical system that solves a constrained nonlinear optimization problem and makes the feasible set forward invariant and asymptotically stable. The invariance of the feasible…
Accurately modeling and verifying the correct operation of systems interacting in dynamic environments is challenging. By leveraging parametric uncertainty within the model description, one can relax the requirement to describe exactly the…
Consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them. Each subsystem is assumed to be a massive point moving with friction on a compact surface,…
We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable…
We study in this paper the behavior of a periodically driven nonlinear mechanical system. Bifurcation diagrams are found which locate regions of quasiperiodic, periodic and chaotic behavior within the parameter space of the system. We also…
We examine the duality theory for a class of non-convex functions obtained by composing a convex function with a continuous one. Using Fenchel duality, we derive a dual problem that satisfies weak duality under general assumptions. To…
We propose a methodology to address two analysis problems concerning complex systems, namely bounding state functionals of stochastic differential equations (SDEs) and verifying set avoidance of systems described by partial differential…
Optimization is finding the best solution, which mathematically amounts to locating the global minimum of some cost function. Optimization is traditionally automated with digital or quantum computers, each having their limitations and none…
This paper presents a framework for abstracting uncertain or non-polynomial components of dynamical systems using polynomial constraints. This enables the application of polynomial-based analysis tools, such as sum-of-squares programming,…
The double pendulum, a simple system of classical mechanics, is widely studied as an example of, and testbed for, chaotic dynamics. In 2016, Maiti et al. studied a generalization of the simple double pendulum with equal point-masses at…
The double coplanar pendulum is an example of the coexistence of regular and chaotic dynamics for equal energy values but different initial conditions. Regular trajectories predominate for low energies; as the energy is increased, the…
We introduce a two-dimensional discrete-time dynamical system which represents the evolution of an angle and angular velocity. While the angle evolves by a fixed amount in every step, the evolution of the angular velocity is governed by a…
We study stochastic systems characterized by difference inclusions. Such stochastic differential inclusions are defined by set-valued maps involving the current state and stochastic input. For such systems, we investigate the problem of…
Considering uncertainties and disturbances is an important, yet challenging, step in successful decision making. The problem becomes more challenging in safety-constrained environments. In this paper, we propose a robust and safe trajectory…
We investigate a variation of the simple double pendulum in which the two point masses are replaced by square plates. The double square pendulum exhibits richer behavior than the simple double pendulum and provides a convenient…