Oscillations in I/O monotone systems under negative feedback

Oscillatory behavior is a key property of many biological systems. The Small-Gain Theorem (SGT) for input/output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions. One advantage of the use of the monotone SGT technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low-dimensional (in many interesting examples, just one-dimensional) model reduction. This robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters. However, verifying the conditions needed in order to apply the SGT is not always easy. This paper provides an approach to the verification of the needed properties, and illustrates the approach through an application to a classical model of circadian oscillations, as a nontrivial ``case study,'' and also provides a theorem in the converse direction of predicting oscillations when the SGT conditions fail.Comment: Related work can be retrieved from second author's websit

Pattern transition in spacecraft formation flying using bifurcating potential field

Many new and exciting space mission concepts have developed around spacecraft formation flying, allowing for autonomous distributed systems that can be robust, scalable and flexible. This paper considers the development of a new methodology for the control of multiple spacecraft. Based on the artificial potential function method, research in this area is extended by considering the new approach of using bifurcation theory as a means of controlling the transition between different formations. For real, safety or mission critical applications it is important to ensure that desired behaviours will occur. Through dynamical systems theory, this paper also aims to provide a step in replacing traditional algorithm validation with mathematical proof, supported through simulation. This is achieved by determining the non-linear stability properties of the system, thus proving the existence or not of desired behaviours. Practical considerations such as the issue of actuator saturation and communication limitations are addressed, with the development of a new bounded control law based on bifurcating potential fields providing the key contribution of this paper. To illustrate spacecraft formation flying using the new methodology formation patterns are considered in low-Earth-orbit utilising the Clohessy-Wiltshire relative linearised equations of motion. It is shown that a formation of spacecraft can be driven safely onto equally spaced projected circular orbits, autonomously reconfiguring between them, whilst satisfying constraints made regarding each spacecraft