Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems

Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. Here, we extend these results to periodically or quasi-periodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two dimensional) time-dependent SSMs. A judicious choice of the parametrization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal-form methods.

Application of Power Geometry and Normal Form Methods to the Study of Nonlinear ODEs

This paper describes power transformations of degenerate autonomous polynomial systems of ordinary differential equations which reduce such systems to a non-degenerative form. Example of creating exact first integrals of motion of some planar degenerate system in a closed form is given.

DYNAMIC CONSEQUENCES OF PREY REFUGIA IN A TWO-PREDATOR–ONE-PREY SYSTEM

We consider a two predator and one prey model with Holling type II functional response incorporating a constant prey refuge. Depending upon the constant prey refuge m, which provides a criterion for protecting m of prey from predation, sufficient conditions for stability and global stability of equilibria are obtained. We find the critical value of this refuge parameter m for which the dynamical system undergoes a Hopf bifurcation and then makes use of center manifold theorem and normal form methods to find the direction of the Hopf bifurcation as well as the stability of the resulting limit cycle. The influence of the prey refuge parameter is also investigated at the interior equilibrium. Numerical simulations were carried out to illustrate and support the analytical results.

ON THE REPRESENTATION OF MAPS BY LIE TRASFORMS

The problem of representing a class of maps in a form suited for application of normal form methods is revisited. It is shown that using the methods of Lie series and of Lie transform a normal form algorithm is constructed in a straightforward manner. The examples of the Schröder–Siegel map and of the Chirikov standard map are included, with extension to arbitrary dimension.

Stability of systems with time-periodic delay: application to variable speed boring process

The stability of systems with fluctuating delay is studied. The aim is to demonstrate that, greater depths of cut may be achieved in a boring process, when the speed of the spindle is modulated sinusoidally instead of being kept constant. Since the variation of spindle speed is small and independent of the tool motion, by expanding the delay terms about a finite mean delay and augmenting the system, the time-dependent delay system can be written as a system of non-linear delay equations with fixed delay. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The centre-manifold and normal form methods are then used to obtain an approximate and simpler four-dimensional system. Analysis of this simpler system shows that periodic variations in the delay lead to larger stability boundaries.

HIGH-ORDER DESCRIPTION OF THE DYNAMICS IN FFAGs AND RELATED ACCELERATORS

In this paper, we describe newly developed tools for the study and analysis of the dynamics in FFAG accelerators based on transfer map methods unique to the code COSY INFINITY. With these new tools, closed orbits, transverse amplitude dependencies and dynamic aperture are determined inclusive of full nonlinear fields and kinematics to arbitrary order. The dynamics are studied at discrete energies, via a high-order energy-dependent transfer map. The order-dependent convergence in the calculated maps allows precise determination of dynamic aperture and detailed particle dynamics. Using normal form methods, and minimal impact symplectic tracking, amplitude- and energy-dependent tune shifts and resonance strengths are extracted. Optimization by constrained global optimization methods further refine and promote robust machine attributes. Various methods of describing the fields will be presented, including representation of fields in radius-dependent Fourier modes, which include complex magnet edge contours and superimposed fringe fields, as well as the capability to interject calculated or measured field data from a magnet design code or actual components, respectively.