312 research outputs found
Nonlinear Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments
International audienceTwo principal concepts of nonlinear normal vibrations modes (NNMs), namely the Kauderer–Rosenberg and Shaw–Pierre concepts, are analyzed. Properties of the NNMs and methods of their analysis are presented. NNMs stability and bifurcations are discussed. Combined application of the NNMs and the Rauscher method to analyze forced and parametric vibrations is discussed. Generalization of the NNMs to continuous systems dynamics is also described
Synchronization problems for unidirectional feedback coupled nonlinear systems
In this paper we consider three different synchronization problems consisting
in designing a nonlinear feedback unidirectional coupling term for two
(possibly chaotic) dynamical systems in order to drive the trajectories of one
of them, the slave system, to a reference trajectory or to a prescribed
neighborhood of the reference trajectory of the second dynamical system: the
master system. If the slave system is chaotic then synchronization can be
viewed as the control of chaos; namely the coupling term allows to suppress the
chaotic motion by driving the chaotic system to a prescribed reference
trajectory. Assuming that the entire vector field representing the velocity of
the state can be modified, three different methods to define the nonlinear
feedback synchronizing controller are proposed: one for each of the treated
problems. These methods are based on results from the small parameter
perturbation theory of autonomous systems having a limit cycle, from nonsmooth
analysis and from the singular perturbation theory respectively. Simulations to
illustrate the effectiveness of the obtained results are also presented.Comment: To appear in Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Ana
Contributions to ida-pbc with adaptive control for underactuated mechanical systems
This master thesis is devoted to developing an adaptive control scheme for the well-
known Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC) technique. The main
objective of this adaptive scheme is to asymptotically stabilize a class of Underactuated
Mechanical Systems (UMSs) in the presence of uncertainties (not necessarily matched). This class of
UMSs is characterized by the solvability of the Partial Differential Equation (PDE) resulting from
the IDA-PBC technique. Two propositions are stated in this work to design the adaptive IDA-PBC. One
of the main properties of these propositions is that even though the parameter estimation conver-
gence is not guaranteed, the adaptive IDA-PBC achieves asymptotic stabilization. To illustrate the
effectiveness of these propositions, this work performs simulations of the Inertia Wheel Inverted
Pendulum (IWIP) system, considering a time-dependent input disturbance, a type of physical damping,
i.e., friction (not considered in the standard
IDA-PBC methodology), and parameter uncertainties in the system (e.g., inertia).Tesi
Time Dependent Saddle Node Bifurcation: Breaking Time and the Point of No Return in a Non-Autonomous Model of Critical Transitions
There is a growing awareness that catastrophic phenomena in biology and
medicine can be mathematically represented in terms of saddle-node
bifurcations. In particular, the term `tipping', or critical transition has in
recent years entered the discourse of the general public in relation to
ecology, medicine, and public health. The saddle-node bifurcation and its
associated theory of catastrophe as put forth by Thom and Zeeman has seen
applications in a wide range of fields including molecular biophysics,
mesoscopic physics, and climate science. In this paper, we investigate a simple
model of a non-autonomous system with a time-dependent parameter and
its corresponding `dynamic' (time-dependent) saddle-node bifurcation by the
modern theory of non-autonomous dynamical systems. We show that the actual
point of no return for a system undergoing tipping can be significantly delayed
in comparison to the {\em breaking time} at which the
corresponding autonomous system with a time-independent parameter undergoes a bifurcation. A dimensionless parameter
is introduced, in which is the curvature
of the autonomous saddle-node bifurcation according to parameter ,
which has an initial value of and a constant rate of change . We
find that the breaking time is always less than the actual point
of no return after which the critical transition is irreversible;
specifically, the relation is analytically obtained. For a system with a small , there exists a significant window of opportunity
during which rapid reversal of the environment can save the system from
catastrophe
On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems
For nonlinear time-delay systems, domains of attraction are rarely studied
despite their importance for technological applications. The present paper
provides methodological hints for the determination of an upper bound on the
radius of attraction by numerical means. Thereby, the respective Banach space
for initial functions has to be selected and primary initial functions have to
be chosen. The latter are used in time-forward simulations to determine a first
upper bound on the radius of attraction. Thereafter, this upper bound is
refined by secondary initial functions, which result a posteriori from the
preceding simulations. Additionally, a bifurcation analysis should be
undertaken. This analysis results in a possible improvement of the previous
estimation. An example of a time-delayed swing equation demonstrates the
various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in
'Nonlinear Dynamics'. The final authenticated version is available online at
https://doi.org/10.1007/s11071-020-05620-8
Synchronization between Bidirectional Coupled Nonautonomous Delayed Cohen-Grossberg Neural Networks
Based on using suitable Lyapunov function and the properties of M-matrix, sufficient conditions for complete synchronization of bidirectional coupled nonautonomous Cohen-Grossberg neural networks are obtained. The methods for discussing synchronization avoid complicated error system of Cohen-Grossberg neural networks. Two numerical examples are given to show the effectiveness of the proposed synchronization method
Dynamics of oscillator populations globally coupled with distributed phase shifts
We consider a population of globally coupled oscillators in which phase
shifts with respect to the global force are different. Such a setup appears for
spatially distributed oscillators with different propagation times of the
global forcing signal. In the presence of independent noises and in the
thermodynamic limit, we show that the dynamics can be reduced, for arbitrary
coupling function, to an effective ensemble of units with identical phase
shifts but with a proper renormalization of the order parameters. The same
reduction is shown to be valid, by virtue of an analysis of Ott-Antonsen
equations, for oscillators with a Cauchy distribution of natural frequencies
and with the first harmonics coupling. However, the reduction to an effective
ensemble may fail if the coupling function is complex enough to ensure the
multistability of locked states
Positivity-preserving methods for ordinary differential equations
[EN] Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.The authors thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme "Geometry, compatibility and structure preservation in computational differential equations" when work on this paper was undertaken. This work was supported by EPSRC grant EP/R014604/1. S.B. has been supported by project PID2019-104927GB-C21 (AEI/FEDER, UE).Blanes Zamora, S.; Iserles, A.; Macnamara, S. (2022). Positivity-preserving methods for ordinary differential equations. ESAIM Mathematical Modelling and Numerical Analysis. 56(6):1843-1870. https://doi.org/10.1051/m2an/20220421843187056
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