594 research outputs found
A non-autonomous kind of Duffing equation
Agraïments: The research of AR is supported by the Swedish Research Council (VR Grant 2010/5905). The authors would like to thank the Göran Gustafsson Foundation UU/KTH for financial support.We study the periodic solutions of the following kind of non-autonomous Duffing differential equation y + ay − εy 3 = εh(t, y, y), with a > 0, ε a small parameter and h a C 2 function in its variables
A non-autonomous kind of Duffing equation
Agraïments: The research of AR is supported by the Swedish Research Council (VR Grant 2010/5905). The authors would like to thank the Göran Gustafsson Foundation UU/KTH for financial support.We study the periodic solutions of the following kind of non-autonomous Duffing differential equation y + ay − εy 3 = εh(t, y, y), with a > 0, ε a small parameter and h a C 2 function in its variables
Bifurcation structure of two Coupled Periodically driven double-well Duffing Oscillators
The bifurcation structure of coupled periodically driven double-well Duffing
oscillators is investigated as a function of the strength of the driving force
and its frequency . We first examine the stability of the steady
state in linear response, and classify the different types of bifurcation
likely to occur in this model. We then explore the complex behaviour associated
with these bifurcations numerically. Our results show many striking departures
from the behaviour of coupled driven Duffing Oscillators with single
well-potentials, as characterised by Kozlowski et al \cite{k1}. In addition to
the well known routes to chaos already encountered in a one-dimensional Duffing
oscillator, our model exhibits imbricated period-doubling of both types,
symmetry-breaking, sudden chaos and a great abundance of Hopf bifurcations,
many of which occur more than once for a given driving frequency. We explore
the chaotic behaviour of our model using two indicators, namely Lyapunov
exponents and the power spectrum. Poincar\'e cross-sections and phase portraits
are also plotted to show the manifestation of coexisting periodic and chaotic
attractors including the destruction of tori doubling.Comment: 16 pages, 8 figure
Can we identify non-stationary dynamics of trial-to-trial variability?"
Identifying sources of the apparent variability in non-stationary scenarios is a fundamental problem in many biological data analysis settings. For instance, neurophysiological responses to the same task often vary from each repetition of the same experiment (trial) to the next. The origin and functional role of this observed variability is one of the fundamental questions in neuroscience. The nature of such trial-to-trial dynamics however remains largely elusive to current data analysis approaches. A range of strategies have been proposed in modalities such as electro-encephalography but gaining a fundamental insight into latent sources of trial-to-trial variability in neural recordings is still a major challenge. In this paper, we present a proof-of-concept study to the analysis of trial-to-trial variability dynamics founded on non-autonomous dynamical systems. At this initial stage, we evaluate the capacity of a simple statistic based on the behaviour of trajectories in classification settings, the trajectory coherence, in order to identify trial-to-trial dynamics. First, we derive the conditions leading to observable changes in datasets generated by a compact dynamical system (the Duffing equation). This canonical system plays the role of a ubiquitous model of non-stationary supervised classification problems. Second, we estimate the coherence of class-trajectories in empirically reconstructed space of system states. We show how this analysis can discern variations attributable to non-autonomous deterministic processes from stochastic fluctuations. The analyses are benchmarked using simulated and two different real datasets which have been shown to exhibit attractor dynamics. As an illustrative example, we focused on the analysis of the rat's frontal cortex ensemble dynamics during a decision-making task. Results suggest that, in line with recent hypotheses, rather than internal noise, it is the deterministic trend which most likely underlies the observed trial-to-trial variability. Thus, the empirical tool developed within this study potentially allows us to infer the source of variability in in-vivo neural recordings
Parameter switching in a generalized Duffing system: Finding the stable attractors
This paper presents a simple periodic parameter-switching method which can
find any stable limit cycle that can be numerically approximated in a
generalized Duffing system. In this method, the initial value problem of the
system is numerically integrated and the control parameter is switched
periodically within a chosen set of parameter values. The resulted attractor
matches with the attractor obtained by using the average of the switched
values. The accurate match is verified by phase plots and Hausdorff distance
measure in extensive simulations
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