Controlling chaos in low and high dimensional systems with periodic parametric perturbations

Mirus, K. A.; Sprott, J. C.

doi:10.2172/663249

Cited by 2 publications

(8 citation statements)

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“…16, the required γ for periodic behaviour, for the smallest value of , seems to be ∼0.17, with the range of γ values increasing with increasing , though for some values of this value of γ does not give a non-chaotic system. Hence this seems not to be in agreement with the findings of Mirius and Sprott (1999) for the real Lorenz system. The reasons for this are not entirely clear and merit further investigation.…”

Section: Ramping γ For Fixedcontrasting

confidence: 86%

“…From the results of Mirius and Sprott (1999) we would expect this to be at the periods of the UPOs or integer multiples of these. The period one UPO has γ f =0.15533, and the main region of stability for all the values of considered here has period one (as seen in the amplitude bifurcation plots), born from a period-two inverse period-doubling cascade.…”

Section: Ramping γ For Fixedmentioning

confidence: 98%

“…Sätherblom, 1997;Ding et al, 1994). Mirius and Sprott (1999) suggest that these predictors are not always reliable, but that the optimum perturbation frequencies more typically correspond to rational multiples of the frequencies of the unstable periodic orbits (UPOs) embedded in the chaotic attractor. The UPOs can, in principle, be extracted from the time series of any state-space variable and hence no specific model for the system dynamics is required to predict optimum perturbation frequencies.…”

Section: Unstable Periodic Orbitsmentioning

confidence: 99%

“…The UPOs can, in principle, be extracted from the time series of any state-space variable and hence no specific model for the system dynamics is required to predict optimum perturbation frequencies. Mirius and Sprott (1999) apply their method to a variety of systems including the Lorenz equations, and we follow them here (see also Lathrop and Kostelich, 1989). The first step is to identify the UPOs present in the attractor.…”

Section: Unstable Periodic Orbitsmentioning

confidence: 99%

“…In other words there are regions of parameter space in which the results are more sensitive to initial conditions. Mirius and Sprott (1999) comment that differing conditions led to differing specific trajectories in their studies on the Lorenz equations, but not different behaviour. Kociuba and Heckenberg (2002), however, found that for some values of perturbation frequency and amplitude control did not occur for all initial conditions in their studies on the Lorenz equations with complex coefficients.…”

Section: Sensitivity To Initial Conditionsmentioning

confidence: 99%

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Synchronization and chaos control in a periodically forced quasi-geostrophic two-layer model of baroclinic instability

Eccles

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Haine

2006

Nonlin. Processes Geophys.

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Abstract.Cyclic forcing on many timescales is believed to have a significant effect on various quasi-periodic, geophysical phenomena such as El Niño, the Quasi-Biennial Oscillation, and glacial cycles. This variability has been investigated by numerous previous workers, in models ranging from simple energy balance constructions to full general circulation models. We present a numerical study in which periodic forcing is applied to a highly idealised, two-layer, quasi-geostrophic model on a β-plane. The bifurcation structure and (unforced) behaviour of this particular model has been extensively examined by Lovegrove et al. (2001) and Lovegrove et al. (2002). We identify from their work three distinct regimes on which we perform our investigations: a steady, travelling wave regime, a quasi-periodic, modulated wave regime and a chaotic regime. In the travelling wave regime a nonlinear resonance is found. In the periodic regime, Arnol'd tongues, frequency locking and a Devil's staircase is seen for small amplitudes of forcing. As the forcing is increased the Arnol'd tongues undergo a period doubling route to chaos, and for larger forcings still, the parameter space we explored is dominated by either period 1 behaviour or chaotic behaviour. In the chaotic regime we extract unstable periodic orbits (UPOs) and add the periodic forcing at periods corresponding to integer multiples of the UPO periods. We find regions of synchronization, similar to Arnol'd tongue behaviour but more skewed and centred approximately on these periods. The regions where chaos suppression took place are smaller than the synchronization regions, and are contained within them.

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Section: Ramping γ For Fixedcontrasting

confidence: 86%

Section: Ramping γ For Fixedmentioning

confidence: 98%

Section: Unstable Periodic Orbitsmentioning

confidence: 99%

Section: Unstable Periodic Orbitsmentioning

confidence: 99%

Section: Sensitivity To Initial Conditionsmentioning

confidence: 99%

See 3 more Smart Citations

Synchronization and chaos control in a periodically forced quasi-geostrophic two-layer model of baroclinic instability

Eccles

Read

Haine

2006

Nonlin. Processes Geophys.

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The Path Towards a Longer Life: On Invariant Sets and the Escape Time Landscape

Bollt

2005

Int. J. Bifurcation Chaos

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Unstable invariant sets are important to understand mechanisms behind many dynamically important phenomenon such as chaotic transients which can be physically relevant in experiments. However, unstable invariant sets are nontrivial to find computationally. Previous techniques such as the PIM triple method [Nusse & Yorke, 1989] and simplex method variant [Moresco & Dawson, 1999], and even the step-and-stagger method [Sweet et al., 2001] have computationally inherent dimension limitations. In the current study, we explicitly investigate the landscape of an invariant set, which leads us to a simple gradient search algorithm to construct points close to the invariant set. While the calculation of the necessary derivatives can be computationally very expensive, the methods of our algorithm are not as dimension dependant as the previous techniques, as we show by examples such as the two-dimensional instability example from [Sweet et al., 2001] followed by a four-dimensional instability example, and then a nine-dimensional flow from the Yoshida equations, with a two-dimensional instability.

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Controlling chaos in low and high dimensional systems with periodic parametric perturbations

Cited by 2 publications

References 0 publications

Synchronization and chaos control in a periodically forced quasi-geostrophic two-layer model of baroclinic instability

Synchronization and chaos control in a periodically forced quasi-geostrophic two-layer model of baroclinic instability

The Path Towards a Longer Life: On Invariant Sets and the Escape Time Landscape

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