Clustering, chaos, and crisis in a bailout embedding map

Thyagu, N. Nirmal; Gupte, Neelima

doi:10.1103/physreve.76.046218

Cited by 11 publications

(19 citation statements)

References 17 publications

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“…We compare our results with earlier work [12] in which the dynamics of inertial particles on the surface of a fluid is modeled by four-dimensional dissipative bailout embedding maps. Here, the 2D standard map is used as the base flow, leading to the following map equations for the embedded map [12]:…”

Section: The Phase Diagram Of the Two-action Casesupporting

confidence: 50%

“…However, in contrast, the embedded ABC map we have studied does not show intermittency and associated power law behavior in the aerosol regime. Prior to the study of [12], it was believed that only bubbles are capable of breaching the elliptical islands in the phase space and targeting periodic structures. However, both the study of the 2D standard map carried out in [12] as well as the 3D ABC map studied here show that such a dynamics is demonstrated by aerosols as well, at certain values of the dissipation parameter.…”

Section: The Phase Diagram Of the Two-action Casementioning

confidence: 99%

“…Prior to the study of [12], it was believed that only bubbles are capable of breaching the elliptical islands in the phase space and targeting periodic structures. However, both the study of the 2D standard map carried out in [12] as well as the 3D ABC map studied here show that such a dynamics is demonstrated by aerosols as well, at certain values of the dissipation parameter. Therefore, it is clear that the dissipation parameter γ has a crucial role to play in the preferential concentration of inertial particles and in the targeting of periodic structures.…”

Section: The Phase Diagram Of the Two-action Casementioning

confidence: 99%

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Dynamics of impurities in a three-dimensional volume-preserving map

Das¹,

Gupte²

2014

Phys. Rev. E

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We study the dynamics of inertial particles in three-dimensional incompressible maps, as representations of volume-preserving flows. The impurity dynamics has been modeled, in the Lagrangian framework, by a six-dimensional dissipative bailout embedding map. The fluid-parcel dynamics of the base map is embedded in the particle dynamics governed by the map. The base map considered for the present study is the Arnold-Beltrami-Childress (ABC) map. We consider the behavior of the system both in the aerosol regime, where the density of the particle is larger than that of the base flow, as well as the bubble regime, where the particle density is less than that of the base flow. The phase spaces in both the regimes show rich and complex dynamics with three types of dynamical behaviors--chaotic structures, regular orbits, and hyperchaotic regions. In the one-action case, the aerosol regime is found to have periodic attractors for certain values of the dissipation and inertia parameters. For the aerosol regime of the two-action ABC map, an attractor merging and widening crisis is identified using the bifurcation diagram and the spectrum of Lyapunov exponents. After the crisis an attractor with two parts is seen, and trajectories hop between these parts with period 2. The bubble regime of the embedded map shows strong hyperchaotic regions as well as crisis induced intermittency with characteristic times between bursts that scale as a power law behavior as a function of the dissipation parameter. Furthermore, we observe a riddled basin of attraction and unstable dimension variability in the phase space in the bubble regime. The bubble regime in the one-action case shows similar behavior. This study of a simple model of impurity dynamics may shed light upon the transport properties of passive scalars in three-dimensional flows. We also compare our results with those seen earlier in two-dimensional flows.

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Section: The Phase Diagram Of the Two-action Casesupporting

confidence: 50%

Section: The Phase Diagram Of the Two-action Casementioning

confidence: 99%

Section: The Phase Diagram Of the Two-action Casementioning

confidence: 99%

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Dynamics of impurities in a three-dimensional volume-preserving map

Das¹,

Gupte²

2014

Phys. Rev. E

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“…At the crisis, at the critical parameter value γ c = 0.40452 (see ref. [4] for a plot to this accuracy in γ), λ max jumps suddenly as seen in figure 1b).…”

Section: The Crisis and Intermittency In The Aerosol Regimementioning

confidence: 77%

“…The four-dimensional standard embedding map M (x n , y n , δ x (n), δ y (n)) [4] is obtained by substituting the standard map for M(x n ) in eq. (3).…”

Section: The Embedding Mapmentioning

confidence: 99%

Crisis and unstable dimension variability in the bailout embedding map

Thyagu

Gupte

2008

Pramana - J Phys

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The dynamics of inertial particles in 2-d incompressible flows can be modeled by 4-d bailout embedding maps. The density of the inertial particles, relative to the density of the fluid, is a crucial parameter which controls the dynamical behaviour of the particles. We study here the dynamical behaviour of aerosols, i.e. particles heavier than the flow. An attractor widening and merging crisis is seen in the phase space in the aerosol case. Crisis-induced intermittency is seen in the time series and the laminar length distribution of times before bursts give rise to a power law with the exponent β = −1/3. The maximum Lyapunov exponent near the crisis fluctuates around zero indicating unstable dimension variability (UDV) in the system. The presence of unstable dimension variability is confirmed by the behaviour of the probability distributions of the finite time Lyapunov exponents.

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Transport and diffusion in the embedding map

Thyagu

Gupte

2009

Phys. Rev. E

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We study the transport properties of passive inertial particles in two-dimensional (2D) incompressible flows. Here, the particle dynamics is represented by the four-dimensional dissipative embedding map of the 2D area-preserving standard map which models the incompressible flow. The system is a model for impurity dynamics in a fluid and is characterized by two parameters, the inertia parameter alpha and the dissipation parameter gamma . The aerosol regime, where the particles are denser than the fluid, and the bubble regime, where they are less dense than the fluid, correspond to the parameter regimes alpha>1 and alpha<1 , respectively. Earlier studies of this system show a rich phase diagram with dynamical regimes corresponding to periodic orbits, chaotic structures, and mixed regimes. We obtain the statistical characterizers of transport for this system in these dynamical regimes. These are the recurrence time statistics, the diffusion exponent, and the distribution of jump lengths. The recurrence time distribution shows a power-law tail in the dynamical regimes, where there is preferential concentration of particles in sticky regions of the phase space, and an exponential decay in mixing regimes. The diffusion exponent shows behavior of three types-normal, subdiffusive, and superdiffusive, depending on the parameter regimes. Phase diagrams of the system are constructed to differentiate different types of diffusion behavior, as well as the behavior of the absolute drift. We correlate the dynamical regimes seen for the system at different parameter values with the transport properties observed at these regimes and in the behavior of the transients. This system also shows the existence of a crisis and unstable dimension variability at certain parameter values. The signature of the unstable dimension variability is seen in the statistical characterizers of transport. We discuss the implications of our results for realistic systems.

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Clustering, chaos, and crisis in a bailout embedding map

Cited by 11 publications

References 17 publications

Dynamics of impurities in a three-dimensional volume-preserving map

Dynamics of impurities in a three-dimensional volume-preserving map

Crisis and unstable dimension variability in the bailout embedding map

Transport and diffusion in the embedding map

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