scholarly journals Breakdown of a 2D Heteroclinic Connection in the Hopf-Zero Singularity (I)

2018 ◽  
Vol 28 (5) ◽  
pp. 1551-1627 ◽  
Author(s):  
I. Baldomá ◽  
O. Castejón ◽  
T. M. Seara
Keyword(s):  
Heteroclinic Connection

Travelling fronts in a food-limited population model with time delay

2002 ◽  
Vol 132 (1) ◽  
pp. 75-89 ◽  
Author(s):  
S. A. Gourley ◽  
M. A. J. Chaplain
Keyword(s):  
Population Model ◽  
Time Delays ◽  
Front Solutions ◽  
Heteroclinic Connection ◽  
Non Local ◽  
Spatial Terms ◽  
Travelling Fronts ◽  
Manifold Theory ◽  
And Diffusion ◽  
Existence Question

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.

scholarly journals Reconfiguring Smart Structures Using Approximate Heteroclinic Connections in a Spring-Mass Model

2015 ◽  
Author(s):  
Jiaying Zhang ◽  
Colin R. McInnes
Keyword(s):  
Optimal Control ◽  
Control Method ◽  
Smart Structures ◽  
Polynomial Method ◽  
Reference Trajectory ◽  
Sensors And Actuators ◽  
Embedded Computing ◽  
Equal Energy ◽  
Optimal Control Method ◽  
Heteroclinic Connection

Several new methods are proposed to reconfigure smart structures with embedded computing, sensors and actuators. These methods are based on heteroclinic connections between equal-energy unstable equilibria in an idealised spring-mass smart structure model. Transitions between equal-energy unstable (but actively controlled) equilibria are considered since in an ideal model zero net energy input is required, compared to transitions between stable equilibria across a potential barrier. Dynamical system theory is used firstly to identify sets of equal-energy unstable configurations in the model, and then to connect them through heteroclinic connection in the phase space numerically. However, it is difficult to obtain such heteroclinic connections numerically in complex dynamical systems, so an optimal control method is investigated to seek transitions between unstable equilibria, which approximate the ideal heteroclinic connection. The optimal control method is verified to be effective through comparison with the results of the exact heteroclinic connection. In addition, we explore the use of polynomials of varying order to approximate the heteroclinic connection, and then develop an inverse method to control the dynamics of the system to track the polynomial reference trajectory. It is found that high order polynomials can provide a good approximation to true heteroclinic connections and provide an efficient means of generating such trajectories. The polynomial method is envisaged as being computationally efficient to form the basis for real-time reconfiguration of real, complex smart structures with embedded computing, sensors and actuators.

Cellular flow in a partially filled rotating drum: regular and chaotic advection

2017 ◽  
Vol 825 ◽  
pp. 631-650 ◽  
Author(s):  
Francesco Romanò ◽  
Arash Hajisharifi ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Chaotic Advection ◽  
Two Dimensional ◽  
Rotating Drum ◽  
Three Dimensional Flow ◽  
Dimensional Flow ◽  
Heteroclinic Connection ◽  
Two Dimensional Flow ◽  
Chaotic Streamlines

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter

1991 ◽  
Vol 2 (2) ◽  
pp. 133-158 ◽  
Author(s):  
John G. Byatt-Smith
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Phase Plane ◽  
Careful Analysis ◽  
Heteroclinic Orbits ◽  
Perturbed System ◽  
Heteroclinic Connection ◽  
Small Dispersion

Low order differential equations typically have solutions which represent homoclinic or heteroclinic orbits between singular points in the phase plane. These orbits occur when the stable manifold of one singular point intersects or coincides with its unstable manifold, or the unstable manifold of another singular point. This paper investigates the persistence of these orbits when small dispersion is added to the system. In the perturbed system the stable manifold of a singular point passes through an exponentially small neighbourhood of a singular point and careful analysis is required to determine whether a homoclinic or heteroclinic connection is achieved.

A Perturbation Procedure for Limit Cycle and Heteroclinic Connection Analysis of Certain Self-Excited Oscillator

2012 ◽  
Vol 204-208 ◽  
pp. 4529-4532
Author(s):  
Yang Yang Chen ◽  
Wei Zhao ◽  
Le Wei Yan
Keyword(s):  
Hopf Bifurcation ◽  
Perturbation Method ◽  
Limit Cycle ◽  
Present Method ◽  
Elliptic Functions ◽  
Hyperbolic Functions ◽  
Perturbation Procedure ◽  
Heteroclinic Connection ◽  
Excited Oscillator

A perturbation procedure, in which the elliptic perturbation method and the hyperbolic perturbation method are applied, is presented for predicting heteroclinic connection of limit cycle or self-excited ocsillator. The limit cycle can be analytically　constructed first by the elliptic perturbation method after Hopf bifurcation, in which the amplitude of limit cycle can be　controlled by the modulus of elliptic functions. The heteroclinic trajectories, which are formed by the heteroclinic connection of limit cycle, can also be constructed by similar perturbation　procedure but adopting the hyperbolic functions instead of　elliptic functions. And the criterion of heteroclinic connection is given in the perturbation procedure. A typical self-excited oscillator is studied in detail to assess the present method.

scholarly journals Heteroclinic connections in plane Couette flow

2009 ◽  
Vol 621 ◽  
pp. 365-376 ◽  
Author(s):  
J. HALCROW ◽  
J. F. GIBSON ◽  
P. CVITANOVIĆ ◽  
D. VISWANATH
Keyword(s):  
Reynolds Number ◽  
Couette Flow ◽  
Invariant Sets ◽  
Navier Stokes ◽  
Lower Branch ◽  
Plane Couette Flow ◽  
Navier Stokes Equation ◽  
Starting Point ◽  
Heteroclinic Connection ◽  
Flow Transitions

Plane Couette flow transitions to turbulence at Re ≈ 325 even though the laminar solution with a linear profile is linearly stable for all Re (Reynolds number). One starting point for understanding this subcritical transition is the existence of invariant sets in the state space of the Navier–Stokes equation, such as upper and lower branch equilibria and periodic and relative periodic solutions, that are distinct from the laminar solution. This article reports several heteroclinic connections between such objects and briefly describes a numerical method for locating heteroclinic connections. We show that the nature of streaks and streamwise rolls can change significantly along a heteroclinic connection.

scholarly journals Koopman mode expansions between simple invariant solutions

2019 ◽  
Vol 879 ◽  
pp. 1-27 ◽  
Author(s):  
Jacob Page ◽  
Rich R. Kerswell
Keyword(s):  
State Space ◽  
Stokes Equations ◽  
Time Window ◽  
Landau Equation ◽  
Dynamic Mode Decomposition ◽  
Navier Stokes ◽  
Dynamic Mode ◽  
Invariant Solutions ◽  
Mode Decomposition ◽  
Heteroclinic Connection

A Koopman decomposition is a powerful method of analysis for fluid flows leading to an apparently linear description of nonlinear dynamics in which the flow is expressed as a superposition of fixed spatial structures with exponential time dependence. Attempting a Koopman decomposition is simple in practice due to a connection with dynamic mode decomposition (DMD). However, there are non-trivial requirements for the Koopman decomposition and DMD to overlap, which mean it is often difficult to establish whether the latter is truly approximating the former. Here, we focus on nonlinear systems containing multiple simple invariant solutions where it is unclear how to construct a consistent Koopman decomposition, or how DMD might be applied to locate these solutions. First, we derive a Koopman decomposition for a heteroclinic connection in a Stuart–Landau equation revealing two possible expansions. The expansions are centred about the two fixed points of the equation and extend beyond their linear subspaces before breaking down at a cross-over point in state space. Well-designed DMD can extract the two expansions provided that the time window does not contain this cross-over point. We then apply DMD to the Navier–Stokes equations near to a heteroclinic connection in low Reynolds number ($Re=O(100)$) plane Couette flow where there are multiple simple invariant solutions beyond the constant shear basic state. This reveals as many different Koopman decompositions as simple invariant solutions present and once more indicates the existence of cross-over points between the expansions in state space. Again, DMD can extract these expansions only if it does not include a cross-over point. Our results suggest that in a dynamical system possessing multiple simple invariant solutions, there are generically places in phase space – plausibly hypersurfaces delineating the boundary of a local Koopman expansion – across which the dynamics cannot be represented by a convergent Koopman expansion.

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