Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition

Dynamical systems containing heteroclinic cycles and networks can be invoked as models of intransitive competition between three or more species. When populations are assumed to be well-mixed, a system of ordinary differential equations (ODEs) describes the interaction model. Spatially extending these equations with diffusion terms creates a system of partial differential equations which captures both the spatial distribution and mobility of species. In one spatial dimension, travelling wave solutions can be observed, which correspond to periodic orbits in ODEs that describe the system in a steady-state travelling frame of reference. These new ODEs also contain a heteroclinic structure. For three species in cyclic competition, the topology of the heteroclinic cycle in the well-mixed model is preserved in the steady-state travelling frame of reference. We demonstrate that with four species, the heteroclinic cycle which exists in the well-mixed system becomes a heteroclinic network in the travelling frame of reference, with additional heteroclinic orbits connecting equilibria not connected in the original cycle. We find new types of travelling waves which are created in symmetry-breaking bifurcations and destroyed in an orbit flip bifurcation with a cycle between only two species. These new cycles explain the existence of "defensive alliances" observed in previous numerical experiments. We further describe the structure of the heteroclitic network for any number of species, and we conjecture how these results may generalise to systems of any arbitrary number of species in cyclic competition

Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control

For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragus. A recent paper by Fiedler et al uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragus-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragus control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.Comment: 21 pages, 8 figures, to appear in PR

Time-delayed feedback control in astrodynamics

In this paper we present time-delayed feedback control (TDFC) for the purpose of autonomously driving trajectories of nonlinear systems into periodic orbits. As the generation of periodic orbits is a major component of many problems in astodynamics we propose this method as a useful tool in such applications. To motivate the use of this method we apply it to a number of well known problems in the astrodynamics literature. Firstly, TDFC is applied to control in the chaotic attitude motion of an asymmetric satellite in an elliptical orbit. Secondly, we apply TDFC to the problem of maintaining a spacecraft in a periodic orbit about a body with large ellipticity (such as an asteroid) and finally, we apply TDFC to eliminate the drift between two satellites in low Earth orbits to ensure their relative motion is bounded

Classification and stability of simple homoclinic cycles in R^5

The paper presents a complete study of simple homoclinic cycles in R^5. We find all symmetry groups Gamma such that a Gamma-equivariant dynamical system in R^5 can possess a simple homoclinic cycle. We introduce a classification of simple homoclinic cycles in R^n based on the action of the system symmetry group. For systems in R^5, we list all classes of simple homoclinic cycles. For each class, we derive necessary and sufficient conditions for asymptotic stability and fragmentary asymptotic stability in terms of eigenvalues of linearisation near the steady state involved in the cycle. For any action of the groups Gamma which can give rise to a simple homoclinic cycle, we list classes to which the respective homoclinic cycles belong, thus determining conditions for asymptotic stability of these cycles.Comment: 34 pp., 4 tables, 30 references. Submitted to Nonlinearit

Spirals and heteroclinic cycles in a spatially extended Rock-Paper-Scissors model of cyclic dominance

Spatially extended versions of the cyclic-dominance Rock-Paper-Scissors model have traveling wave (in one dimension) and spiral (in two dimensions) behaviour. The far field of the spirals behave like traveling waves, which themselves have profiles reminiscent of heteroclinic cycles. We compute numerically a nonlinear dispersion relation between the wavelength and wavespeed of the traveling waves, and, together with insight from heteroclinic bifurcation theory and further numerical results from 2D simulations, we are able to make predictions about the overall structure and stability of spiral waves in 2D cyclic dominance models

An explicit state-space approach to the one-block super-optimal distance problem

An explicit state-space approach is presented for solving the super-optimal Nehari-extension problem. The approach is based on the all-pass dilation technique developed in (Jaimoukha and Limebeer in SIAM J Control Optim 31(5):1115–1134, 1993) which offers considerable advantages compared to traditional methods relying on a diagonalisation procedure via a Schmidt pair of the Hankel operator associated with the problem. As a result, all derivations presented in this work rely only on simple linear-algebraic arguments. Further, when the simple structure of the one-block problem is taken into account, this approach leads to a detailed and complete state-space analysis which clearly illustrates the structure of the optimal solution and allows for the removal of all technical assumptions (minimality, multiplicity of largest Hankel singular value, positive-definiteness of the solutions of certain Riccati equations) made in previous work (Halikias et al. in SIAM J Control Optim 31(4):960–982, 1993; Limebeer et al. in Int J Control 50(6):2431–2466, 1989). The advantages of the approach are illustrated with a numerical example. Finally, the paper presents a short survey of super-optimization, the various techniques developed for its solution and some of its applications in the area of modern robust control

Requirement of Podocalyxin in TGF-Beta Induced Epithelial Mesenchymal Transition

Epithelial mesenchymal transition (EMT) is characterized by the development of mesenchymal properties such as a fibroblast-like morphology with altered cytoskeletal organization and enhanced migratory potential. We report that the expression of podocalyxin (PODXL), a member of the CD34 family, is markedly increased during TGF-β induced EMT. PODXL is enriched on the leading edges of migrating A549 cells. Silencing of podocalyxin expression reduced cell ruffle formation, spreading, migration and affected the expression patterns of several proteins that normally change during EMT (e.g., vimentin, E-cadherin). Cytoskeletion assembly in EMT was also found to be dependent on the production of podocalyin. Compositional analysis of podocalyxin containing immunoprecipitates revealed that collagen type 1 was consistently associated with these isolates. Collagen type 1 was also found to co-localize with podocalyxin on the leading edges of migrating cells. The interactions with collagen may be a critical aspect of podocalyxin function. Podocalyxin is an important regulator of the EMT like process as it regulates the loss of epithelial features and the acquisition of a motile phenotype