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We give the first proof of the existence of a renormalisation fixed-point for period-doubling in pairs of maps of two variables lying in the so-called Feigenbaum-Summation (FS) universality class. The first map represents a subsystem that is unimodal with an extremum of degree two. The dynamics of the second map accumulates an integral characteristic of the dynamics of the first, via a particular form of unidirectional coupling. We prove the existence of the corresponding renormalisation fixed point by rigorous computer-assisted means and gain tight rigorous bounds on the associated universal constants. Our work provides the first step in establishing rigorously the picture conjectured by Kuznetsov et al of the birth, from the FS-type fixed point, of the so-called C-type two-cycle via a period doubling in the dynamics of the renormalisation group transformation itself.

Prostate cancer is the fifth most common cause of death from cancer, and the second most common diagnosed cancer in men. In the last few years many mathematical models have been proposed to describe the dynamics of prostate cancer under treatment. So far one of the major challenges has been the development of mathematical models that would represent in vivo conditions and therefore be suitable for clinical applications, while being mathematically treatable. In this paper, we take a step in this direction, by proposing a nonlinear distributed-delay dynamical system that explores neuroendocrine transdifferentiation in human prostate cancer in vivo. Sufficient conditions for the existence and the stability of a tumour-present equilibrium are given, and the occurrence of a Hopf bifurcation is proven for a uniform delay distribution. Numerical simulations are provided to explore differences in behaviour for uniform and exponential delay distributions. The results suggest that the choice of the delay distribution is key in defining the dynamics of the system and in determining the conditions for the onset of oscillations following a switch in the stability of the tumour-present equilibrium.

Using a hybrid cellular automaton with stochastic elements, we investigate the effectiveness of multiple drug therapies on prostate cancer (PCa) growth. The ability of Androgen Deprivation Therapy to reduce PCa growth represents a milestone in prostate cancer treatment, nonetheless most patients eventually become refractory and develop castration-resistant prostate cancer. In recent years, a "second generation" drug called enzalutamide has been used to treat advanced PCa, or patients already exposed to chemotherapy that stopped responding to it. However, tumour resistance to enzalutamide is not well understood, and in this context, preclinical models and in silico experiments (numerical simulations) are key to understanding the mechanisms of resistance and to assessing therapeutic settings that may delay or prevent the onset of resistance. In our mathematical system, we incorporate cell phenotype switching to model the development of increased drug resistance, and consider the effect of the micro-environment dynamics on necrosis and apoptosis of the tumour cells. The therapeutic strategies that we explore include using a single drug (enzalutamide), and drug combinations (enzalutamide and everolimus or cabazitaxel) with different treatment schedules. Our results highlight the effectiveness of alternating therapies, especially alternating enzalutamide and cabazitaxel over a year, and a comparison is made with data taken from TRAMP mice to verify our findings.

Prostate cancer represents the second most common cancer diagnosed in men and the fifth most common cause of death from cancer worldwide. In this paper, we consider a nonlinear mathematical model exploring the role of neuroendocrine transdifferentiation in human prostate cancer cell dynamics. Sufficient conditions are given for both the biological relevance of the model’s solutions and for the existence of its equilibria. By means of a suitable Liapunov functional the global asymptotic stability of the tumour-free equilibrium is proven, and through the use of sensitivity and bifurcation analyses we identify the parameters responsible for the occurrence of Hopf and saddle-node bifurcations. Numerical simulations are provided highlighting the behaviour discovered, and the results are discussed together with possible improvements to the model.

We gain tight rigorous bounds on the renormalization fixed point for period doubling in families of unimodal maps with degree 4 critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearization of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over 320 significant figures.

We investigate the interplay between chaotic and integrable Hamiltonian systems. In detail, a fully connected four-site lattice system associated with the discrete nonlinear Schrodinger equation is studied. On an embedded two-site segment (dimer) of the four-site system (tetramer) the coupling element between its two sites is time-periodically modified by an external driving term rendering the dimer dynamics chaotic, along with delocalisation of initially single-site excitations. Starting from an isolated dimer system the strength of the coupling to the remaining two sites of the tetramer is treated as a control parameter. It is striking that when the dimer interacts globally with the remaining two sites, thus constituting a fully connected tetramer, a non-trivial dependence of the degree of localisation on the strength of the coupling is found. There even exist ranges of optimal coupling strengths for which the driven tetramer dynamics becomes not only regular but also restores complete single-site localisation. We relate the re-establishment of complete localisation with transitions from permanent chaos via regular transients to permanent stable motion on a torus in the higher-dimensional phase space. In conclusion, increasing the dimension of a system can have profound effects on the character of the dynamics in higher-dimensional mixed phase spaces such that even full stabilisation of motion can be accomplished.

We construct a prototypical example of a spatially-open autonomous Hamiltonian system in which localised, but otherwise unbiased, ensembles of initial conditions break spatio-temporal symmetries in the subsequent ensemble dynamics, despite time reversal symmetry of the equations of motion. Together with transient chaos, this provides the mechanism for the occurrence of a current. Transporting trajectories pass through transient chaos and subsequently cross surfaces of no-return, after which they perform solely regular motion so that the current is of continual ballistic nature.

We study the dynamics of particles evolving in a two-dimensional periodic, spatially-symmetric potential landscape. The system is subjected to weak external time-periodic forces rocking the potential in either direction which, inter alia, breaks integrability. In particular, chaotic layers arising around separatrices which connect unstable equilibria constitute a network of interlinked paths along which trajectories can diffuse. Attention is paid to the emergence of directed motion for an ensemble of particles being trapped initially inside one well of the two-dimensional potential landscape. Interestingly, with properly chosen values of the strength, frequencies, and phases, of the external modulation forces, the particle motion can not only be initiated but also targeted in a desired direction. We demonstrate that, although unstable motion in the form of Arnold diffusion is possible, non-biased diffusion of an ensemble of particles is restricted to certain regions in phase space such that the resulting net motion proceeds directedly. This gives evidence that higher-dimensional tori play the role of partial barriers impeding unhindered motion in phase space and, in addition, serve to capture trajectories in transporting channels.

We consider the dynamics of a chain of coupled units evolving in a periodic substrate potential. The chain is initially in a flat state and situated in a potential well. A bias force, acting as a weak driving mechanism, is applied at a single unit of the chain. We study the instigation of directed transport in two types of system: (i) a microcanonical situation associated with deterministic and conservative dynamics and (ii) the Langevin dynamics when the system is in contact with a heat bath. Interestingly, for the deterministic and conservative dynamics the directed transport is drastically enhanced compared with its Langevin counterpart. In particular, in the deterministic and conservative regime a self-organised redistribution of energy triggers huge-sized avalanches yielding ultimately accelerated transport of the chain. In contrast, in the thermally-assisted process between avalanches the chain settles always into a pinned metastable state impeding continual accelerated chain motion.

We study the conservative and deterministic dynamics of two nonlinearly interacting particles evolving in a one-dimensional spatially periodic washboard potential. A weak tilt of the washboard potential is applied biasing one direction for particle transport. However, the tilt vanishes asymptotically in the direction of bias. Moreover, the total energy content is not enough for both particles to be able to escape simultaneously from an initial potential well; to achieve transport the coupled particles need to interact cooperatively. For low coupling strength the two particles remain trapped inside the starting potential well permanently. For increased coupling strength there exists a regime in which one of the particles transfers the majority of its energy to the other one, as a consequence of which the latter escapes from the potential well and the bond between them breaks. Finally, for suitably large couplings, coordinated energy exchange between the particles allows them to achieve escapes -- one particle followed by the other -- from consecutive potential wells resulting in directed collective motion. The key mechanism of transport rectification is based on the asymptotically vanishing tilt causing a symmetry breaking of the non-chaotic fraction of the dynamics in the mixed phase space. That is, after a chaotic transient, only at one of the boundaries of the chaotic layer do resonance islands appear. The settling of trajectories in the ballistic channels associated with transporting islands provides long-range directed transport dynamics of the escaping dimer.

In this paper we use recently developed phase-space transport theory coupled with a so-called classical spectral theorem to develop a dynamically exact and computationally efficient procedure for studying escape from a planetary neighbourhood. The ‘planetary neighbourhood’ is a bounded region of phase space where entrance and escape are only possible by entering or exiting narrow ‘bottlenecks’ created by the influence of a saddle point. The method therefore immediately applies to, for example, the circular restricted three-body problem and Hill's lunar problem (which we use to illustrate the results), but it also applies to more complex, and higher-dimensional, systems possessing the relevant phase-space structure. It is shown how one can efficiently compute the mean passage time through the planetary neighbourhood, the phase-space flux in, and out, of the planetary neighbourhood, the phase-space volume of initial conditions corresponding to trajectories that escape from the planetary neighbourhood, and the fraction of initial conditions in the planetary neighbourhood corresponding to bound trajectories. These quantities are computed for Hill's problem. We study the dependence of the proportions of these quantities on energy and dimensionality (two-dimensional planar and three-dimensional spatial Hill's problem). The methods and quantities presented are of central interest for many celestial and stellar dynamical applications such as, for example, the capture and escape of moons near giant planets, the formation of binaries in the Kuiper belt and the escape of stars from star clusters orbiting about a galaxy.

We examine the problem of extending, in a natural way, order-preserving maps that are defined on the interior of a closed cone K1 (taking values in another closed cone K2) to the whole of K1.We give conditions, in considerable generality (for cones in both finite- and infinite-dimensional spaces), under which a natural extension exists and is continuous. We also give weaker conditions under which the extension is upper semi-continuous.Maps f defined on the interior of the non-negative cone K in RN, which are both homogeneous of degree 1 and order preserving, are non-expanding in the Thompson metric, and hence continuous. As a corollary of our main results, we deduce that all such maps have a homogeneous order-preserving continuous extension to the whole cone. It follows that such an extension must have at least one eigenvector in K – {0}. In the case where the cycle time χ(f) of the original map does not exist, such eigenvectors must lie in ∂K – {0}.We conclude with some discussions and applications to operator-valued means. We also extend our results to an ‘intermediate’ situation, which arises in some important application areas, particularly in the construction of diffusions on certain fractals via maps defined on the interior of cones of Dirichlet forms.

The effective moment of inertia of a CO impurity molecule in ${^{4}\mathrm{He}}_{N}$ and $p\text{\ensuremath{-}}({\mathrm{H}}_{2}{)}_{N}$ solvent clusters initially increases with $N$ but then commences a nonclassical decrease at $N=4$ ($^{4}\mathrm{He}$) or $N=6$ ($p\text{\ensuremath{-}}{\mathrm{H}}_{2}$). This suggests molecule-solvent decoupling and a transition to microscopic superfluidity. However, the quantum decoupling mechanism has not been elucidated. To understand the decoupling mechanism, a one-dimensional model is introduced in which the $^{4}\mathrm{He}$ atoms are confined to a ring. This model captures the physics and shows that decoupling happens primarily because of bosonic solvent-solvent repulsion. Quantum Monte Carlo and basis set calculations suggest that the system can be modeled as a stirred Tonks-Girardeau gas. This allows the $N$-particle time-dependent Schr\"odinger equation to be solved directly. Computations of the integrated particle current reveal a threshold for stirring and current generation, indicative of superfluidity.

We study the escape of a chain of coupled units over the barrier of a metastable potential. It is demonstrated that a very weak external driving field with a suitably chosen frequency suffices to accomplish speedy escape. The latter requires passage through a transition state, the formation of which is triggered by permanent feeding of energy from a phonon background into humps of localized energy and elastic interaction of the arising breather solutions. In fact, cooperativity between the units of the chain entailing coordinated energy transfer is shown to be crucial for enhancing the rate of escape in an extremely effective and low-energy cost way where the effects of entropic localization and breather coalescence conspire.

We consider a system of two coupled particles evolving in a periodic and spatially symmetric potential under the influence of external driving and damping. The particles are driven individually in such a way that in the uncoupled regime, one particle evolves on a chaotic attractor, while the other evolves on regular periodic attractors. Notably only the latter supports coherent particle transport. The influence of the coupling between the particles is explored, and in particular how it relates to the emergence of a directed current. We show that increasing the (weak) coupling strength subdues the current in a process, which in phase-space, is related to a merging crisis of attractors forming one large chaotic attractor in phase-space. Further, we demonstrate that complete current suppression coincides with a chaos-hyperchaos transition.

Given a closed, convex and pointed cone K in R^n, we present a result which infers K-irreducibility of sets of K-quasipositive matrices from strong connectedness of certain bipartite digraphs. The matrix-sets are defined via products, and the main result is relevant to applications in biology and chemistry. Several examples are presented., A few corrections and clarifications have been made

We prove, with the assistance of rigorous computer calculations, that Widom's renormalization fixed point (1983), and any pair in its domain of attraction, has a Holder continuous invariant curve through the origin. We deduce that any complex analytic map attracted by Widom's fixed point has a Siegel disc bounded by a Holder continuous Jordan curve passing through a critical point of the map. It is known empirically that the stable manifold of Widom's fixed point has real codimension two. This would imply that the above holds on an open set of maps having a golden mean Siegel disc.

We explore the scattering of particles evolving in a two-degree-of-freedom Hamiltonian system, in which both degrees of freedom are open. Particles, initially having all kinetic energy, are sent into a so-called ``interaction region,'' where there will be an exchange of energy with particles that are initially at rest. The open nature of both components of this system eliminates any restrictions on which particles can escape from the interaction region. Notably, it is shown that two particles can cooperate in a mutual exchange of energy allowing both particles to escape and travel large distances. It is also shown that this level of cooperation is highly sensitive to the coupling strength between both components of the system. Indeed, large fluctuations of the magnitude and direction of the current are observed for small changes of this coupling parameter. Further, it is seen that current reversals are a prominent feature of this model. Another interesting observation is that even with the presence of chaotic scattering, it is possible that the system, for certain parameter regimes, will express a vanishing current, suggesting that there is a restoration of symmetry which, due to the initial setup, is broken. For an explanation of the different features of particle motion, we relate the phase-space dynamics to the various regimes of particle current.

We consider the damped and driven dynamics of two interacting particles evolving in a symmetric and spatially periodic potential. The latter is exerted to a time-periodic modulation of its inclination. Our interest is twofold: Firstly we deal with the issue of chaotic motion in the higher-dimensional phase space. To this end a homoclinic Melnikov analysis is utilised assuring the presence of transverse homoclinic orbits and homoclinic bifurcations for weak coupling allowing also for the emergence of hyperchaos. In contrast, we also prove that the time evolution of the two coupled particles attains a completely synchronised (chaotic) state for strong enough coupling between them. The resulting `freezing of dimensionality' rules out the occurrence of hyperchaos. Secondly we address coherent collective particle transport provided by regular periodic motion. A subharmonic Melnikov analysis is utilised to investigate persistence of periodic orbits. For directed particle transport mediated by rotating periodic motion we present exact results regarding the collective character of the running solutions entailing the emergence of a current. We show that coordinated energy exchange between the particles takes place in such a manner that they are enabled to overcome - one particle followed by the other - consecutive barriers of the periodic potential resulting in collective directed motion.

We propose a mechanism to rectify charge transport in the semiclassical Holstein model. It is shown that localized initial conditions associated with a polaron solution, in conjunction with static electron on-site potential not having inversion symmetry, constitute minimal prerequisites for the emergence of a directed current in the underlying periodic lattice system. In particular, we demonstrate that for unbiased spatially localized initial conditions (constituted by kicked static polaron states), violation of parity prevents the existence of pairs of counterpropagating trajectories, thus allowing for a directed current despite the time reversibility of the equations of motion. Nevertheless, propagating polaron solutions associated with sets of unbiased localized initial conditions which eventually leave the region of localized initial conditions do not exhibit time reversibility. Since the initial conditions belonging to the corresponding counterpropagating, current-compensating polaron solutions are not contained in the set, this gives rise to the emergence of a current. Occurrence of long-range coherent charge transport is demonstrated.

We propose a minimal model for the emergence of a directed flow in autonomous Hamiltonian systems. It is shown that internal breaking of the spatio-temporal symmetries, via localised initial conditions, that are unbiased with respect to the transporting degree of freedom, and transient chaos conspire to form the physical mechanism for the occurrence of a current. Most importantly, after passage through the transient chaos, trajectories perform solely regular transporting motion so that the resulting current is of continual ballistic nature. This has to be distinguished from the features of transport reported previously for driven Hamiltonian systems with mixed phase space where transport is determined by intermittent behaviour exhibiting power-law decay statistics of the duration of regular ballistic periods.

Using renormalization techniques, we provide rigorous computer-assisted bounds on the Hausdorff dimension of the boundary of Siegel discs. Specifically, for Siegel discs with golden mean rotation number and quadratic critical points we show that the Hausdorff dimension is less than 1.08523. This is done by exploiting a previously found renormalization fixed point and expressing the Siegel disc boundary as the attractor of an associated Iterated Function System.

A procedure is presented for computing the phase space volume of initial conditions for trajectories that escape or ‘‘react’’ from a multidimensional potential well. The procedure combines a phase space transition state theory, which allows one to construct dividing surfaces that are free of local recrossing and that minimize the directional flux, and a classical spectral theorem. The procedure gives the volume of reactive initial conditions in terms of a sum over each entrance channel of the well of the product of the phase space flux across the dividing surface associated with the channel and the mean residence time in the well of trajectories which enter through the channel. This approach is illustrated for HCN isomerization in three dimensions, for which the method is several orders of magnitude more efficient than standard Monte Carlo sampling.

A computational procedure that allows the detection of a new type of high-dimensional chaotic saddle in Hamiltonian systems with three degrees of freedom is presented. The chaotic saddle is associated with a so-called normally hyperbolic invariant manifold (NHIM). The procedure allows to compute appropriate homoclinic orbits to the NHIM from which we can infer the existence a chaotic saddle. NHIMs control the phase space transport across an equilibrium point of saddle-centre-...-centre stability type, which is a fundamental mechanism for chemical reactions, capture and escape, scattering, and, more generally, ``transformation'' in many different areas of physics. Consequently, the presented methods and results are of broad interest. The procedure is illustrated for the spatial Hill's problem which is a well known model in celestial mechanics and which gained much interest e.g. in the study of the formation of binaries in the Kuiper belt., Comment: 12 pages, 6 figures, pdflatex, submitted to JPhysA

The three-dimensional hydrogen cyanide/isocyanide isomerization problem is taken as an example to present a general theory for computing the phase space structures which govern classical reaction dynamics in systems with an arbitrary (finite) number of degrees of freedom. The theory, which is algorithmic in nature, comprises the construction of a dividing surface of minimal flux which is locally a ‘‘surface of no return.’’ The theory also allows for the computation of the global phase space transition pathways that trajectories must follow in order to react. The latter are enclosed by the stable and unstable manifolds of a so-called normally hyperbolic invariant manifold (NHIM). A detailed description of the geometrical structures and the resulting constraints on reaction dynamics is given, with particular emphasis on the three degrees of freedom case. A procedure is given which uses these structures to compute orbits homoclinic to, and heteroclinic between, NHIMs. The role of homoclinic and heteroclinic orbits in global recrossings of dividing surfaces and transport in complex systems is explained. The complete description provided here is inherently one within phase space; it cannot be inferred from a configuration space picture. A complexification of the classical phase space structures to incorporate quantum effects is also discussed. The results presented here call into question certain assumptions routinely made on the global dynamics; this paper provides methods that enable one to understand and quantify the phase space dynamics of reactions without making such assumptions.

It has been thought1,2,3 that the capture of irregular moons—with non-circular orbits—by giant planets occurs by a process in which they are first temporarily trapped by gravity inside the planet's Hill sphere (the region where planetary gravity dominates over solar tides4). The capture of the moons is then made permanent by dissipative energy loss (for example, gas drag3) or planetary growth2. But the observed distributions of orbital inclinations, which now include numerous newly discovered moons5,6,7,8, cannot be explained using current models. Here we show that irregular satellites are captured in a thin spatial region where orbits are chaotic9, and that the resulting orbit is either prograde or retrograde depending on the initial energy. Dissipation then switches these long-lived chaotic orbits10 into nearby regular (non-chaotic) zones from which escape is impossible. The chaotic layer therefore dictates the final inclinations of the captured moons. We confirm this with three-dimensional Monte Carlo simulations that include nebular drag3,4,11, and find good agreement with the observed inclination distributions of irregular moons at Jupiter7 and Saturn8. In particular, Saturn has more prograde irregular moons than Jupiter, which we can explain as a result of the chaotic prograde progenitors being more efficiently swept away from Jupiter by its galilean moons.

We study the autonomous Hamiltonian dynamics of non-interacting particles trapped initially in one well of a symmetric multiple-well washboard potential. The particles interact locally with an anharmonic oscillator acting as an energy deposit. For a range of interaction strengths, the particles gain sufficient energy during a chaotic transient to escape from the well and afterwards settle onto regular (rotational) dynamics. Strikingly, microcanonical ensembles of initial conditions that are unbiased with respect to the washboard coordinate nevertheless give rise to net directed motion. We demonstrate that for unbiased spatially localized initial conditions, violation of parity prevents the existence of pairs of counter-propagating trajectories within the ensemble, despite the time-reversibility symmetry of the equations of motion, allowing for a nonzero current. Recent studies have shown that particle current may be induced in other systems with preserved spatial symmetry by an external periodic but asymmetric driving force, however, averaging over the phase of the latter yields zero current. The system we propose is novel in that averaging over the phase of the oscillator still yields nonzero current. Furthermore, no mixed phase space is required, and chaos is needed only in an initial stage of the dynamics to guide trajectories from the interior of separatrices onto sustained integrable rotational motion.

We present the formal proof of a procedure to compute the phase-space volume of initial conditions for trajectories that, for a constant energy, escape or ‘react’ from a multi-dimensional potential well with one or several exit/entrance channels. The procedure relies on a phase-space formulation of transition state theory. It gives the volume of reactive initial conditions as the sum over the exit/entrance channels where each channel contributes by the product of the phase-space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel. An example is given to demonstrate the computational efficiency of the procedure.

Maps f defined on the interior of the standard non-negative cone K in R N which are both homogeneous of degree 1 and order-preserving arise naturally in the study of certain classes of Discrete Event Systems. Such maps are non-expanding in Thompson's part metric and continuous on the interior of the cone. It follows from more general results presented here that all such maps have a homogeneous orderpreserving continuous extension to the whole cone. It follows that the extension must have at least one eigenvector in K - {0}. In the case where the cycle time X ( f ) of the original map does not exist, such eigenvectors must lie in ∂K - {0}.