Fixed Point Solution Research Articles

Public debt dynamics under ambiguity by means of iterated function systems on density functions

<p style='text-indent:20px;'>We analyze a purely dynamic model of public debt stabilization under ambiguity. We assume that the debt to GDP ratio is described by a random variable, and thus it can be characterized by investigating the evolution of its density function through iteration function systems on mappings. Ambiguity is associated with parameter uncertainty which requires policymakers to respond to such an additional layer of uncertainty according to their ambiguity attitude. We describe ambiguity attitude through a simple heuristic rule in which policymakers adjust the available vague information (captured by the empirical distribution of the debt ratio) with a measure of their ignorance (captured by the uniform distribution). We show that such a model generates fractal-type objects that can be characterized as fixed-point solutions of iterated function systems on mappings. Ambiguity is a source of unpredictability in the long run outcome since it introduces some singularity features in the steady state distribution of the debt ratio. However, the presence of some ambiguity aversion removes such unpredictability by smoothing out the singularities in the steady state distribution.</p>

Phase transitions in TGFT: functional renormalization group in the cyclic-melonic potential approximation and equivalence to O(N) models

In the group field theory approach to quantum gravity, continuous spacetime geometry is expected to emerge via phase transition. However, understanding the phase diagram and finding fixed points under the renormalization group flow remains a major challenge. In this work we tackle the issue for a tensorial group field theory using the functional renormalization group method. We derive the flow equation for the effective potential at any order restricting to a subclass of tensorial interactions called cyclic melonic and projecting to a constant field in group space. For a tensor field of rank r on U(1) we explicitly calculate beta functions and find equivalence with those of O(N) models but with an effective dimension flowing from r − 1 to zero. In the r − 1 dimensional regime, the equivalence to O(N) models is modified by a tensor specific flow of the anomalous dimension with the consequence that the Wilson-Fisher type fixed point solution has two branches. However, due to the flow to dimension zero, fixed points describing a transition between a broken and unbroken phase do not persist and we find universal symmetry restoration. To overcome this limitation, it is necessary to go beyond compact configuration space.

Deciphering chaos in evolutionary games.

A discrete-time replicator map is a prototype of evolutionary selection game dynamical models that have been very successful across disciplines in rendering insights into the attainment of the equilibrium outcomes, like the Nash equilibrium and the evolutionarily stable strategy. By construction, only the fixed-point solutions of the dynamics can possibly be interpreted as the aforementioned game-theoretic solution concepts. Although more complex outcomes like chaos are omnipresent in nature, it is not known to which game-theoretic solutions they correspond. Here, we construct a game-theoretic solution that is realized as the chaotic outcomes in the selection monotone game dynamic. To this end, we invoke the idea that in a population game having two-player-two-strategy one-shot interactions, it is the product of the fitness and the heterogeneity (the probability of finding two individuals playing different strategies in the infinitely large population) that is optimized over the generations of the evolutionary process.

A Code-Agnostic Driver Application for Coupled Neutronics and Thermal-Hydraulic Simulations

While the literature has numerous examples of Monte Carlo (MC) and computational fluid dynamics (CFD) coupling, most are hardwired codes intended primarily for research rather than as stand-alone, general-purpose applications. In this work, we describe an open source application, the Exascale Nuclear Reactor Investigative COde (ENRICO), which enables coupled neutronic and thermal-hydraulic simulations between multiple codes that can be chosen at run time (as opposed to a coupling between two specific codes). The application has been designed such that the control flow logic, domain mapping, nonlinear fixed-point iteration, solution transfers, and convergence checks are all agnostic to the underlying physics solvers used. Special emphasis has also been placed on enabling efficient execution on distributed-memory computing environments. The transfer of solution fields between solvers is performed in memory rather than through filesystem input/output. Additionally, solvers can be configured to run on overlapping or disjoint sets of processes. To date, coupling with the OpenMC and Shift MC codes, the Nek5000 CFD code, and a simplified heat diffusion and subchannel solver has been implemented in ENRICO. We present results for coupled simulations of a single light water reactor fuel assembly based on the NuScale reactor using various combinations of the physics solvers. For this problem, the coupled simulations are shown to converge in about four Picard iterations. A comparison of the heat source and temperature distributions computed by ENRICO using OpenMC coupled with Nek5000 and Shift coupled with Nek5000 illustrates remarkable agreement between the codes.

Maxing Out Entropy: A Conditioning Approach

Entropy and entropy-like measures of pricing kernel dispersion emerge as useful tools in asset pricing research. We develop a systematic approach to bounding entropy by incorporating conditioning information. Our bounds feature a fixed-point solution to a dynamic asset allocation problem. Similar to how the ratio provides variance bound in the L2-space, our solution is interpret-able as generalized Sharpe ratios in the entropy space. Importantly, as opposed to relying on physical moments alone, our bounds strike a balance in exploiting physical return predictability and hedging risk-neutral higher order moments. Applying our approach to recently proposed return predictors, we document enhanced entropy restrictions that more than double the benchmark equity risk premium. We highlight the implications of our results in diagnosing leading macro-finance models.

Orbital angular momentum dynamics of Bose-Einstein condensates trapped in two stacked rings

We investigate the stability and dynamics of the orbital angular momentum modes of a repulsive Bose-Einstein condensate trapped in two tunnel-coupled rings in a stack configuration. Within mean-field theory, we derive a two-state model for the system in the case in which we populate equally both rings with a single orbital angular momentum mode and include small perturbations in other modes. Analyzing the fixed point solutions of the model and the associated classical Hamiltonian, we characterize the destabilization of the stationary states and the subsequent dynamics. By populating a single orbital angular momentum mode with an arbitrary population imbalance between the rings, we derive analytically the boundary between the regimes of Josephson oscillations and macroscopic quantum self-trapping and study numerically the stability of these solutions.

Continuous-wave stability and multipulse structures in a universal complex Ginzburg–Landau model for passively mode-locked lasers with a saturable absorber

The stability of continuous waves and the dynamics of multipulse structures are studied theoretically for a generic model of passively mode-locked fiber lasers with arbitrary nonlinearity. The model is characterized by a complex Ginzburg–Landau equation with a nonlinear term I m / ( 1 + Γ I ) n , where I is the field intensity, m and n are real numbers, and Γ is the saturation power. Fixed-point solutions of the governing equations reveal an interesting loci of singular points in the amplitude-frequency plane consisting of zero, one, or two fixed points, depending on the values of m and n . The continuous-wave stability is analyzed within the framework of the modulational-instability theory, and the results demonstrate a bifurcation in the continuous-wave amplitude growth rate and propagation constant characteristic of multiperiodic wave structures. In the full nonlinear regime, these multiperiodic wave structures turn out to be multipulse trains, unveiled via numerical simulations of the model nonlinear equation, the rich variety of which is highlighted by considering different combinations of values for the pair ( m , n ). The results are consistent with previous analyses of the dynamics of multipulse structures in several contexts of passively mode-locked lasers with a saturable absorber, as well as with predictions about the existence of multipulse structures and bound-state solitons in optical fibers with strong optical nonlinearity, such as cubic-quintic and saturable nonlinearities.

Hardware Implementation of Streaming Logarithm Computing Unit for Fixed-Point Data

The work aims to create hardware implementation of the streaming computing unit for logarithm calculation in fixed-point. Logarithms are widely used in telecommunications, particularly in radio intelligence to convert power spectrum values to decibels for further processing of spectrum data, e.g. for radio signals detection, range-finding, or direction-finding. For spectrum analysis of high sampling rate wideband signals, it is expedient to utilize hardware computing units for streaming logarithm calculation, implemented inside FPGA or ASIC chips. The market offers a large amount of IP cores for logarithm calculation in floating point. Floating-point calculation units offer a high dynamic range but also consume a large number of hardware resources that could diminish the maximum clock frequency of devices. In the proposed work, different approaches for logarithm calculating are considered, including CORDIC, Taylor series, and table-based methods. Authors proposed a mathematical model and architecture of streaming computing unit for base 2 logarithm calculation in fixed-point that can be easily adapted to any other base, simply multiplying the result by a constant. The proposed computing unit utilizes a table-based approach and counting leading zeroes in the argument. Based on the mathematical model, the high-level computational model in MATLAB® Simulink® was created. All the components of the mentioned model are compatible with HDL Coder. The proposed MATLAB® Simulink® model is parameterizable, one can set word and fraction width for input/output data, and memory size for table-based part. Using HDL Coder, Verilog HDL implementation for the proposed logarithm computing unit was synthesized. Utilizing HDL Verifier authors created the testbench in Verilog language for the verification of created computing unit on RTL level of abstraction based on reference data collected during simulation in Simulink. Running generated testbench in ModelSim simulator for one million clock cycles proved that there are no differences in operation between the Simulink Model and generated HDL design. The authors were synthesized HDL implementation of the created computing unit in Quartus Prime for the Stratix IV FPGA chip to evaluate the hardware cost of the proposed solution. The developed logarithm calculation unit was compared to the existing CORDIC, Taylor series, and table-based implementations in terms of calculation error and hardware costs. Additionally, for comparison purposes the authors created a hardware implementation for the base 2 logarithm calculation unit in a single-precision floating-point. During the evaluation of the calculation error, the double-precision floating-point logarithm computing unit from Simulink was chosen as the source for reference results. The comparison showed that created computing unit provides less calculation error compared to the existing fixed-point solutions, requires fewer hardware resources for implementation and can operate on higher clock frequencies. All the created models and source codes are open for utilization and can be downloaded from GitHub.

The cavity master equation: average and fixed point of the ferromagnetic model in random graphs

The cavity master equation (CME) is a closure scheme to the usual master equation representing the dynamics of discrete variables in continuous time. In this work we explore the CME for a ferromagnetic model in a random graph. We first derive and average equation of the CME that describes the dynamics of mean magnetization of the system. We show that the numerical results compare remarkably well with the Monte Carlo simulations. Then, we show that the stationary state of the CME is well described by BP-like equations (independently of the dynamic rules that let the system toward the stationary state). These equations may be rewritten exactly as the fixed point solutions of the cavity equation if one also assumes that the stationary state is well described by a Boltzmann distribution.

Invariant density of intermittent nonlinear maps descriptive of coherent quantum transport through disorderless lattices

Weakly chaotic attractors of an intermittent map defined on the complex unit circle arise in an analogy with a recurrence relation of the scattering matrix associated with wave transport through locally periodic structures of consecutive sizes. It is demonstrated that the fixed-point solution (infinite iteration time or scattering structure size) of the relation corresponds to an average of the scattering matrix over a set, or “ensemble”, of systems of all sizes. This ergodic property implies the analyticity of the scattering matrix S and the existence of its “ensemble” average 〈S〉, called the optical S-matrix. We find that the invariant density of the map that governs the sample-to-sample fluctuations of coherent transport is given by the Poisson kernel of potential theory, and consequently the distribution of S is uniquely determined by 〈S〉 which depends only on the transport properties of a single scattering cell. The theoretical distribution, closely related to the Cauchy distribution, shows perfect agreement with numerical results for a chain of delta potentials. A consequence of our findings is the a priori knowledge of 〈S〉 without the customary resort to experimental data.

Hysteresis in linearly polarized nonresonantly driven exciton-polariton condensates

It is shown that a condensate of microcavity exciton-polaritons exhibits bistability between its two orthogonal linearly polarized components for varying nonresonant excitation power. A non-Hermitian splitting of these components results in two parity-symmetric fixed-point solutions of opposite polarization, making up the hysteresis branches. These solutions correspond to the lowest threshold linearly polarized mode, which dominates at low pump powers, and a cross-polarized mode, which appears due to nonlinear pinning at higher excitation powers. The spin symmetry of the solutions extends the nonresonant bistable operation of polariton condensates to the entire equatorial plane of the Poincare sphere. The results pave the way towards development of tailored bistable, linearly polarized, coherent sources of light.

Stability of Spin–Torque Oscillators With Dual Free Layers

A system of differential equations describing the stability of a spin–torque oscillator (STO) with two free layers is introduced. The system is derived from the Landau–Lifshitz–Gilbert (LLG) equation for two freely rotating magnetic moments interacting via spin–torque. In the case of two free magnetic layers, the magnetization of each layer can precess at its own frequency, which is directly proportional to the local effective field in the layer. The spin–torque depends on the relative angle between two layers. Therefore, if the magnetization in two magnetic layers rotates with different frequencies, the spin–torque will vary in time. This can lead to instabilities of the precession angle in both the layers. When two layers oscillate at equal frequencies, instabilities due to spin–torque variations are eliminated. The requirement for the synchronization leads to a system of three differential equations for three unknowns. The stable oscillation orbits for each layer are obtained by finding fixed point solutions of the system. The results were confirmed using finite-element method (FEM) micromagnetic simulations.

The poor man’s magnetohydrodynamic (PMMHD) equations

We present a mathematical derivation of a discrete dynamical system by following a Fourier-Galerkin approximation of the 3-D incompressible magnetohydrodynamic (MHD) equations. In this way, a 6-D map, depending on 12 bifurcation parameters, is derived as a truncated set of nonlinear ordinary differential equations (ODEs) to characterize incompressible plasma dynamical behaviors, also conserving total energy and cross-helicity in the ideal MHD approximation. Moreover, three different subspaces, associated with long-living non-trivial solutions (e.g., fixed point solutions), have been found like the fluid, magnetic, and the Alfvenic fixed points. Our set can be seen as a Lorenz-like model to investigate MHD phenomena.

Diversity increases the stability of ecosystems.

In 1972, Robert May showed that diversity is detrimental to an ecosystem since, as the number of species increases, the ecosystem is less stable. This is the so-called diversity-stability paradox, which has been derived by considering a mathematical model with linear interactions between the species. Despite being in contradiction with empirical evidence, the diversity-stability paradox has survived the test of time for over 40+ years. In this paper we first show that this paradox is a conclusion driven solely by the linearity of the model employed in its derivation which allows for the neglection of the fixed point solution in the stability analysis. The linear model leads to an ill-posed solution and along with it, its paradoxical stability predictions. We then consider a model ecosystem with nonlinear interactions between species, which leads to a stable ecosystem when the number of species is increased. The saturating non linear term in the species interaction is analogous to a Hill function appearing in systems like gene regulation, neurons, diffusion of information and ecosystems The exact fixed point solution of this model is based on k-core percolation and shows that the paradox disappears. This theoretical result, which is exact and non-perturbative, shows that diversity is beneficial to the ecosystem in agreement with analyzed experimental evidence.

Periodic orbit can be evolutionarily stable: Case Study of discrete replicator dynamics

In evolutionary game theory, it is customary to be partial to the dynamical models possessing fixed points so that they may be understood as the attainment of evolutionary stability, and hence, Nash equilibrium. Any show of periodic or chaotic solution is many a time perceived as a shortcoming of the corresponding game dynamic because (Nash) equilibrium play is supposed to be robust and persistent behaviour, and any other behaviour in nature is deemed transient. Consequently, there is a lack of attempt to connect the non-fixed point solutions with the game theoretic concepts. Here we provide a way to render game theoretic meaning to periodic solutions. To this end, we consider a replicator map that models Darwinian selection mechanism in unstructured infinite-sized population whose individuals reproduce asexually forming non-overlapping generations. This is one of the simplest evolutionary game dynamic that exhibits periodic solutions giving way to chaotic solutions (as parameters related to reproductive fitness change) and also obeys the folk theorems connecting fixed point solutions with Nash equilibrium. Interestingly, we find that a modified Darwinian fitness—termed heterogeneity payoff—in the corresponding population game must be put forward as (conventional) fitness times the probability that two arbitrarily chosen individuals of the population adopt two different strategies. The evolutionary dynamics proceeds as if the individuals optimize the heterogeneity payoff to reach an evolutionarily stable orbit, should it exist. We rigorously prove that a locally asymptotically stable period orbit must be heterogeneity stable orbit—a generalization of evolutionarily stable state.

High Level Design of a Flexible PCA Hardware Accelerator Using a New Block-Streaming Method

Principal Component Analysis (PCA) is a technique for dimensionality reduction that is useful in removing redundant information in data for various applications such as Microwave Imaging (MI) and Hyperspectral Imaging (HI). The computational complexity of PCA has made the hardware acceleration of PCA an active research topic in recent years. Although the hardware design flow can be optimized using High Level Synthesis (HLS) tools, efficient high-performance solutions for complex embedded systems still require careful design. In this paper we propose a flexible PCA hardware accelerator in Field-Programmable Gate Arrays (FPGA) that we designed entirely in HLS. In order to make the internal PCA computations more efficient, a new block-streaming method is also introduced. Several HLS optimization strategies are adopted to create an efficient hardware. The flexibility of our design allows us to use it for different FPGA targets, with flexible input data dimensions, and it also lets us easily switch from a more accurate floating-point implementation to a higher speed fixed-point solution. The results show the efficiency of our design compared to state-of-the-art implementations on GPUs, many-core CPUs, and other FPGA approaches in terms of resource usage, execution time and power consumption.

Abrupt phase transition of epidemic spreading in simplicial complexes

Recent studies on network geometry, a way of describing network structures as geometrical objects, are revolutionizing our way to understand dynamical processes on networked systems. Here, we cope with the problem of epidemic spreading, using the Susceptible-Infected-Susceptible (SIS) model, in simplicial complexes. In particular, we analyze the dynamics of SIS in complex networks characterized by pairwise interactions (links), and three-body interactions (filled triangles, also known as 2-simplices). This higher-order description of the epidemic spreading is analytically formulated using a microscopic Markov chain approximation. The analysis of the fixed point solutions of the model, reveal an interesting phase transition that becomes abrupt with the infectivity parameter of the 2-simplices. Our results pave the way for network theorists to advance in our physical understanding of epidemic spreading in real scenarios where diseases are transmitted among groups as well as among pairs, and to better understand the behaviour of dynamical processes in simplicial complexes.

Gene-mating dynamic evolution theory: fundamental assumptions, exactly solvable models and analytic solutions

Fundamental properties of macroscopic gene-mating dynamic evolutionary systems are investigated. A model is studied to describe a large class of systems within population genetics. We focus on a single locus, any number of alleles in a two-gender dioecious population. Our governing equations are time-dependent continuous differential equations labeled by a set of parameters, where each parameter stands for a population percentage carrying certain common genotypes. The full parameter space consists of all allowed parameters of these genotype frequencies. Our equations are uniquely derived from four fundamental assumptions within any population: (1) a closed system; (2) average-and-random mating process (mean-field behavior); (3) Mendelian inheritance; and (4) exponential growth and exponential death. Even though our equations are nonlinear with time-evolutionary dynamics, we have obtained an exact analytic time-dependent solution and an exactly solvable model. Our findings are summarized from phenomenological and mathematical viewpoints. From the phenomenological viewpoint, any initial parameter of genotype frequencies of a closed system will eventually approach a stable fixed point. Under time evolution, we show (1) the monotonic behavior of genotype frequencies, (2) any genotype or allele that appears in the population will never become extinct, (3) the Hardy-Weinberg law and (4) the global stability without chaos in the parameter space. To demonstrate the experimental evidence for our theory, as an example, we show a mapping from the data of blood type genotype frequencies of world ethnic groups to our stable fixed-point solutions. From the mathematical viewpoint, our highly symmetric governing equations result in continuous global stable equilibrium solutions: these solutions altogether consist of a continuous curved manifold as a subspace of the whole parameter space of genotype frequencies. This fixed-point manifold is a global stable attractor known as the Hardy-Weinberg manifold, attracting any initial point in any Euclidean fiber bounded within the genotype frequency space to the fixed point where this fiber is attached. The stable base manifold and its attached fibers form a fiber bundle, which fills in the whole genotype frequency space completely. We can define the genetic distance of two populations as their geodesic distance on the equilibrium manifold. In addition, the modification of our theory under the process of natural selection and mutation is addressed.

Optimal Measurement Policy for Linear Measurement Systems With Applications to UAV Network Topology Prediction

Dynamic network topology can pose important challenges to communication and control protocols in networks of autonomous vehicles. For instance, maintaining connectivity is a key challenge in unmanned aerial vehicle (UAV) networks. However, tracking and computational resources of the observer module might not be sufficient for constant monitoring of all surrounding nodes in large-scale networks. In this article, we propose an optimal measurement policy for network topology monitoring under constrained resources. To this end, We formulate the localization of multiple objects in terms of linear networked systems and solve it using Kalman filtering with intermittent observation. The proposed policy includes two sequential steps. We first find optimal measurement attempt probabilities for each target using numerical optimization methods to assign the limited number of resources among targets. The optimal resource allocation follows a waterfall-like solution to assign more resources to targets with lower measurement success probability. This provides a 10% to 60% gain in prediction accuracy. The second step is finding optimal on-off patterns for measurement attempts for each target over time. We show that a regular measurement pattern that evenly distributed resources over time outperforms the two extreme cases of using all measurement resources either in the beginning or at the end of the measurement cycle. Our proof is based on characterizing the fixed-point solution of the error covariance matrix for regular patterns. Extensive simulation results confirm the optimality of the most alternating pattern with up to 10-fold prediction improvement for different scenarios. These two guidelines define a general policy for target tracking under constrained resources with applications to network topology prediction of autonomous systems.

Functional renormalization group approach to color superconducting phase transition

We investigate the order of the color superconducting phase transition using the functional renormalization group approach. We analyze the Ginzburg-Landau effective theory of color superconductivity and more generic scalar SU(Nc) gauge theories by calculating the β function of the gauge coupling in arbitrary dimension d based on two different regularization schemes. We find that in d = 3, due to gluon fluctuation effects, the β function never admits an infrared fixed point solution. This indicates that, unlike the ordinary superconducting transition, color superconductivity can only show a first-order phase transition.