Normal Form Equations Research Articles

Small-amplitude synchronization in driven Potts models

We study driven q-state Potts models with thermodynamically consistent dynamics and global coupling. For a wide range of parameters, these models exhibit a dynamical phase transition from decoherent oscillations into a synchronized phase. Starting from a general microscopic dynamics for individual oscillators, we derive the normal form of the high-dimensional Hopf bifurcation that underlies the phase transition. The normal-form equations are exact in the thermodynamic limit and close to the bifurcation. Exploiting the symmetry of the model, we solve these equations and thus uncover the intricate stable synchronization patterns of driven Potts models, characterized by a rich phase diagram. Making use of thermodynamic consistency, we show that synchronization reduces dissipation in such a way that the most stable synchronized states dissipate the least entropy. Close to the phase transition, our findings condense into a linear dissipation-stability relation that connects entropy production with phase-space contraction, a stability measure. At finite system size, our findings suggest a minimum-dissipation principle for driven Potts models that holds arbitrarily far from equilibrium. Published by the American Physical Society 2024

Terms of recurrence sequences in the solution sets of norm form equations

The structure as well as several arithmetic properties of the solution sets of norm form equations are of classical and recent interest. In this paper, we give a finiteness result for terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Our main theorem yields a common generalization of certain recent results from the literature.

Asymptotic solutions of singularly perturbed linear differential-algebraic equations with periodic coefficients

The paper deals with the problem of constructing asymptotic solutions for singular perturbed linear differential-algebraic equations with periodic coefficients. The case of multiple roots of a characteristic equation is studied. It is assumed that the limit pencil of matrices of the system has one eigenvalue of multiplicity n, which corresponds to two finite elementary divisors and two infinite elementary divisors whose multiplicity is greater than 1.A technique for finding the asymptotic solutions is developed and n formal linearly independent solutions are constructed for the corresponding differential-algebraic system. The developed algorithm for constructing formal solutions of the system is a nontrivial generalization of the corresponding algorithm for constructing asymptotic solutions of a singularly perturbed system of differential equations in normal form, which was used in the case of simple roots of the characteristic equation.The modification of the algorithm is based on the equalization method in a special way the coefficients at powers of a small parameter in algebraic systems of equations, from which the coefficients of the formal expansions of the searched solution are found. Asymptotic estimates for the terms of these expansions with respect to a small parameter are also given.For an inhomogeneous differential-algebraic system of equations with periodic coefficients, existence and uniqueness theorems for a periodic solution satisfying some asymptotic estimate are proved, and an algorithm for constructing the corresponding formal solutions of the system is developed. Both critical and non-critical cases are considered.

Gaussian dynamics equation in normal product form

Gaussian dynamics equation in normal product form

Cascade-enhanced transport efficiency of biochemical systems.

Recent developments in nonequilibrium thermodynamics, known as thermodynamic uncertainty relations, limit the system's accuracy by the amount of free-energy consumption. A transport efficiency, which can be used to characterize the capacity to control the fluctuation by means of energy cost, is a direct result of the thermodynamic uncertainty relation. According to our previous research, biochemical systems consume much lower energy cost by noise-induced oscillations to keep almost equal efficiency to maintain precise processes than that by normal oscillations. Here, we demonstrate that the performance of noise-induced oscillations propagating can be further improved through a cascade reaction mechanism. It has been discovered that it is possible to considerably enhance the transport efficiency of the biochemical reactions attained at the terminal cell, allowing the cell to use the cascade reaction mechanism to operate more precisely and efficiently. Moreover, an optimal reaction coupling strength has been predicted to maximize the transport efficiency of the terminal cell, uncovering a concrete design strategy for biochemical systems. By using the local mean field approximation, we have presented an analytical framework by extending the stochastic normal form equation to the system perturbed by external signals, providing an explanation of the optimal coupling strength.

The elliptic sieve and Brauer groups

AbstractA theorem of Serre states that almost all plane conics over have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations.

Modeling frog saccular hair-bundle dynamics using systems of differential equations

Hair cells of the inner ear are specialized cells that detect mechanical motion induced by incoming sound waves and transmit that information to innervating neurons. They are found in the auditory and vestibular systems of all vertebrate species. As these cells exhibit active, energy-consuming limit-cycle oscillations, as well as a highly nonlinear response to mechanical signals, they provide an experimental testing ground for nonlinear dynamics theory. In particular, models based on the normal form equation for the Hopf bifurcation have been shown to explain a number of experimental measurements. The nonlinear response, the amplification of low-amplitude signals, and the presence of spontaneous oscillations are all elegantly captured by this simple equation. Further, experiments have shown that the response plotted over the full physiological range of frequencies and amplitudes displays the classic Arnold Tongue, predicted by theory. In this work, we conducted a comprehensive computational study, with systems of nonlinear equations that describe the cellular process in the hair cell, including mechanotransduction, adaptation, and others, and showed that they produce the Arnold Tongue response, consistent with experiments and dynamical systems theory. The model then allowed us to vary a number of parameters in the model, and deduce the range of validity for this description. Next, we were able to explore the reverse process, to probe how the systems of ion channels in the cell body impacts the hair bundle mechanics. This numerical study was compared to measurements performed under voltage-clamp conditions, which allowed direct manipulation of resting potential of the cell in conjunction with mechanical stimuli. Our numerical model was able to reproduce the data, explore internal cellular processes not currently accessible experimentally, and suggest future experiments.

Algebraic Persistent Fault Analysis of SKINNY_64 Based on S_Box Decomposition

Algebraic persistent fault analysis (APFA), which combines algebraic analysis with persistent fault attacks, brings new challenges to the security of lightweight block ciphers and has received widespread attention since its introduction. Threshold Implementation (TI) is one of the most widely used countermeasures for side channel attacks. Inspired by this method, the SKINNY block cipher adopts the S_box decomposition to reduce the number of variables in the set of algebraic equations and the number of Conjunctive Normal Form (CNF) equations in this paper, thus speeding up the algebraic persistent fault analysis and reducing the number of fault ciphertexts. In our study, we firstly establish algebraic equations for full-round faulty encryption, and then analyze the relationship between the number of fault ciphertexts required and the solving time in different scenarios (decomposed S_boxes and original S_box). By comparing the two sets of experimental results, the success rate and the efficiency of the attack are greatly improved by using S_box decomposition. In this paper, We can recover the master key in a minimum of 2000s using 11 pairs of plaintext and fault ciphertext, while the key recovery cannot be done in effective time using the original S_box expression equations. At the same time, we apply S_box decomposition to another kind of algebraic persistent fault analysis, and the experimental results show that using S_box decomposition can effectively reduce the solving time and solving success rate under the same conditions.

Effective methods for norm-form equations

While effective resolution of Thue equations has been well understood since the work of Baker in the 1960s, similar results for norm-form equations in more than two variables have proven difficult to achieve. In 1983, Vojta was able to address the case of three variables over totally complex and Galois number fields. In this paper, we extend his results to effectively resolve several new classes of norm-form equations. In particular, we completely and effectively settle the question of norm-form equations over totally complex Galois sextic fields.

Effective estimates for the interger solutions of norm form and discriminant form equations

Effective estimates for the interger solutions of norm form and discriminant form equations

Approximation of dissipative systems by elastic chains: Numerical evidence

An old and debated problem in Mechanics concerns the capacity of finite dimensional Lagrangian systems to describe dissipation phenomena. It is true that Helmholtz conditions determine not-always verifiable conditions establishing when a system of n second-order ordinary differential equations in normal form (nODEs) be the Lagrange equations deriving from an nth dimensional Lagrangian. However, it is also true that one could conjecture that, given nODEs it is possible to find a (n+k)th dimensional Lagrangian such that the evolution of suitably chosen n Lagrangian parameters allows for the approximation of the solutions of the nODEs. In fact, while it is well known that the ordinary differential equations (ODEs) usually introduced for describing some dissipation phenomena do not verify Helmholtz conditions, in this paper, we give some preliminary evidence for a positive answer to the conjecture that a dissipative system having n degrees of freedom (DOFs) can be approximated, in a finite time interval and in a suitable norm, by an extended Lagrangian system, having a greater number of DOFs. The theoretical foundation necessary to formulate such a conjecture is here laid and three different examples of extended Lagrangians are shown. Finally, we give some computational results, which encourage to deepen the study of the theoretical aspects of the problem.

A note on pencils of norm-form equations

We find all solutions to the parametrized family of norm-form equations $$ x^3-(t^3-1)y^3+3(t^3-1)xy+(t^3-1)^2 = \pm 1, $$ where $t \gt 1$ is an integer, studied by Amoroso, Masser and Zannier. Our proof relies upon an appeal to lower bounds for linear fo

Norm form equations and linear divisibility sequences

Finding integer solutions to norm form equations is a classical Diophantine problem. Using the units of the associated coefficient ring, we can produce sequences of solutions to these equations. It is known that these solutions can be written as tuples of linear recurrence sequences. We show that for certain families of norm forms defined over quartic fields, there exist integrally equivalent forms making any one fixed coordinate sequence a linear divisibility sequence.

PENCILS OF NORM FORM EQUATIONS AND A CONJECTURE OF THOMAS

We introduce a new method to deal with families of norm form equations. These generalize the Thue equations studied first by Thomas using Baker's Method (which, however, we do not use here). We show that for all large integer values of the parameter t, every solution over Z arises from specializing a solution over Z [ T ] by T = t . The results are completely effective.

Spatial localization beyond steady states in the neighbourhood of the Takens–Bogdanov bifurcation

Abstract Double-zero eigenvalues at a Takens–Bogdanov (TB) bifurcation occur in many physical systems such as double-diffusive convection, binary convection and magnetoconvection. Analysis of the associated normal form, in 1D with periodic boundary condition, shows the existence of steady patterns, standing waves, modulated waves (MW) and travelling waves, and describes the transitions and bifurcations between these states. Values of coefficients of the terms in the normal form classify all possible different bifurcation scenarios in the neighbourhood of the TB bifurcation (Dangelmayr, G. & Knobloch, E. (1987) The Takens–Bogdanov bifurcation with O(2)-symmetry. Phil. Trans. R. Soc. Lond. A, 322, 243-279). In this work we develop a new and simple pattern-forming partial differential equation (PDE) model, based on the Swift–Hohenberg equation, adapted to have the TB normal form at onset. This model allows us to explore the dynamics in a wide range of bifurcation scenarios, including in domains much wider than the lengthscale of the pattern. We identify two bifurcation scenarios in which coexistence between different types of solutions is indicated from the analysis of the normal form equation. In these scenarios, we look for spatially localized solutions by examining pattern formation in wide domains. We are able to recover two types of localized states, that of a localized steady state (LSS) in the background of the trivial state (TS) and that of a spatially localized travelling wave (LTW) in the background of the TS, which have previously been observed in other systems. Additionally, we identify two new types of spatially localized states: that of a LSS in a MW background and that of a LTW in a steady state (SS) background. The PDE model is easy to solve numerically in large domains and so will allow further investigation of pattern formation with a TB bifurcation in one or more dimensions and the exploration of a range of background and foreground pattern combinations beyond SSs.

Normal form approach to the one-dimensional periodic cubic nonlinear Schr\xf6dinger equation in almost critical Fourier-Lebesgue spaces

In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces ℱLp({cal F}{L^p}(mathbb{T})), 1 ≤ p < ∞. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in ℱLp({cal F}{L^p}(mathbb{T})) for 1 leq p leq {3 over 2}.

Five-Wave Resonances in Deep Water Gravity Waves: Integrability, Numerical Simulations and Experiments

In this work we consider the problem of finding the simplest arrangement of resonant deep-water gravity waves in one-dimensional propagation, from three perspectives: Theoretical, numerical and experimental. Theoretically this requires using a normal-form Hamiltonian that focuses on 5-wave resonances. The simplest arrangement is based on a triad of wavevectors K1+K2=K3 (satisfying specific ratios) along with their negatives, corresponding to a scenario of encountering wavepackets, amenable to experiments and numerical simulations. The normal-form equations for these encountering waves in resonance are shown to be non-integrable, but they admit an integrable reduction in a symmetric configuration. Numerical simulations of the governing equations in natural variables using pseudospectral methods require the inclusion of up to 6-wave interactions, which imposes a strong dealiasing cut-off in order to properly resolve the evolving waves. We study the resonance numerically by looking at a target mode in the base triad and showing that the energy transfer to this mode is more efficient when the system is close to satisfying the resonant conditions. We first look at encountering plane waves with base frequencies in the range 1.32–2.35 Hz and steepnesses below 0.1, and show that the time evolution of the target mode’s energy is dramatically changed at the resonance. We then look at a scenario that is closer to experiments: Encountering wavepackets in a 400-m long numerical tank, where the interaction time is reduced with respect to the plane-wave case but the resonance is still observed; by mimicking a probe measurement of surface elevation we obtain efficiencies of up to 10% in frequency space after including near-resonant contributions. Finally, we perform preliminary experiments of encountering wavepackets in a 35-m long tank, which seem to show that the resonance exists physically. The measured efficiencies via probe measurements of surface elevation are relatively small, indicating that a finer search is needed along with longer wave flumes with much larger amplitudes and lower frequency waves. A further analysis of phases generated from probe data via the analytic signal approach (using the Hilbert transform) shows a strong triad phase synchronisation at the resonance, thus providing independent experimental evidence of the resonance.

Unconditional uniqueness for the periodic modified Benjamin–Ono equation by normal form approach

Abstract We show that the solution (in the sense of distribution) to the Cauchy problem with the periodic boundary condition associated with the modified Benjamin–Ono equation is unique in $L^\infty _t(H^s(\mathbb{T} ))$ for $s&amp;gt;1/2$. The proof is based on the analysis of a normal form equation obtained by infinitely many reduction steps using integration by parts in time after a suitable gauge transform.

Stability analysis of alternating wave solution in a Stuart-Landau system with time delay

In this paper, the profile and stability of alternating wave solution, which arises as a bifurcated periodic solution of equivariant Hopf bifurcation with amazing properties, are investigated for a Stuart-Landau system consisting of three oscillators. The method of multiple scales is used to compute the normal form equation up to fifth order. The Floquet theory is introduced because it is difficult to directly analyze the stability of the alternating wave solution. By applying a time-varying complex coordinate transformation which does not change the stability of the solution of normal form that represents the alternating wave, the multipliers that completely determine the stability of alternating wave solution are explicitly solved. As a result, the criteria on parameters such that stable alternating wave solutions can be observed are provided. Based on studies through examples, we show that the proposed scheme of analysis is effective and some results on how parameters influence the stability of the alternating wave solution can be summarized. Our analysis confirms Golubitsky's assertion that the alternating wave solution will not be stable immediately after the equivariant Hopf bifurcation. We also find that a large time delay and a complex nonlinear gain will enhance the stability of alternating wave solution.

Norm form equations with solutions taking values in a multi-recurrence

We are interested in solutions of a norm form equation that takes values in a given multi-recurrence. We show that among the solutions there are only finitely many values in each component which lie in the given multi-recurrence unless the recurrence is of precisely described exceptional shape. This gives a variant of the question on arithmetic progressions in the solution set of norm form equations.