Simple Normal Form Research Articles

Amplitude response and singularity behavior of circadian clock to external stimuli

Amplitude changes caused by environmental cues are universal in the circadian clock and associated with various diseases. Singularity behavior, characterized by the disruption of circadian rhythms due to critical stimuli, has been observed across various species. Several mathematical models of the circadian clock have replicated this phenomenon. A comprehensive understanding of the amplitude response remains elusive due to experimental limitations. In this study, we address this question by utilizing a simple normal form model that accurately fits previous experimental data, thereby presenting a general mechanism. We employ a geometric framework to illustrate the dynamics in different stimuli of light-induced transcription (LIT) and light-induced degradation (LID), highlighting the core role of invisible instability in amplitude response. Our model systematically elucidates how stimulus mode, phase, and strength determine amplitude responses. The results show that external stimuli induce alterations in both the amplitudes of individual oscillators and the synchronization among oscillators, collectively influencing the overall amplitude response. While experimental methods impose constraints resulting in limited outcomes under specific conditions, our model provides a comprehensive and three-dimensional mechanistic explanation. A comparison with existing experimental findings demonstrates the consistency of our proposed mechanism. Considering the response direction, the framework enables the identification of phases that lead to increased circadian amplitude. Based on this mechanism derived from the framework, stimulus strategies for resetting circadian rhythms with reduced side effects could be designed. Our results demonstrate that the framework has great potential for understanding and applying stimulus responses in the circadian clock and other limit cycle oscillations.

On Resilient Boolean and Vectorial Boolean Functions with High Nonlinearity

Boolean functions and vectorial Boolean functions are the most important nonlinear components of stream ciphers. They should satisfy several criteria such as high nonlinearity, proper resiliency and so on to guarantee the security of the whole system. However, there are some constraints among the criteria, and how to achieve a trade-off between them is an important issue. In this paper, some nonlinear Boolean functions possessing simple algebraic normal form with special Walsh spectrum are proposed. By using these functions, we provide two construction methods on balanced and resilient Boolean functions with high nonlinearity. In addition, based on the disjoint linear codes and vector matrices with special properties, some resilient vectorial Boolean functions with currently best-known nonlinearity have also been given.

Formula omitted] is equivalent to [formula omitted

The organization of temporal properties into a Temporal Hierarchy proposed by Manna and Pnueli is central in the study of the expressive power of fragments of Linear Temporal Logic with Past (LTL+P). A crucial role is played by the Reactivity class, which itself forms a hierarchy consisting of the classes Reactivity(1), Reactivity(2), and so on, and it is proved to be expressively equivalent to LTL+P. The logic of Generalized Reactivity(1) (GR(1) for short), introduced in the context of reactive synthesis, expands the Reactivity(1) class (abbreviated R(1)) with safety formulas and multiple fairness conditions.In this paper, we prove that GR(1) is expressively equivalent to R(1). We first define a simple normal form for GR(1) formulas, and then we prove the expressive equivalence between GR(1) (in such a normal form) and R(1). In addition, we show that all the involved translations require only a linear increase in size. Finally, we lift these results to the general case proving that GR(N) is equivalent to R(N), for all N≥1.

A flow box theorem for 2d slow-fast vector fields and diffeomorphisms and the slow log-determinant integral

We prove a flow box theorem for smooth 2-dimensional slow-fast vector fields, providing a simple normal form that is obtained by smooth coordinate changes, without having to change the time. We introduce a notion of 2d slow-fast diffeomorphism, define the log-determinant integral and prove a normal form theorem similar to the flow box theorem for slow-fast vector fields. The normal form is used in proving the relevance of the log-determinant integral.

The lower bound of the weightwise nonlinearity profile of a class of weightwise perfectly balanced functions

Boolean functions satisfying good cryptographic criteria when restricted to the set of vectors with constant Hamming weight play an important role in the recent FLIP stream cipher. Although the nonlinearity of Boolean functions can be calculated rapidly by Walsh transform, the nonlinearity of the Boolean functions with restricted input seems to be very difficult to calculate. In this paper, a class of weightwise perfectly balanced Boolean functions with very simple algebraic normal form is proposed. Then, the cryptographic properties of the weightwise perfectly balanced Boolean functions are discussed. Especially, the lower bound of the weightwise nonlinearity of these 2m-variable functions is given for any positive integer m.

An approximation for nonlinear differential-algebraic equations via singular perturbation theory

In this paper, we study jumps of nonlinear DAEs caused by inconsistent initial values. First, we propose a simple normal form called the index-1 nonlinear Weierstrass form (INWF) for nonlinear DAEs. Then we generalize the notion of consistency projector for linear DAEs to the nonlinear case. By an example, we compare our proposed nonlinear consistency projectors with two existing consistent initialization methods to show that the two existing methods are not coordinate-free, i.e., the consistent points calculated by the two methods are not invariant under nonlinear coordinates transformations. Next we propose a singular perturbed system approximation for nonlinear DAEs, which is an ordinary differential equation (ODE) with a small perturbation parameter, we show that the solutions of the proposed perturbation system approximate both the jumps resulting from the nonlinear consistency projectors and the $\mathcal C^1$-solutions of the DAE. At last, we use a numerical simulation of a nonlinear DAE model arising from an electric circuit to illustrate the effectiveness of the proposed singular perturbed system approximation of DAEs.

Pairs of commuting nilpotent operators with one-dimensional intersection of kernels and matrices commuting with a Weyr matrix

I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A,B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t-tuple of linear operators. Moreover, it contains the problem of classifying representations of an arbitrary quiver and an arbitrary finite-dimensional algebra, and so it is considered as hopeless.If (A,B) is such a pair, then KerA∩KerB≠0. We give a simple normal form (Anor,Bnor) of the matrices of (A,B) if KerA∩KerB is one-dimensional. We do not know whether it is canonical; i.e., whether (Anor,Bnor) is uniquely determined by (A,B). We prove its uniqueness only if the Jordan canonical form of A is a direct sum of Jordan blocks of the same size and the field is of zero characteristic.The matrix Anor is the Weyr canonical form of A, and Bnor commutes with Anor. In order to describe the structure of (Anor,Bnor), we describe explicitly all matrices commuting with a given Weyr matrix.

Integrability and linearizability of cubic Z2 systems with non-resonant singular points

In this paper, complete integrability and linearizability of cubic Z2 systems with two non-resonant and elementary singular points are investigated. First of all, four simple normal forms are obtained based on the coefficients and eigenvalues of cubic Z2 systems. Then, the integrable and linearizable conditions are classified according to the four different cases respectively, and the problem is solved thoroughly for cubic Z2 systems with two non-resonant singular points.

Multipulse Homoclinic Orbits and Chaotic Dynamics of a Reinforced Composite Plate with Carbon Nanotubes

The multipulse homoclinic orbits and chaotic dynamics of a reinforced composite plate with the carbon nanotubes (CNTs) under combined in-plane and transverse excitations are studied in the case of 1 : 1 internal resonance. The method of multiple scales is adopted to derive the averaged equations. From the averaged equations, the normal form theory is applied to reduce the equations to a simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method proposed by Haller and Wiggins is utilized to examine the global bifurcations and chaotic dynamics of the CNT-reinforced composite plate. The analytical results demonstrate that the multipulse Shilnikov-type homoclinic orbits and chaotic motions exist in the system. Homoclinic trees are constructed to illustrate the repeated bifurcations of multipulse solutions. In order to verify the theoretical results, numerical simulations are given to show the multipulse Shilnikov-type chaotic motions in the CNT-reinforced composite plate. The results obtained here imply that the motion is chaotic in the sense of the Smale horseshoes for the CNT-reinforced composite plate.

Bogdanov-Takens bifurcation in predator-prey systems

In this paper, we consider Bogdanov-Takens bifurcation in two predator-prey systems. It is shown that in the full parameter space, Bogdanov-Talens bifurcation can be codimension \begin{document}$ 2 $\end{document} , \begin{document}$ 3 $\end{document} or \begin{document}$ 4 $\end{document} . First, the simplest normal form theory is applied to determine the codimension of the systems as well as the unfolding terms. Then, bifurcation analysis is carried out to describe the dynamical behaviour and bifurcation property of the systems around the critical point.

Perturbation analysis of a matrix differential equation ẋ = ABx

Abstract Two complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S −1 AR, R −1 BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + A͠, B + B͠) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of A͠ and B͠. Each perturbation (A͠, B͠) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + A͠)(B + B͠) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations ẋ = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S −1 CS and (A, B) by contragredient equivalence transformations (S −1 AR, R −1 BS).

Sensitivity of asymmetric rate-dependent critical systems to initial conditions: Insights into cellular decision making.

The work reported here aims to address the effects of time-dependent parameters and stochasticity on decision making in biological systems. We achieve this by extending previous studies that resorted to simple bifurcation normal forms, although in the present case we focus primarily on the issue of the system's sensitivity to initial conditions in the presence of two different noise distributions, Gaussian and Lévy. In addition, we also assess the impact of two-way sweeping at different rates through the critical region of a canonical Pitchfork bifurcation with a constant external asymmetry. The parallel with decision making in biocircuits is performed on this simple system since it is equivalent in its available states and dynamics to more complex genetic circuits published previously. Overall we verify that rate-dependent effects, previously reported as being important features of bifurcating systems, are specific to particular initial conditions. Processing of each starting state, which for the normal form underlying this study is akin to a classification task, is affected by the balance between sweeping speed through critical regions and the type of fluctuations added. For the heavy-tailed noise, two-way dynamic bifurcations are more efficient in processing the external signals, here understood to be jointly represented by the critical parameter profile and the external asymmetry amplitude, when compared to the system relying on escape dynamics. This is particular to the case when the system starts at an attractor not favored by the asymmetry and, in conjunction, when the sweeping amplitude is large.

A nonlinear Luenberger-like observer for nonlinear singular systems

This paper investigates observer design problem for a large class of nonlinear singular systems with multiple outputs. We first regularize the singular system by injecting the derivative of outputs into the system. Then differential geometric method is applied to transform the regularized system into a simple normal form, for which a Luenberger-like observer is proposed.

Favard theory and fredholm alternative for disconjugate recurrent second order equations

We discuss the existence of a Fredholm–type Alternative for a recurrent second order linear equation, which is disconjugate in a strong sense. The basic result is about bounded solutions of equations with bounded coefficients: it depends on kinematic similarities that allow to reduce the problem to a pair of very simple normal forms. Then the result is specialized to recurrent equations, by means of Favard theory.

Intersection type calculi of bounded dimension

A notion of dimension in intersection typed λ-calculi is presented. The dimension of a typed λ-term is given by the minimal norm of an elaboration (a proof theoretic decoration) necessary for typing the term at its type, and, intuitively, measures intersection introduction as a resource. Bounded-dimensional intersection type calculi are shown to enjoy subject reduction, since terms can be elaborated in non-increasing norm under β-reduction. We prove that a multiset interpretation (corresponding to a non-idempotent and non-linear interpretation of intersection) of dimensionality corresponds to the number of simultaneous constraints required during search for inhabitants. As a consequence, the inhabitation problem is decidable in bounded multiset dimension, and it is proven to be EXPSPACE-complete. This result is a substantial generalization of inhabitation for the rank 2-fragment, yielding a calculus with decidable inhabitation which is independent of rank. Our results give rise to a new criterion (dimensional bound) for subclasses of intersection type calculi with a decidable inhabitation problem, which is orthogonal to previously known criteria, and which should have immediate applications in synthesis. Additionally, we give examples of dimensional analysis of fragments of the intersection type system, including conservativity over simple types, rank 2-types, and normal form typings, and we provide some observations towards dimensional analysis of other systems. It is suggested (for future work) that our notion of dimension may have semantic interpretations in terms of of reduction complexity.

Traveling Pulses in a Nonlocal Equation Arising Near a Saddle-Node Infinite Cycle Bifurcation

We formally derive a simple normal form for the dynamics of a nonlocally coupled neural field model when the local dynamics is near a saddle-node infinite cycle (SNIC) bifurcation. The derivation produces a nonlocally coupled scalar model which does not satisfy the comparison principle (ordered initial data produces ordered dynamical solutions). We prove the existence of unique traveling waves for the corresponding nonlocal evolution problem with a new tool that does not use the comparison principle. We obtain sharp estimates for the speed of fast and slow waves and compare these to numerical results.

Neighborhood radius estimation for Arnold's miniversal deformations of complex and p-adic matrices

V.I. Arnold (1971) constructed a simple normal form to which all complex matrices B in a neighborhood U of a given square matrix A can be reduced by similarity transformations that smoothly depend on the entries of B. We calculate the radius of the neighborhood U. A.A. Mailybaev (1999, 2001) constructed a reducing similarity transformation in the form of Taylor series; we construct this transformation by another method. We extend Arnold's normal form to matrices over the field Qp of p-adic numbers and the field F((T)) of Laurent series over a field F.

Polynomial Normal Forms for Some Germs of Nonstrongly 1-Resonant Diffeomorphisms

We study in this paper a class of smoothly finite determined germs of diffeomorphisms which have infinitely many resonant relations near a nonstrongly 1-resonant fixed point on [Formula: see text] and derive the simplest normal forms of such germs with nondegenerated nonlinear parts.

The Effect Of Randomness On The Stability Of Capillary Gravity Waves In The Presence Of Air Flowing Over Water

A nonlinear spectral transport equation for the narrow band Gaussian random surface wave trains is derived from a fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves. The effect of randomness on the stability of deep water capillary gravity waves in the presence of air flowing over water is investigated. The stability is then considered for an initial homogenous wave spectrum having a simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained; in which a higher order contribution comes from the fourth order term in the evolution equation, which is responsible for wave induced mean flow. This higher order contribution produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order term in the evolution equation.

Miniversal deformations of matrices under *congruence and reducing transformations

Arnold (1971) [1] constructed a miniversal deformation of a square complex matrix under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We give miniversal deformations of matrices of sesquilinear forms; that is, of square complex matrices under *congruence, and construct an analytic reducing transformation to a miniversal deformation. Analogous results for matrices under congruence were obtained by Dmytryshyn, Futorny, and Sergeichuk (2012) [11].