superstatistical distributions from a maximum entropy principle we deal with a generalized statistical description of nonequilibrium complex systems based on least biased distributions given some prior information a maximum entropy principle is introduced that allows for the determination of the distribution of the fluctuating intensive parameter beta of a superstatistical system given certain constraints on the complex system under consideration we apply the theory to three examples the superstatistical quantum mechanical harmonic oscillator the superstatistical classical ideal gas and velocity time series as measured in a turbulent taylor couette flow

the 1 1 resonance in hamiltonian systems abstract two degree of freedom hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation considering the frequencies as parameters the system undergoes a bifurcation when the frequencies pass through a resonance these bifurcations are well understood for most resonances k l but not the semisimple cases 1 1 and 1 1 a two degree of freedom hamiltonian system can be approximated to any order by an integrable normal form the reason is that the normal form of a hamiltonian system has an additional integral due to the normal form symmetry the latter is intimately related to the ratio of the frequencies for a rational frequency ratio this leads to s 1 symmetric systems the question we wish to address is about the co dimension of such a system in 1 1 resonance with respect to left right equivalence where the right action is s 1 equivariant the result is a co dimension five unfolding of the central singularity two of the unfolding parameters are moduli and the remaining non modal parameters are the ones found in the linear unfolding of this system

heteroclinic bifurcations near hopf zero bifurcation in reversible vector fields in r 3 abstract we study the dynamics near a symmetric hopf zero also known as saddle node hopf or fold hopf bifurcation in a reversible vector field in r 3 with involutory an reversing symmetry whose fixed point subspace is one dimensional we focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle foci we study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets horseshoes independent of the eigenvalues of the saddle foci as well as cascades of bifurcations of periodic heteroclinic and homoclinic orbits finally we discuss the application of our results to the michelson system describing stationary states and travelling waves of the kuramoto sivashinsky pde

normal forms and versal deformations of linear hamiltonian systems abstract equivalence classes of time independent linear real hamiltonian systems can be identified up to canonical transformations with the orbits of the adjoint action of the real symplectic group on its lie algebra a new set of representatives also called normal forms for these orbits is given versal deformations of systems in normal form are constructed applications of versal deformations to the study of bifurcations of linear systems with small codimension are indicated

versal deformations of elements of classical jordan algebras every involution defining by x x the elements of one of the classical lie algebras o n c and sp 2n c over c and o p q sp 2n r u p q o 2n and sp 2p 2q over r also defines symmetric matrices y y spanning the classical jordan algebras the paper contains an explicit description of all perturbations of symmetric matrices considered up to equivalence under the action of the corresponding classical lie group such exhaustive perturbations are called versal deformations

superstatistics thermodynamics and fluctuations a thermodynamiclike formalism is developed for superstatistical systems based on conditional entropies this theory takes into account large scale variations of intensive variables of systems in nonequilibrium stationary states ordinary thermodynamics is recovered as a special case of the present theory and corrections to it can systematically be evaluated a generalization of einstein s relation for fluctuations is presented using a maximum entropy condition

dynamic aspects of the flip annihilation process a one dimensional interacting particle system is revisited it has discrete time and its components are located in the set of integers these components can disappear in the functioning process each component assumes two possible states called plus and minus and interacts at every time step only with its nearest neighbors the following two transformations happen the first one is called flip under its action a component in state minus turns into a plus with probability the second one is called annihilation under its action whenever a component in state plus is a left neighbor of a component in state minus both components disappear with probability let us consider a set of initial measures to the process for these measures we show the upper bound for the mean time of convergence which is a function of the initial measure moreover we obtain the upper bound to the mean quantity of minuses on the process in each time step considering the initial measure concentrated at the configuration whose components are in the state minus we improved a well known result that the process is non ergodic when 2 250 now we are able to offer non ergodicity when 9 2 1000 we also established new conditions to the ergodicity of the process finally we performed some monte carlo simulations for this process

reversible equivariant linear systems in this paper we classify the structure of linear reversible systems vector fields on rn that are equivariant with respect to a linear representation of a compact lie group h we assume the time reversal symmetry r also acts linearly and is such that the group g that is generated by h and r is again a compact lie group the main tool for the classification is the representation theory of compact lie groups the results are applied to some generic eigenvalue movements of linear reversible equivariant systems

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

fluctuations of entropy and log normal superstatistics nonequilibrium complex systems are often effectively described by the mixture of different dynamics on different time scales superstatistics which is statistics of statistics with two largely separated time scales offers a consistent theoretical framework for such a description here a theory is developed for log normal superstatistics based on the fluctuation theorem for entropy changes as well as the maximum entropy method this gives novel physical insight into log normal statistics other than the traditional multiplicative random processes a comment is made on a possible application of the theory to the fluctuating energy dissipation rate in turbulence

reversible relative periodic orbits abstract we study the bundle structure near reversible relative periodic orbits in reversible equivariant systems in particular we show that the vector field on the bundle forms a skew product system by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves we also discuss possibilities for drifts along group orbits our results extend those recently obtained in the equivariant context by b sandstede et al 1999 j nonlinear sci 9 439 478 and c wulff et al 2001 ergodic theory dynam systems 21 605 635

every continuous operator has an invariant measure we prove the existence of an invariant measure for a large class of random processes with discrete time without assuming their linearity our main examples are processes with variable length in which components may appear and disappear in the course of functioning one of these examples displays non uniqueness of invariant measure in a 1 d process

variable length analog of stavskaya process a new example of misleading simulation this article presents a new example intended to showcase limitations of computer simulations in the study of random processes with local interaction for this purpose we examine a new version of the well known stavskaya process which is a discrete time analog of the well known contact processes like the bulk of random processes studied till now the stavskaya process is constant length that is its components do not appear or disappear in the course of its functioning the process which we study here and call variable stavskaya vs is similar to stavskaya it is discrete time its states are bi infinite sequences whose terms take only two values denoted here as minus and plus and the measure concentrated in the configuration all pluses is invariant however it is a variable length which means that its components also called particles may appear and disappear under its action the operator vs is a composition of the following two operators the first operator called birth depends on

reversible equivariant hopf bifurcation in this paper we study codimension one hopf bifurcation from symmetric equilibrium points in reversible equivariant vector fields such bifurcations are characterized by a doubly degenerate pair of purely imaginary eigenvalues of the linearization of the vector field at the equilibrium point the eigenvalue movements near such a degeneracy typically follow one of three scenarios splitting from two pairs of imaginary eigenvalues to a quadruplet on the complex plane passing on the imaginary axis or crossing a quadruplet crossing the imaginary axis we give a complete description of the behaviour of reversible periodic orbits in the vicinity of such a bifurcation point for non reversible periodic solutions in the case of hopf bifurcation with crossing eigenvalues we obtain a generalization of the equivariant hopf theorem

singularities on the boundary of the stability domain near 1 1 resonance abstract we study the linear differential equation x l x in 1 1 resonance that is x r 4 and l is 4 4 matrix with a semi simple double pair of imaginary eigenvalues i i i i we wish to find all perturbations of this linear system such that the perturbed system is stable since linear differential equations are in one to one correspondence with linear maps we translate this problem to gl 4 r in this setting our aim is to determine the stability domain and the singularities of its boundary the dimension of gl 4 r is 16 therefore we first reduce the dimension as far as possible here we use a versal unfolding of l i e a transverse section of the orbit of l under the adjoint action of gl 4 r repeating a similar procedure in the versal unfolding we are able to reduce the dimension to 4 a 3 sphere in this 4 dimensional space contains all information about the neighborhood of l in gl 4 r considering the 3 sphere as two 3 discs glued smoothly along their common boundary we find that the boundary of the stability domain is contained in two right conoids one in each 3 disc the singularities of this surface are transverse self intersections whitney umbrellas and an intersection of self intersections where the surface has a self tangency a whitney stratification of the 3 sphere such that the eigenvalue configurations of corresponding matrices are constant on strata allows us to describe the neighborhood of l and in particular identify the stability domain

non ergodicity in a 1 d particle process with variable length we present a 1 d random particle process with uniform local interaction which displays some form of non ergodicity similar to contact processes but more unexpected particles enumerated by integer numbers interact at every step of the discrete time only with their nearest neighbors every particle has two possible states called minus and plus at every time step two transformations occur the first one turns every minus into plus with probability independently from what happens at other places and thereby favors pluses against minuses the second one is impartial under its action whenever a plus is a left neighbor of a minus both disappear with probability independently from presence and fate of other pairs of this sort if is small enough by comparison with 2 and we start with all minuses the minuses never die out

tsallis distributions and 1 f noise from nonlinear stochastic differential equations probability distributions that emerge from the formalism of nonextensive statistical mechanics have been applied to a variety of problems in this article we unite modeling of such distributions with the model of widespread 1 f noise we propose a class of nonlinear stochastic differential equations giving both the q exponential or q gaussian distributions of signal intensity revealing long range correlations and 1 f behavior of the power spectral density the superstatistical framework to get 1 f noise with q exponential and q gaussian distributions of the signal intensity is proposed as well

maximum entropy approach to h theory statistical mechanics of hierarchical systems a formalism called h theory is applied to the problem of statistical equilibrium of a hierarchical complex system with multiple time and length scales in this approach the system is formally treated as being composed of a small subsystem representing the region where the measurements are made in contact with a set of nested heat reservoirs corresponding to the hierarchical structure of the system where the temperatures of the reservoirs are allowed to fluctuate owing to the complex interactions between degrees of freedom at different scales the probability distribution function pdf of the temperature of the reservoir at a given scale conditioned on the temperature of the reservoir at the next largest scale in the hierarchy is determined from a maximum entropy principle subject to appropriate constraints that describe the thermal equilibrium properties of the system the marginal temperature distribution of the innermost reservoir is obtained by integrating over the conditional distributions of all larger scales and the resulting pdf is written in analytical form in terms of certain special transcendental functions known as the fox h functions the distribution of states of the small subsystem is then computed by averaging the quasiequilibrium boltzmann distribution over the temperature of the innermost reservoir this distribution can also be written in terms of h functions the general family of distributions reported here recovers as particular cases the stationary distributions recently obtained by macedo et al phys rev e 95 032315 2017 10 1103 physreve 95 032315 from a stochastic dynamical approach to the problem

normal forms and unfoldings of linear systems in eigenspaces of anti automorphisms of order two abstract in this article we classify normal forms and unfoldings of linear maps in eigenspaces of anti automorphisms of order two our main motivation is provided by applications to linear systems of ordinary differential equations general and hamiltonian which have both time preserving and time reversing symmetries however the theory gives a uniform method to obtain normal forms and unfoldings for a wide variety of linear differential equations with additional structure we give several examples and include a discussion of the phenomenon of orbit splitting as a consequence of orbit splitting we observe passing and splitting of eigenvalues in unfoldings

a long range memory stochastic model of the return in financial markets abstract we present a nonlinear stochastic differential equation sde which mimics the probability density function pdf of the return and the power spectrum of the absolute return in financial markets absolute return as a measure of market volatility is considered in the proposed model as a long range memory stochastic variable the sde is obtained from the analogy with an earlier proposed model of trading activity in the financial markets and generalized within the nonextensive statistical mechanics framework the proposed stochastic model generates time series of the return with two power law statistics i e the pdf and the power spectral density reproducing the empirical data for the one minute trading return in the nyse

versal deformations and normal forms for reversible and hamiltonian linear systems the problem of this article is the characterization of equivalence classes and their versal deformations for reversible and reversible hamiltonian matrices in both cases the admissible transformations form a subgroup g of gl m therefore the gl m orbits of a given matrix may split into several g orbits these orbits are characterized by signs for each sign we have a normal form and a corresponding versal deformation the main tool in the characterization is reduction to the semi simple case c 1996 academic press inc

chaos and monte carlo approximations of the flip annihilation process the flip annihilation process is a random particle process with one dimensional local interaction in discrete time initially presented by one of us namely toom in 2004 its components are enumerated by integer numbers and every component has two states minus and plus at every time step two transformations occur the first one called flip independently turns every minus into plus with probability the second one called annihilation acts thus whenever a plus is a left neighbor of a minus both disappear with probability independently from other components what is interesting about this process is that it is ergodic for 2 and non ergodic for 2 250 it is natural to conjecture that there is some transition curve which we call the true curve and denote by beta mathsf true alpha which separates the areas of ergodicity and non ergodicity of this process from each other the estimates mentioned above albeit rigorous leave a large gap between them and the present article s purpose is to obtain some closer albeit non rigorous approximations of the true curve we do it in two ways one of which is a chaos approximation and the other is a monte carlo simulation thus we obtain two curves which are much closer to each other than the rigorous estimations also we fill in albeit only numerically another shortcoming of the rigorous estimation 2 250 namely that it leaves us uncertain whether the true curve has a zero or positive slope at the point 0 both approximate curves have a positive slope at 0 as we hoped

normal linear stability of quasi periodic tori abstract we consider families of dynamical systems having invariant tori that carry quasi periodic motions our interest is the persistence of such tori under small nearly integrable perturbations this persistence problem is studied in the dissipative the hamiltonian and the reversible setting as part of a more general kam theory for classes of structure preserving dynamical systems this concerns the parametrized kam theory as initiated by moser j k moser on the theory of quasiperiodic motions siam rev 8 2 1966 145 172 j k moser convergent series expansions for quasi periodic motions math ann 169 1967 136 176 and further developed in g b huitema unfoldings of quasi periodic tori phd thesis university of groningen 1988 h w broer g b huitema f takens unfoldings of quasi periodic tori mem amer math soc 83 421 1990 1 82 h w broer g b huitema unfoldings of quasi periodic tori in reversible systems j dynam differential equations 7 1 1995 191 212 the corresponding nondegeneracy condition involves certain trans versality conditions on the normal linear leading part at the invariant tori we show that as a consequence a cantor family of diophantine tori with positive hausdorff measure is persistent under nearly integrable perturbations this result extends the above references since presently the case of multiple floquet exponents is included our leading example is the normal 1 1 resonance which occurs a lot in applications both hamiltonian and reversible as an illustration of this we briefly describe the lagrange top coupled to an oscillator