nullity invariance for pivot and the interlace polynomial we show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given square matrix is described by the symmetric difference operator for sets we consider its consequences for graphs and in particular generalize the recursive relation of the interlace polynomial and simplify its proof

feigenvalues for mandelsets the authors discuss generalizations of feigenbaum s constants alpha and delta to complex polynomial maps of degree higher than 2 and present some numerical estimates universality classes are found to depend on the nature of the critical points of the polynomial

renormalization and transition to chaos in area preserving nontwist maps abstract the problem of transition to chaos i e the destruction of invariant circles or kam kolmogorov arnold moser curves in area preserving nontwist maps is studied within the renormalization group framework nontwist maps are maps for which the twist condition is violated along a curve known as the shearless curve in renormalization language this problem is that of finding and studying the fixed points of the renormalization group operator r that acts on the space of maps a simple period two fixed point of r whose basin of attraction contains the nontwist maps for which the shearless curve exists is found also a critical period 12 fixed point of r with two unstable eigenvalues is found the basin of attraction of this critical fixed point contains the nontwist maps for which the shearless curve is at the threshold of destruction this basin defines a new universality class for the transition to chaos in area preserving maps

new universal scenarios for the onset of chaos in lorenz type flows it is shown that in simple three dimensional flows there are uncountably many different codimension 2 scenarios for the onset of chaos we analyze these scenarios via one dimensional poincar e maps and show how to construct renormalization groups to obtain new universal results comparisons with numerical studies are given

a physical interpretation of the he non map abstract the orientation preserving henon map is shown to be equivalent to the period one return map of a periodically kicked harmonic oscillator with a suitable nonlinear coupling to the kicking term the conservative dissipative and extremely dissipative logistic limit cases are each considered all can be derived from a single differential equation describing the kicked oscillator system in each case explicit coordinate and parameter transformations are given relating the return map of the kicked harmonic oscillator to a standard form of the henon map in the dissipative case two studies are presented with periodic kicking replaced by a smooth periodic function each is found to yield a henon like chaotic attractor

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

rank preserving diagonal completions of a matrix abstract suppose a is an n by n matrix over a field f we prove that it is possible to complete the diagonal entries of a so that the resulting rank of a is as small as possible when n 3r where r is the off diagonal rank of a and n r 3 1

farey level separation in mode locking structure of circle mappings abstract we consider various characteristic scales both on the x axis and in the parameter space for critical and near critical circle mappings juxtaposition of different scales for the set of all rational mode lockings exposes the effect of separation into layers this layered structure is shown to reproduce farey tree levels of the corresponding rotation numbers the separation is studied numerically for maps with various orders of singularity and interpreted in terms of coefficients of rotation function expansion

structures preserved by generalized inversion and schur complementation in this paper we investigate the inheritance of certain structures under generalized matrix inversion these structures contain the case of rank structures and the case of displacement structures we do this in an intertwined way in the sense that we develop an argument that can be used for deriving the results for displacement structures from thoses for rank structures we pay particular attention to the moore penrose generalized inverse showing that for the cases of most interest the ranks of the structure satisfied by the moore penrose inverse can at most double with respect to the original ranks we consider also the case of inheritance of structure by generalized schur complements

a variety of period doubling universality classes in multi parameter analysis of transition to chaos abstract in multi parameter analysis of the onset of chaos non feigenbaum period doubling behavior may occur at some special paths in the parameter space there are two possibilities i the dynamics at the onset of chaos remains essentially one dimensional but the one dimensional map is distorted in such a way that leaves feigenbaum s universality class ii a new mode comes to the threshold of instability and increases the effective dimension of the dynamics we submit a list of one dimensional and two dimensional maps which represent distinct classes of the period doubling universality discuss the properties of the associated types of critical behavior at the border of chaos and demonstrate pictures of the universal parameter space arrangement near the critical points

chaotic phase similarities and recurrences in a damped driven duffing oscillator we report strong evidence of remarkably close periodic repetitions of the structuring of the parameter space of a damped driven duffing oscillator as the amplitude of the drive increases families of period adding cascades and some intricate networks of periodic oscillations embedded in chaotic phases are also found to recur closely as the driving force grows such surprising regularities suggest that some hitherto unknown renormalization mechanism may be operating in higher codimension controlling the alternation of chaos and order in parameter space of certain flows

nullities of submatrices of the moore penrose inverse let a be a complex m n matrix of rank r and with moore penrose inverse a if a a11a21a12a22 a b11b21b12b22 are partitioned so that aii is mi ni and bii is ni mi with m m1 m2 n n1 n 2 and if the row nullity of a matrix is denoted by then m r b22 a11 n r

multiparameter critical situations universality and scaling in two dimensional period doubling maps we review critical situations linked with period doubling transition to chaos which require using at least two dimensional maps as models representing the universality classes each of them corresponds to a saddle solution of the two dimensional generalization of feigenbaum cvitanovic equation and is characterized by a set of distinct universal constants analogous to feigenbaum s and one type of criticality designated h was discovered by several authors in 80 th in the context of period doubling in conservative dynamics but occurs as well in dissipative dynamics as a phenomenon of codimension 2 second is bicritical behavior which takes place in systems allowing decomposition onto two dissipative period doubling subsystems each of which is brought by parameter tuning onto a threshold of chaos types of criticality designated as fq and c occur in non invertible two dimensional maps we present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems in particular we indicate a possibility for realization of the h type criticality without vanishing dissipation but with its compensation in a self oscillatory system next we present a number of examples coupled henon like maps coupled driven oscillators coupled chaotic self oscillators which manifest bicritical behavior for fq type we indicate possibility to arrange it in non symmetric systems of coupled period doubling subsystems e g in henon like maps and in chua s circuits for c type we present examples of its appearance in a driven rossler oscillator at the period doubling accumulation on the edge of syncronization tongue and in a model map with the neimark sacker bifurcation

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

bounds on the unstable eigenvalue for the asymmetric renormalization operator for period doubling we establish rigorous bounds for the unstable eigenvalue of the period doubling renormalization operator for asymmetric unimodal maps herglotz function techniques and cone invariance ideas are used our result generalizes an established result for conventional period doubling

generalized inverses of morphisms with kernels abstract let x y be a morphism with kernel k x in an additive category with an involution then has a moore penrose inverse with respect to iff is invertible in this case 1 if x y then has a group inverse iff has a cokernel x k and 2 is invertible in this case 2 1

bifurcation structure of the c type period doubling transition abstract the period doubling transition to chaos along the edge of an arnold tongue is known to display unusual organization and scaling behavior kuznetsov et al 2005 7 it is also known that forced period doubling systems may be associated with the appearance of so called period doubled tori arneodo et al 1983 15 using the rossler system as an example we present a detailed analysis of the bifurcation structure associated with the forcing of a three dimensional period doubling system we explain how this structure is related to the recently discovered phenomenon of multi layered tori and discuss different bifurcation scenarios that transform a resonance torus into a period doubled ergodic torus similar bifurcation phenomena have recently been observed in a biologically relevant model of kidney blood flow regulation in response to fluctuations in arterial pressure

completing a symmetric 2 2 block matrix and its inverse abstract we consider the following completion problems suppose n1 n2 are nonnegative integers such that n 1 n 2 n 0 let a11 a12 a21 b22 be matrices with dimensions n1 n1 n1 n2 n2 n1 and n2 n2 respectively we determine necessary and sufficient conditions so that there exists an n2 n2 matrix a22 such that a a 11 a 21 a 12 a 22 and i a is nonsingular and symmetric and b22 is the lower right block of a partitioning of a 1 ii a is symmetric positive definite and b22 is the lower right block of a partitioning of a 1

the nullity theorem for principal pivot transform abstract we generalize the nullity theorem of gustafson 1984 8 from matrix inversion to principal pivot transform several special cases of the obtained result are known in the literature such as a result concerning local complementation on graphs as an application we show that a particular matrix polynomial the so called nullity polynomial is invariant under principal pivot transform

period doubling for two dimensional non invertible maps renormalization group analysis and quantitative universality two types of universal behavior associated with period doubling in two dimensional non invertible maps are investigated the first called c type may occur as a generic two parameter phenomenon when the map possesses a fold singularity the second kind of universal behavior called fq type is associated with the singularity arising from projecting whitney s umbrella typically this kind of period doubling scaling behavior appears as a phenomenon of codimension three i e at some critical point in the three dimensional parameter space numerical results are presented to illustrate the self similarity of the parameter space topography near the critical points c and fq the possibility of observing these new types of critical behavior in physical systems is discussed

degenerate routes to chaos we present examples of simple dynamical systems containing alternative routes to chaos which are degenerate with a period doubling route the existence of degenerate routes implies the possibility of having more entangled dynamical behaviors hidden in observations based on a single variable time series possibly requiring therefore special care to discriminate them a model system allowing the investigation of transitions between degenerate routes is also considered

degrees of freedom of interference channels with comp transmission and reception we study the degrees of freedom dof of the k user interference channel with coordinated multipoint comp transmission and reception each message is jointly transmitted by m t successive transmitters and is jointly received by m r successive receivers we refer to this channel as the comp channel with a transmit cooperation order of m t and receive cooperation order of m r since the channel has a total of k transmit antennas and k receive antennas the maximum possible dof is equal to k we show that the comp channel has k dof if and only if m t m r geq k 1 the key idea is that the zero forcing of the interference corresponding to the i rm th message at the decoder of the j rm th message where j ne i can be viewed as a shared responsibility between the m t transmitters carrying the i rm th message and the m r receivers decoding the j rm th message for the general case we derive an outer bound that states that the dof is bounded above by left lceil k m t m r 2 2 right rceil for the special case with only comp transmission i e m r 1 we propose a scheme that can achieve k m t 1 2 dof for all k and conjecture that the result holds true for all k in the proposed coding scheme the m t transmitters carrying each message are used to cancel the interference introduced by this message at the first m t 1 receivers thereby allowing each of these receivers to enjoy 1 dof and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space the achievability proofs are based on the notion of algebraic independence from algebraic geometry

fast qr iterations for unitary plus low rank matrices some fast algorithms for computing the eigenvalues of a block companion matrix have recently appeared in the literature in this paper we generalize the approach to encompass unitary plus low rank matrices of the form a u xy h where u is a general unitary matrix three important cases for applications are u unitary diagonal u unitary block hessenberg and u unitary in block cmv form our extension exploits the properties of a larger matrix hat a obtained by a certain embedding of the hessenberg reduction of a suitable to maintain its structural properties we show that hat a can be factored as product of lower and upper unitary hessenberg matrices possibly perturbed in the first k rows and moreover such a data sparse representation is well suited for the design of fast eigensolvers based on the qr iteration the resulting eigenvalue algorithm is fast and backward stable

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

renormalization in a circle map with two inflection points abstract an extended version of the sine circle map becomes non invertible by developing simultaneously two critical points in part of parameter space universality in the corresponding transition from quasiperiodicity to chaos manifests itself in the organisation of doubly superstable orbits four operators are needed to generate all such orbits the existence of non trivial equivalence relations for operation sequences makes the present theory different from other multi operator renormalisation approaches possible implications of the theory in higher dimensional systems are discussed

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

universality and scaling of period doubling bifurcations in a dissipative distributed medium abstract a homogeneous medium consisting of nonlinear elements demonstrating transition to chaos via period doubling bifurcations is considered the coupling between the elements is supposed to be of a dissipative type i e it tends to equalize their instantaneous states using the renormalization group approach the following scaling law for weakly inhomogeneous states near the critical point is obtained at each period doubling the spatial scale increases by 2 on the basis of this law the scaling hypotheses for the transition to chaos in the semi infinite and finite systems are proposed the scaling properties are verified by the numerical calculations with a simple model

bounds on the unstable eigenvalue for period doubling bounds are given for the unstable eigenvalue of the period doubling operator for unimodal maps of the interval these bounds hold for all types of behaviour x r of the interval map near its critical point they are obtained by finding cones in function space which are invariant under the tangent map to the doubling operator at its fixed point

feigenbaum theory for unimodal maps with asymmetric critical point rigorous results we apply the methods of h epstein to prove the existence of a line of period 2 solutions of the feigenbaum period doubling renormalisation transformation these solutions govern the universal behaviour of families of unimodal maps with asymmetric critical points of degree d for which the d th derivative has differing left and right limits

different types of critical behavior in conservatively coupled henon maps we study the dynamics of two conservatively coupled henon maps at different levels of dissipation it is shown that the decrease of dissipation leads to changes in the structure of the parameter plane and the scenarios of transition to chaos compared to the case of infinitely strong dissipation particularly the feigenbaum line becomes divided into several fragments some of these fragments have critical points of different types namely of c and h type as their terminal points also the mechanisms of formation of these feigenbaum line ruptures are described

completion of an orthogonal projector and a related inverse mapping problem we study the completion problem for an orthogonal projector a of r n from knowledge of one or several of its columns as well as the related matrix inverse problem of finding an orthogonal projector a with ax y for two given n by k matrices x and y using geometric properties of orthogonal projections and of the singular value decomposition we derive theoretical and numerical solutions for these problems we give algorithms and test their matlab implementations

on the structure rank of a class of structure matrices abstract we introduce a class of structure matrices called transpose complementary structure matrices we give a characterization of the n extendable transpose complementary structure matrices with trace k where k 0 2 we also prove that a nonsingular matrix and its inverse have the same s rank for this class of structure matrices s

death of period doublings locating the homoclinic doubling cascade abstract this paper studies a natural mechanism called a homoclinic doubling cascade for the disappearance of period doubling cascades in vector fields simply put an entire period doubling cascade collides with a saddle type equilibrium homoclinic doubling cascades are known to have self similar structure in contrast to the well known feigenbaum constant the scaling constants for homoclinic doubling depend on the eigenvalues of the saddle equilibrium specifically we present here for the first time a detailed study of homoclinic doubling cascades in a smooth vector field namely a three dimensional polynomial model proposed by sandstede a numerical algorithm is presented for computing homoclinic doubling cascades in general vector fields which makes use of the program auto homcont this allows us to compute two types of homoclinic doubling cascades one where the primary homoclinic orbit undergoes an inclination flip bifurcation and one where it undergoes an orbit flip bifurcation our results bring out the scaling constants in good agreement with analytical estimates obtained from one dimensional maps

on the quadratic mapping z z2 μ for complex μ and z the fractal structure of its m set and scaling abstract for each complex denote by f the largest bounded set in the complex plane that is invariant under the action of the mapping z z2 mandelbrot 1980 1982 chap 19 reported various remarkable properties of the m set the set of those values of the complex for which f contains domains and of the closure m of m the goals of the present work are as follows a to restate some previously reported properties of f m and m in new ways and to report new observations b to deduce some known properties of the mapping f for real and z with f 1 4 2 and z 1 2 1 2 1 4 1 2 1 2 1 4 in many ways the properties of the transformation f are easier to grasp in the complex plane than in an interval this exemplifies the saying that when one wishes to simplify a theory one should complexify the variables c to serve as introduction to some recent pure mathematical work triggered by mandelbrot 1980 further pure mathematical work is strongly urged

completing a block diagonal matrix with a partially prescribed inverse abstract a complete solution of the matrix completion problem a b 1 c d is obtained in terms of solutions of a riccati type matrix equation several special cases are described in detail

the complete set of hamiltonian intermittency scaling behaviors renormalization group equations describing the phenomenon of intermittency in hamiltonian systems are presented all solutions satisfying certain physical constraints are obtained they are the complete set of simple singularities further considerations lead to precise predictions for scaling behavior at the onset of intermittency

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

on chip matched filtering and discrete sufficient statistics for asynchronous band limited cdma systems the problem of generating discrete sufficient statistics for signal processing in code division multiple access cdma systems is considered in the context of underlying channel bandwidth restrictions discretization schemes are identified for approximately bandlimited cdma systems and a notion of approximate sufficiency is introduced the role of chip matched filtering in generating accurate discrete statistics is explored the impact of approximate sufficiency on performance is studied in three cases conventional matched filter mf detection minimum mean squared error detection and delay acquisition it is shown that for waveforms limited to a chip interval sampling the chip mf output at the chip rate can lead to a significant degradation in performance then with equal bandwidth and equal rate constraints the performance with different chip waveforms is compared in all three cases above it is demonstrated that multichip waveforms that approximate nyquist sine pulses achieve the best performance with the commonly used rectangular chip pulse being severely inferior however the results also indicate that it is possible to approach the best performance with well designed chip waveforms limited to a chip interval as long as the chip mf output is sampled above the nyquist rate

birth of a new class of period doubling scaling behavior as a result of bifurcation in the renormalization equation it is found that a fixed point of the renormalization group equation corresponding to a system of a unimodal map with extremum of power and a map summarizing values of a function of the dynamical variable of the first subsystem undergoes a bifurcation in the course of increase of it occurs at c 1 984396 and results in a birth of the period 2 stationary solution of the rg equation at 2 this period 2 solution corresponds to the universal period doubling behavior discovered earlier and denoted as the c type criticality kuznetsov and sataev in phys lett a 162 236 242 1992 by combination of analytical methods and numerical computations we obtain and analyze an asymptotic expansion of the period 2 solution in powers of c the developed approach resembles the e expansion in the phase transition theory in which a trivial stationary point of the rg transformation undergoes a bifurcation that gives rise to a new fixed point responsible for the critical behavior with nontrivial critical indices

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

on the existence of unimodular matrices with a prescribed submatrix abstract we study the existence of unimodular matrices with a prescribed submatrix over a ring r in particular when either r is hermite and dedekind finite or r has stable range one we give necessary and sufficient conditions for the existence of these unimodular completions

two parameter analysis of the scaling behavior at the onset of chaos tricritical and pseudo tricritical points abstract we discuss the so called tricritical points at the border of the period doubling transition to chaos and examine to what extent the associated universality applies to 2d dissipative maps as a concrete example the ikeda map is studied together with its 1d analog for the approximate 1d map the tricritical points appear as the terminal points of feigenbaum s critical curves in the parameter plane for the 2d map the same type of critical behavior does not occur in a rigorous sense it may be observed as a kind of intermediate asymptotics however when one considers a finite number of period doublings we refer to the associated points in the parameter plane as pseudo tricritical for the ikeda map we present estimates of the number of period doublings after which the departure from the tricritical universality becomes essential

period doubling bifurcations for families of maps on ℝ n infinite sequences of period doubling bifurcations in one parameter families 1 pf of maps enjoy very strong universality properties this is known numerically in a multitude of cases and has been shown rigorously for certain 1 pf of maps on the interval these bifurcations occur in 1 pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 4 6692 in this paper we indicate the main steps of a proof that the same is true for 1 pf of analytic maps from n to n whose restriction to n is real

the interplay of ranks of submatrices a banded invertible matrix t has a remarkable inverse all upper and lower submatrices of t 1 have low rank depending on the bandwidth in t the exact rank condition is known and it allows fast multiplication by full matrices that arise in the boundary element method we look for the right proof of this property of t 1 ultimately it reduces to a fact that deserves to be better known complementary submatrices of any t and t 1 have the same nullity the last figure in the paper when t is tridiagonal shows that on and above the diagonal of t 1 all rows are proportional

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

renormalization group analysis of bifurcations in area preserving maps abstract two dimensional area preserving maps can be represented by a generating function the action high orders of successive period doubling bifurcations are studied by writing a renormalization group scheme for this action fixed points and eigenvalues of this scheme are found and interpreted

critical point of tori collision in quasiperiodically forced systems we report on a type of scaling behavior in quasiperiodically forced systems on the parameter plane the critical point appears as a terminal point of the tori collision bifurcation curve its location is found numerically with high precision for two basic models the forced supercritical circle map and the forced quadratic map the hypothesis of universality based on renormalization group arguments is advanced to explain the observed scaling properties for the critical attractor and for the parameter plane arrangement in the neighborhood of the criticality