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This article aims at providing a new theoretical insight into the fundamental question of the origin of truncated fractals in biological systems. It is well known that fractal geometry is one of the characteristics of living organisms. However, contrary to mathematical fractals which are self-similar at all scales, the biological fractals are truncated, i.e. their self-similarity extends at most over a few orders of magnitude of separation. We show that nonlinear coupled oscillators, modeling one of the basic features of biological systems, may generate truncated fractals: a truncated fractal pattern for basin boundaries appears in a simple mathematical model of two coupled nonlinear oscillators with weak dissipation. This fractal pattern can be considered as a particular hidden fractal property. At the level of sufficiently fine precision technique the truncated fractality acts as a simple structure, leading to predictability, but at a lower level of precision it is effectively fractal, limiting the predictability of the long-term behavior of biological systems. We point out to the generic nature of our result.

In Part 1 of this paper we have investigated numerically the quasiperiodic and frequency locked solutions of mathematical model of a robot with one degree of freedom. In this paper we extend our investigations to the region of transient chaos. The zones of chaotic transients are very broad and lie beyond the parameter range of engineering significance. Transiently chaotic zones exhibit a complex structure, fractally intertwined with tongues of regular pattern and cover a broad range of control parameter L. The crisis point for the onset of sustained chaos lies extremely far from the point of onset of transient chaos.

A numerical experiment was performed for a distribution of lifetimes of chaotic transients in the weakly dissipative single-well Duffing oscillator. It is shown that the results of computations with a uniform distribution of precision of numerical algorithm at a fixed initial condition cannot be distinguished from the well-known exponential distribution of lifetimes computed with an uniform distribution of initial conditions at fixed numerical precision. The same exponential distribution of lifetimes is also obtained if a noise term is added to the driving force in the fixed precision case.

Large-scale combinatorial calculations of level densities were performed for selected nuclei using Gaussian polynomial generating function method (GPM). Contrary to the results of previous combinatorial calculations, we find a good agreement of the combinatorial total level densities and Bethe formula. Combinatorial GPM calculations were performed also for spin-dependent level densities and comparison is made with various algebraic formulas. On the basis of our GPM calculations we propose a new phenomenological spin-dependent level density formula, which in the low-spin limit reduces to the spin-dependent formula of Bethe. The new formula (24) is for all spins factorized into the product of total level density and spin distribution function, where the spin distribution function (25) contains an additional spin correction parameter.

In the framework of the cluster-vibration model we discuss a quasi-f7/2 pattern of low-lying states in143Ce with theI=j—2 anomaly, and its consequences on the spectroscopic factors for this nucleus.

The study of the 137Ba level scheme has been extended in the framework of the interacting boson-fermion model. Two categories of negative-parity states up to excitation energies of 5 MeV were found. One is associated with the h112hole-type family and another with the particle-type configurations placed above the N= 82 boson core. The calculated SU(5) energy spectrum as well as cumulative (d, p) cross sections are in good agreement with experimental data.

Energy spectrum and electromagnetic properties of205Pb have been studied in the cluster-vibration model (CVM). The lifetime of 0.3 ms was predicted for the 2.3 keV level which is of significance for considerations of the205Tl solar neutrino detector.

Gamma rays of199Au obtained after double neutron capture in197Au were measured at the ILL high flux reactor. A level scheme up to 1770 keV excitation energy is established. The result is compared with IBFM and Boson-Fermion-Symmetry calculations.

The properties of the triplet of low-lying states in101Mo have been studied through spectroscopy of the? radiation following thermal neutron capture in100Mo and ß- decay of101Nb and through a measurement of the proton angular distributions in the100Mo(d,p) reaction with 14 MeV deuteron energy. The half-lives of the 13.5 keV state and the 57.0 keV 5/2+ state have been measured as 226(7) and 133(7)ns, respectively. These values and the quadrupole/dipole mixing ratios of the 13.5 keV and 43.5 keV transitions yield spin and parity 3/2+ for the 13.5 keV level. The E2 components in the 13.5 (3/2+ ?1/2+) and 43.5 keV (5/2+?3/2+) transitions are ? 8·10-4 and 54(9)%, respectively. The possibility of an additional state near to the 57.0 keV level is discussed. IBFM/PTQM calculations, taking into consideration the transitional character of the100Mo boson core, account for the electromagnetic-transition and transfer-reaction pattern of the triplet of states.

Level density fluctuations are calculated within the generalized interacting boson model proposed for even-even nuclei, in dependence on the truncation parameter k. For the case ϰ=2 corresponding to the SU(3) dynamical symmetry of the interacting boson model the fluctuation pattern is close to Poissonian. For cases ϰ≠2, including the anharmonic vibrator model for which ϰ=∞, a rapid transition to the fluctuation pattern close to GOE is obtained.

Ground state symmetric rotor bands are described as coherent-state structures of deformation in a quadrupole-phonon model. These structures are shown to be very different from the phonon composition of pseudobands which are culled from the excited state spectrum of nuclei such as75Se,75Kr, and77Kr. The pseudobands are shown to be only a subset of more complete excitation spectra, which occur in most unique-parity odd-nucleon nuclei with a transitional core and can be expected to occur throughout the chart of the nuclides.

The quantal analogs of the Gallagher-Moszkowski bands in the Interacting boson-fermion-fermion model (IBFFM) withSU (3) boson core are associated with the exact quantum numberKjp+jn and approximate quantum numberK-¦jp-jn¦ of IBFFM. For characteristic ratios?/d it is shown that theKc=0 band in even-even IBM system, theK=jp andK=jn bands in the corresponding odd-even (IBFM) systems and theK=jp+j band in the corresponding odd-odd IBFFM system exhibit the supermultiplet pattern.

Life-times of low-lying levels in3791Rb54 and3793Rb56 have been determined fromß--? coincidence measurements at the fission-product separator JOSEF. Values oft1/2=17.0(8) ns and 2.0(2) ns have been obtained for the levels at 1,134 keV in91Rb and 267 keV in93Rb, respectively, and upper limits could be deduced for several other states. Calculations in the frame of IBFM/PTQM have been performed for91Rb, the results of which allow an interpretation of the low-lying levels of this nucleus. For the 1,134 keV level the configuration [pg9/2?2+] 7/2+ is suggested which lies close to or even below the 9/21+ level. The calculated half-life of 14 ns for the 1,134 keV level reproduces well the experimental value. The half-life of the 267 keV level in93Rb favours the assignment ofIp=1/2- to this state over the alternative 3/2-.

The results of a?-? angular correlation and a low energy?-ray singles measurement using thermal neutron capture in39K are presented. The properties of low-lying levels of40K are interpreted in a comparison of the experimental data with the Interacting-Boson-Fermion-Fermion Model (IBFFM). The level fluctuation properties of the IBFFM spectrum are analysed and a sizeable deviation from chaoticity is found.

The level density of the odd-odd nucleus 132Pr is investigated in the interacting boson-fermion-fermion model (IBFFM) which accounts for collectivity and complex interaction between quasiparticle and collective modes. The IBFFM total level density is fitted by a Gaussian and its tail is also fitted by Bethe formula and constant temperature Fermi gas model. The IBFFM spin-dependent level densities show high-spin reductions with respect to Bethe formula. This can be well accounted for by a modified spin-dependent level density formula.

A band with a rotational pattern based on a state at 585.1 keV has been identified in theN=59 neutron-rich nucleus97Sr. Its properties lead to the [422] 3/2 Nilsson-orbital assignment for the band head. There is evidence for a second band with the head at 644.7 keV and the configuration [541] 3/2. Since the ground state and the lowest excited levels are spherical, shape coexistence is established for97Sr. A deformed nature of several levels at 500–600 keV results also from QRPA-model calculations. The structure of the low-lying spherical levels has been studied in the frame of the IBF model. The results of the present investigations lead to a better understanding of theN=59 isotones which constitute the link between the spherical and deformed nuclei atA~100 as a species with shape coexistence but without any indications of particular softness.

A level scheme has been established for3794Rb57 based on studies at the fission-product separator JOSEF of?? coincidences following theß- decay of94Kr. For the 190.7 keV level a half-life of 2.5(8) ns has been determined through the centroid-shift method. The majorß-decay branch of94Kr proceeds with log ft=3.7 into the 988.4 keV state in94Rb. This transition must be of the 0+?1+ GT type involving the transition?g7/2?pg9/2. IBFFM calculations have been performed for94Rb, starting from the parametrization used in the previous IBM investigations for the isotope93Rb and the isotone96Y. A theoretical interpretation of the low-lying levels including propositions for the spins is given in a comparison to the available data.

The structure of the low-lying states with negative parity of3793Rb56 has been studied in the framework of the Interacting Boson-Fermion Model (IBFM) using the even-even neighbour3894Sr56 as a core. A good description of the properties of the levels has been obtained. A sensitive test of the quality of the theoretical studies is the decay pattern of the second excited state at 267 keV with a half-life of 2.0(2) ns. The calculations givet1/2=2.26 ns and they predict the existence of a 13 keV transition into the first excited state, the existence of which has been confirmed afterwards by experiments.

Unique-parity states of97Y are studied in the framework of the interacting boson-fermion model (IBFM). Spin assignments for the levels of the unique-parity yrast band are proposed on the basis of nuclear systematics and the IBFM band pattern associated with coupling of a particle-type quasiparticle (g9/2) to the SU(5) core3896Sr58. The good fit of the experimental data with the IBFM calculations confirms the single particle character of this immediate neighbour of the doubly sub-magic nucleus96Zr.

Spectral rigidity?3 (L) is calculated for the energy spectra of131Te,135Ba and137Ce, obtained in the IBFM calculation. Results reveal an intermediate situation between regularity and chaos, but remarkably closer to the chaotic limit. Influence of the model parameters on the fluctuation pattern is discussed.

The structure of the nucleus3996Y57 is discussed in the framework of the parabolic rule for the splitting of members of proton-neutron coupled configurations and of the interacting boson-fermion-fermion model (IBFFM). A good description of the available experimental data including the existence of theß- decaying 8+ isomer with a half-life of 10 s has been obtained which demonstrates the influence of the subshell closures in the neighbourhood of4096Zr56.

In the Interacting Boson-Fermion-Fermion Model (IBFFM) for odd-odd nuclei the yrast states are calculated for , proton-neutron configurations coupled to theO(6) andSU(3) boson core. The results are discussed in terms of the strongly coupled/ decoupled band type criterion and in comparison to the peaceful and conflicting case of asymmetric rotor model.

The results obtained in this paper show that the SU(3) pattern in the spectra and transition rates of even-even nuclei appear qualitatively as a general feature of a large class of quadrupole-phonon models, independently of SU(3) symmetry. This pattern reflects the truncation of the quadrupole phonon space, incorporated via truncation operator in the fourth-order anharmonic quadrupole-phonon Hamiltonian. Three particular forms of the truncation operator are considered, corresponding to three models, namely the truncated quadrupole-phonon model (TQM), equivalent to the well-known interacting-boson model (IBM), the step-truncated quadrupole-phonon model (STQM), equivalent to a standard anharmonic quadrupole-phonon model and the linearly truncated quadrupole-phonon model (LTQM), introduced here. The results obtained provide a basis for a broader scope for the considerations in the SU(3) limit.

Low lying energy levels and transition probabilities of63,65,67Zn are calculated in the Quasicluster-vibration model (QCVM). This model is an extension of the Clustervibration model (CVM) for three-particle-cluster vibration coupling. Instead of a three-particle cluster, now a one broken pair (number projected three quasiparticle) cluster is coupled to phonon states. It turns out that the sequence of isotopes may be described with one set of parameters, along the same line as previously used in the CVM. The resulting spectra and electromagnetic properties, for positive and negative parity states of odd Zn isotopes, compare with experimental data equally well as large shell-model calculations; this seems to justify the extension of the CVM to these nuclei with more than three particles or holes beyond a closed shell.

γ-spectra and excitation functions of the100Mo(p, nγ)100Tc reaction were measured in the 1.2–3.6 MeV proton energy range by using thick, enriched targets, Ge(Li) and low energy photon (hyperpure Ge) spectrometers. These detectors were used inγγ-coincidence experiments, too. Conversion electron spectrum measurements were performed by means of a superconducting magnet transporter Si(Li) spectrometer (SMS) atEp =4 MeV and multipolarities of some transitions have been determined. Based on the experimental results a level scheme of100Tc has been constructed. Level energies of100Tc were calculated on the basis of the parabolic rule, derived from the cluster-vibration model.

The properties of low-lying states in133Xe are described by coupling a three-neutronhole cluster to the quadrupole vibrational field. The results obtained by diagonalization of the model Hamiltonian and qualitative predictions based on Generalized Vibrational Intensity and Selection Rules (GVISR) are discussed. The results obtained in the present paper are compatible with recent experimental data on133Xe. Further experimental tests of the model are suggested.

The properties of94Mo and95Mo nuclei, which are typical representatives of N=52 and N=53 nuclei, respectively, are described in the cluster-vibrational field coupling model in natural representation (without parameter fitting). The interplay of few valenceshell neutrons and vibrational degree of freedom gives rise to the coexistence of quasivibrational, quasirotational, and clustering phenomena.

The level density of the odd-odd nucleus132Pris investigated in the interacting boson-fermion-fermion model (IBFFM) which accounts for collectivity and complex interaction between quasiparticle and collective modes. The IBFFM total level density is fitted by a Gaussian and its tail is also fitted by Bethe formula and constant temperature Fermi gas model. The IBFFM spin-dependent level densities show high-spin reductions with respect to Bethe formula. This can be well accounted for by a modified spin-dependent level density formula.

Statistical properties of the realistic energy spectra of the odd-odd nuclei106Ag,198Au,134Cs,40K and94Rb, calculated within the Interacting Boson Fermion Fermion Model, are investigated by means of theΔ3statistics and the Nearest Neighbor Spacing Distribution method. New probability distribution function, which describes well the calculated results and enables the characterization of chaos with a physically meaningfull parameter, is proposed. Level spacing fluctuations of the examined nuclei exhibit the transitional behavior between Poisson and GOE limits, revealing different degrees of chaoticity in their dynamics.

Statistical properties of the realistic energy spectra of the odd-odd nuclei 106Ag, 198Au, 134Cs, 40K and 94Rb, calculated within the Interacting Boson Fermion Fermion Model, are investigated by means of the Δ3statistics and the Nearest Neighbor Spacing Distribution method. New probability distribution function, which describes well the calculated results and enables the characterization of chaos with a physically meaningfull parameter, is proposed. Level spacing fluctuations of the examined nuclei exhibit the transitional behavior between Poisson and GOE limits, revealing different degrees of chaoticity in their dynamics.

Boson-fermion dynamical symmetry associated withU (5) boson core is introduced for odd-odd nuclei. An algorithm is constructed for the subsequent reduction appropriate for the boson-fermion group chain associated with an odd proton and neutron inj=3/2 configurations and the corresponding energy formula is given and exemplified for62Cu in the case of an associated supersymmetry.

Excited states of 65Ge were populated via the 12C + 58Ni (261 MeV) reaction using the NORDBALL detector array equipped with charged-particle and neutron detector systems for reaction channel separation. On the basis of γγ-coincidence relations and angular distribution ratios a significantly extended level scheme was constructed up to Ex= 9 MeV and Jπ= (33/2−). The low-energy states of the nucleus are discussed in the framework of the interacting boson-fermion model.

The K=3/2

The levels of198Au have been calculated in IBFFM as a first application of the new model for odd-odd nuclei.

Extrapolating the Interacting Boson-Model treatment from lighter neighbours to98Y we predict the properties of low-lying states of spherical nature in IBFFM. This odd-odd nucleus which presents one of the rare cases of coexisting spherical and deformed shapes is of fundamental importance for understanding nuclear phase transitions.

Boson-fermion dynamical symmetry is introduced for hypernuclei with odd proton, odd neutron and hyperon in j=3/2 configurations and with SO(6) boson core.

We introduce the SU(6) model IBOM/OTQM for odd-odd nuclei and discuss the bands in O(6) limit. Bands are characterized by a particular type of phase correlations in the wave functions and exhibit splitting of the SU(3) analogs of Gallagher-Moszkowski bands.

In TQM a ?n=2 term is introduced in the two-particle transfer operator, which gives a sizeable shift of the L=0 transfer strength to the excited 0+ state in SU(3) and SU(5) limits of IBM/TQM.

For odd-odd nuclei we introduce three dynamical boson-fermion symnetries associated with jp=3/2, j?=3/2 and UB(5) symmetry of the boson core. The corresponding energy formulas and quantum numbers for low-lying states are explicitly given.