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Recently, an analytical solution of a quadratic nonlinear oscillator has been presented based on the harmonic balance method. By introducing a small parameter, a set of nonlinear algebraic equations have been solved which usually appear among unknown coefficients of several harmonic terms. But the method is not suitable for all quadratic oscillators. Earlier, introducing a small parameter to the frequency series, Cheung et al. modified the Lindstedt–Poincare method and used it to solve strong nonlinear oscillators including a quadratic oscillator. But due to some limitations of both parameters, a changed form of frequency-related parameter (introduced by Cheung et al.) has been presented for solving various quadratic oscillators.

Two techniques (mobile seedling nursery and detached leaf) were used to study the pathogenic variability of Erysiphe gramznis DC. ex Merat f.sp. hordei Em. Marchal in four barley growing areas of South Australia (S.A.). The mobile nurseries were conducted over 5 years (1981-1985) to monitor changes in the spectrum of virulence and individual gene frequencies. The race-specific resistance genes M1-a6, M1-k, M1-v and M1-ra were found to be susceptible to the pathogen populations in all surveyed areas. The same virulence genes spectrum (V-a6, V-k, V-v and V-ra) was present in the pathogen populations in the surveyed areas throughout the period of 1981-1984. During the 1985 season, one new virulence matching the resistance gene in cv. Forrest was detected. The resistance in cv. Galleon did not 'break down' over the cultivation period when its cultivation rose to 55% of the barley areas of S.A. The probable reasons for apparent durability of Galleon resistance are discussed. The relative frequencies of the matching virulence genes varied only slightly over time and space. The relative frequency of V-k was always almost 100%. The relative frequencies of V-a6, V-ra and V-v occurred at a higher rate than expected, since the matching resistance genes were not deployed in barley cultivars. The results of the detached leaf experiments (1982-1984) confirmed the virulence spectrum of the pathogen populations found in the mobile nursery experiments. Three individual single colony isolates from different cultivars (Clipper, Sonja and Goldmarker) were isolated and purified. Each isolate can produce a susceptible infection on other cultivars having different resistance genes in addition to its matching cultivar. Thus, each isolate carries more than one virulence gene. In the mobile nursery tests, the resistance gene M1-v (Varunda and LaMi) showed variable infection types (I.T.l-3), but in most cases it was moderately susceptible (I.T.3). Under the controlled conditions of the detached leaf tests, it gave an I.T.3 indicating the presence of virulence gene for M1-v in the pathogen populations. Reasons for this variable reaction are discussed. Midas with an ineffective gene (M1-a6) produced a resistant reaction to all isolates, suggesting the presence of additional resistance genes in the cultivar. The Mildew resistance genes M1-a, M1-a7, M1-a9, 141-al2, M1-g, M1-h, M1-(CP) and ml-o3 were found to be resistant in all surveyed areas in both experiments throughout S.A. Either the matching virulence genes for these resistance genes were absent or they were present at a very low frequency in the pathogen populations that could not be detected by the sampling techniques used.

Two Lupinus species, L, angustifolius L. and L. cosentinii Guss., were grown in pots in the glasshouse on a virgin lateritic soil, with two soil temperatures attained by immersion in water-baths at 10 and 20C, and four rates of superphosphate ranging from 0 to 1792 kg ha-l. Air temperatures were not controlled. Dry matter, and phosphorus and nitrogen concentrations, were measured separately for tops and roots after harvest at 7 weeks. In a second experiment, the rate of early radicle elongation was measured for the two species in complete nutrient solutions at 10 and 20. Water-bath temperature markedly influenced the growth of both roots and tops, growth with 20 being approximately double that with 10 in both species. At both bath temperatures, responses to superphosphate closely paralleled those observed previously on the same soil in the field. There was a slight positive interaction between bath temperature and superphosphate rate, but in the main the effects of the two factors were independent and additive. The results did not support the suggestion that reduced growth with low root temperature may be due primarily to reduced phosphorus uptake. Nor did they indicate reductions in nodulation, nitrogen fixation, or translocation of either nitrogen or phosphorus from roots to tops as likely causes. It is suggested that temperature may have directly affected the growth and/or metabolism of the roots, influencing their production of growth substances, which in turn influenced processes controlling top growth.

Eight varieties from four lupin species were grown on a virgin lateritic soil with six rates of superphosphate, ranging from 0 to 1,792 kg ha-1. Dry matter yield and phosphorus and nitrogen contents of the tops were measured at about full flowering, together with days to flowering, pods set per primary inflorescence, and phosphorus and nitrogen contents of the mature seeds. All species gave sigmoid yield responses, reaching 90 per cent of maximum yield at superphosphate rates between 600 and 900 kg ha-l, depending on species. Varieties within species responded uniformly. The order of tolerance to deficiency was L. cosentini > L. luteus > L. angustifolius = L. albus. Tolerance was based on uptake rather than differing tissue requirement. Phosphorus and nitrogen concentrations in the seeds varied markedly with species, the rankings being L. luteus > L. cosentini = L. albus > L. angustifolios for phosphorus, and L. luteus > L. albus > L. angustifolius > L. cosentini for nitrogen. Superphosphate application caused parallel changes in phosphorus concentrations in the tops and seeds of each species. It slightly increased nitrogen concentrations in the tops, but not in the seeds.

The effects of photoperiod and temperature and/or insolation on flowering and yield components were measured for 24 genotypes of five Lupinus species in the field. Environmental factors were varied by sowing on three dates (April 2, May 17, and July 1) and by extending natural photoperiod with artificial light. Lack of vernalizing temperature (<10C) following early April sowing retarded flower initiation substantially in the midseason to late genotypes of Lupinus angustifolius and L. pilosus and to a lesser degree other species. Genotypes with no vernalization requirement flowered fastest following April sowing. Flowering of all genotypes was accelerated considerably by 24-hour photoperiod, L. luteus being the most responsive species. Number of flowers per primary inflorescence was reduced, and percentage of flowers setting pods increased, by treatments that led to earlier flowering. Number of seeds per pod and 20-seed weight were not affected by the treatments, except that higher temperatures increased 20-seed weight significantly in L. luteus and L. cosentini. Number of flowers, percentage flower set, and 20-seed weight were found to be the most important yield components in the primary inflorescences of lupins.

Controlled environment studies of the effects of vernalization, photoperiod, and growing period temperatures were carried out on selected cultivars of four annual Lupinus species. All responded to both vernalization and photoperiod to varying degrees, and in at least two there were indications of an additional effect of growing period temperatures specifically on flower initiation. Flower initiation in L. angustifolius was found to be controlled mainly by its vernalization requirement, with subsidiary control by photoperiod. In L. cosentini vernalization, photoperiod, and an acceleration of initiation by high temperatures all appeared to play important roles, with critical control by photoperiod under short days. L. luteus responded strongly to both vernalization and photoperiod, but long days were able to substitute for vernalization to a marked degree. The results are discussed in the context of the ecology of lupins and the breeding of new crop cultivars.