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We study a family of nonlinear oscillators governed by the equation(d2x)/(dt2)= -βxp, β > 0 (1)Here β is a constant and p=1,3,5,... is necessarily odd to be an oscillator (the phase portrait needs to be closed and bounded). The initial condition is x(0) = A, x'(0) = 0. Our goal is to find the frequency of the oscillator and its trajectory for each p. We apply the Lindstedt-Poincaré method to a convex homotopy with the parameter λ:x''(t) + x(t) = λ[x(t) − βx(t)p], 0 ≤ λ ≤ 1. (2)Substituting the expansionx(t) = ∑i=0∞ λ ixi(t)and the expression1 = ω2 − λα1 − λ2α2 − ... (3)into 1, the coefficient of x(t) on the left side of (2), we obtain the first few ODE’s for xi(t) such as Eq’s (4.3.4)-(4.3.6) below after equating the coefficients of the powers of λ. The final target solution x(t) and the frequency ω are obtained through a sequence of xi(t), αi, i = 1, 2, ... and setting λ to 1. It is necessary to kill the secular terms to have oscillatory behavior in the model. To achieve this goal, we study about the homotopy perturbation method for nonlinear oscillators which have been demonstrated and discussed in chapter 2. Then we apply He’s homotopy per-turbation method for conservative truly nonlinear oscillators and find their improved approximate solutions [9]. By this approach, we can find a truly periodic solution and the period of the motion as a function of the amplitude of oscillation. We take p = 3 in (1), so-called cubic oscillator and find that this approach works very well for the whole range of parameters and obtain excellent result match of frequencies between approximate and an exact one [9].

Community forestry in Nepal is intended to facilitate both improved forest conditions and the alleviation of poverty, by increasing community participation in forest resource management. While this programme has been successful in improving forest conditions, the economic returns to local people from community forests are not as apparent, especially in the case of the Sal forests in the Terai and Inner-Terai regions of Nepal. Despite having several advantages; high value forest, fertile land, connection with the transportation network, and being close to the regional markets, there is very little timber production from these forests. This research has analysed the problems inherent in the institutional structure of the community forests that would explain why such a high value timber was not being fully utilized. In particular, the research has looked at two aspects of the structure. The first is, whether the size of community forests has a negative effect on the efficiency of commercial timber production, and thus whether there is a need to develop cooperative or joint management structures with other user groups. The second aspect of the research is to look at whether the internal management structures of community forest user groups, and relationships with government institutions, affect their ability to make production decisions within the user groups, or affect their ability to make external commercial contracts with other user groups, log buyers, or contractors. The findings of this research are expected to increase the understanding of the timber production process and the institutional problems of community forestry in the Terai and Inner-Terai regions. Research findings may also be helpful in other regions of Nepal, other parts of the world, and other sectors, such as fisheries, where similar resource management settings exist.

Rimu is one of the main timber species used in the furniture industry in New Zealand. The main source of rimu timber is the West Coast forests of the South Island. In May 2000, the Government of New Zealand announced a logging ban on all public forests, including the West Coast forests that will take effect from March 2002. Because of this ban, there is a substantial decrease in the supply of rimu timber in the market. This shortfall creates uncertainty among furniture manufacturers regarding what to use as alternative timbers to rimu. In order to understand what alternatives might be viable for manufacturers, it's important to know the preference of furniture consumers for timber. This study determines manufacturers' strategies for alternatives to rimu and examines consumer preferences for timber used in furniture. In addition, this study determines wood-specific attributes and other factors that influence consumer preferences. Finally, this study investigates whether there are unique market segments for furniture timber and develops a profile of these segments. This study is conducted in the Canterbury region of New Zealand. Data were collected in two stages: a manufacturer survey and a consumer survey. Eight manufacturers were interviewed using a personal interview method and an open-ended questionnaire. Eighty two furniture show visitors were interviewed for the consumer survey, using Q methodology and a questionnaire. Seventeen timber samples, which were selected from the manufacturers' survey, were used for Q sorting to examine consumer preferences for furniture timber. Data were analysed using factor analysis and simple statistical tools. This study indicates that there is a significant impact of the logging ban on manufacturers in the region. Manufacturers are applying a range of strategies to cope with the rimu shortfall problem. Based on consumer preference for furniture timber this study identified five consumer segments. The colour and grain of the timber are the key attributes for consumers when choosing furniture. In addition to these attributes, matching of furniture timber with existing furniture and origin of timber are other important factors that influence consumer preference for furniture timber. Profiles of these consumer segments are not as clear as anticipated. However, there is statistical evidence that these segments are different in terms of gender and income.