Example 2.7: stable, unstable, and semi-stable limit cycles Consider the following nonlinear systems (a) x₁ = x₂-x₁ (x²+x₂² - 1) x₂ = x₁-x₂(x₁²+x₂² - 1) (2.12)
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- 23. Consider a simple economy with just two industries: farming and manufacturing. Farming consumes 1/2 of the food and 1/3 of the manufactured goods. Manufacturing consumes 1/2 of the food and 2/3 of the manufactured goods. Assuming the economy is closed and in equilibrium, find the relative outputs of the farming and manufacturing industries.A forest has a population of lynx and a population of rabbits. Let xx represent the number of lynx (in hundreds) above some level, denoted with 0. So x=−3x=-3 corresponds NOT to an absence of lynx, but to a population that is 300 below the designated level of lynx. Similarly, let yy represent the number of rabbits (in hundreds) above a level designated by zero. The following system models the two populations over time:x'=−0.25x+yx′=-0.25x+yy'=−x+4yy′=-x+4ySolve the system using the initial conditions x(0)=0x(0)=0 and y(0)=1y(0)=1.x(t)x(t) = y(t)y(t) = Choose the graph that best represents the solution curve.Deep in the redwood forest of California, dusky-footed wood rats provide up to 80% of thediet for the spotted owl, the main predator of the wood rat, which is being modelled usingthe following discrete dynamical system. Denote the owl and wood rat populations at timek by xk = [OkRk], where k is the time in months. Ok+1 = 0.5Ok + 0.4RkRk+1 = −0.104Ok + 1.1RkDetermine the evolution of this system after 5 years. Discuss the evolution of systemafter large time has been passed.
- A forest has a population of cougars and a population of squirrels. Let xx represent the number of cougars (in hundreds) above some level, denoted with 0. So x=−3x=-3 corresponds NOT to an absence of cougars, but to a population that is 300 below the designated level of cougars. Similarly, let yy represent the number of squirrels (in hundreds) above a level designated by zero. The following system models the two populations over time:x'=−0.5x+yx′=-0.5x+yy'=−x−2.5yy′=-x-2.5ySolve the system using the initial conditions x(0)=0x(0)=0 and y(0)=1y(0)=1.x(t)x(t) = y(t)y(t) = Choose the graph that best represents the solution curve.A first order nonlinear system is described by the equationẋ = −f(x)where f(x) is a continuous and differentiable nonlinear function thatsatisfies the following:f(0) = 0;f(x) > 0 for x > 0;f(x) < 0 for x < 0.Use the Lyapunov function V(x) = x2/2 to show that the system isstable near the origin.question 5Estimate the equlibrium price and quantity of the market whose demand and supply functions are pd = −(q + 4)2 + 100 and ps = (q + 2) 2 respectively. Showing all supporting working: If the region A (shaded grey) in the diagram above represents a solution set, derive the system of inequalities which define that region.
- raph the following discrete-time dynamical systems, find the equilibria algebraically, and check whether the stability derived from the Slope Criterion for stability matches that found with cobwebbing. f (x) = x² for 0 ≤ x ≤ 2Question.1. A manufacturing manufactures two items X1 and X2 which are handled in a machine shop and assembly shop. Item X1 requires 2 hours of work in a machine shop and 4 hours of work in the assembly shop to manufacture while product X2 requires 3 hours of work in the machine shop and 2 hours of work in the assembly shop. In one day, the industry cannot use more than 16 hours of machine shop and 22 hours of assembly shop. It earns a profit of rupees 3 per unit of product X1 and rupees. 4 per unit of product X2. Give a mathematical formulation of the problem as to maximise profit.Q-RAM Inc. manufactures solid-state drives (SSDs) of two sizes, 500 GB and 1.5 TB. The company can make a total of 60 drives per day, and it has available 90 labor-hours per day. It takes 1 labor-hour to make a 500 GB drive and 3 hours to make a 1.5 TB drive. The profits are $40 per 500 GB drive and $60 per 1.5 TB drive. (Let x equal the number of 500 GB drives. Let y equal the number of 1.5 TB drives.)
- Q-RAM Inc. manufactures solid-state drives (SSDs) of two sizes, 500 GB and 1.5 TB. The company can make a total of 50 drives per day, and it has available 60 labor-hours per day. It takes 1 labor-hour to make a 500 GB drive and 3 hours to make a 1.5 TB drive. The profits are $40 per 500 GB drive and $60 per 1.5 TB drive. (Let x equal the number of 500 GB drives. Let y equal the number of 1.5 TB drives.) (a) Write the initial simplex matrix to maximize the daily profit. x y s1 s2 f 50 60 (b) Find the maximum profit.$ Find the number of each type of drive that will give the maximum profit. 500 GB drives 1.5 TB drivesIf = X and v = x (x - 1) (y + 1) then the stream line is y = e ^ ((x ^ 2/2 - x) + c) -11.2.4. Consider the spreading of a highly communicable disease on an isolated island with population size N. A portion of the population travels abroad and returns to the island infected with the disease. Formulate a dynamical system to approximate the change in the number of people in the population who have the disease.
