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- Evaluate the stability of the following system using the Lyapunov method. (Treat the constant delta as 0)Consider the following. x1' = 3x1 − 2x2, x1(0) = 3 x2' = 2x1 − 2x2, x2(0) = 1 2 (a) Transform the given system into a single equation of second order by solving the first equation for x2 and substitute into the second equation, thereby obtaining a second order equation for x1.' (b) Find x1 and x2 that also satisfy the initial conditions.Consider the dynamical system Vk+1 = AVk where V0 = 0 3 A= 0 1 6 -1 Find a formula in terms of k for the (1,1) entry of Vk.
- Consider: Lyapunov indirect method for stability: A nonlinear state-space systemis stable if and only ifthe real parts of all eigenvalues of Jacobian of the system have negative real parts. J={ afi / axj } evaluated at steady state. Take in to account the following non linear system: dx1 / dt = f1 (x1 , x2) = 2x1 - x2 dx2 / dt = f2 (x1 , x2)= -x1 - 1/2 x13 - x22 is the system stable in Lyapunov sense?What situation arises in accordance with the Fundamental Theorem of linear algebra for the systems given below when p=4:10.The average sizes of the prey and predator populations are defined as ¯x=1T∫A+TAx(t)dt,¯y=1T∫A+TAy(t)dt,x¯=1T∫AA+Txtdt,y¯=1T∫AA+Tytdt, respectively, where T is the period of a full cycle, and A is any nonnegative constant. a.Using the approximation (24), which is valid near the critical point, show that ¯x=c/γx¯=c/γ and ¯y=a/αy¯=a/α. b.For the solution of the nonlinear system (2) shown in Figure 9.5.3, estimate ¯xx¯ and ¯yy¯ as well as you can. Try to determine whether ¯xx¯ and ¯yy¯ are given by c/γ and a/α, respectively, in this case. Hint: Consider how you might estimate the value of an integral even though you do not have a formula for the integrand. c.Calculate other solutions of the system (2)—that is, solutions satisfying other initial conditions—and determine ¯xx¯ and ¯yy¯ for these solutions. Are the values of ¯xx¯ and ¯yy¯ the same for all solutions? In Problems 11 and 12, we consider the effect of modifying the equation for the prey x by including a term −σx2 so that…
- Given the following R2→R function:b) f(x,y)= x2+5y2-2x-20y+24Find and analyze the nature of the critical points using the algebraic method and Hessian matrix and show that both methods lead to the same results.Two interacting populations of hares and foxes can be modeled by the recursive equations h(t + 1) = 4h(t) − 2 f (t) f(t+1)=h(t)+ f(t). For each of the initial populations given in parts (a) through (c), find closed formulas for h(t) and f (t). a. h(0)= f(0)=100 b. h(0)=200, f(0)=100 c. h(0)=600, f(0)=500Find the general solution in terms of real functions. (b) From the roots of the characteristic equation, determine whether each critical point of the corresponding dynamical system is asymptotically stable, stable, or unstable, and classify it as to type. (c) Use the general solution obtained in part (a) to find a two-parameter family of trajectories x=x1i+x2j=yi+y′j of the corresponding dynamical system. Then sketch by hand, or use a computer, to draw a phase portrait, including any straight-line orbits, from this family of trajectories.