STABILITY 5/4 



Ex. 4: Find all the states of equilibrium of the transformation: 



dxidt = e^*' sin x, cly/dt = x^. 



Ex. 5: If .x' = 2x — > + j\ y' = x + y + A, find values for j and k that will 

 give a state of equilibrium at (1,1). (Hint: First modify the equations to 

 represent the state of equilibriimi.) 



Ex. 6: If T{b) = b, must T2{b), T\b\ etc., all also equal bl 



Ex. 7: Can an absolute system have more states of equilibrium than it has 

 basins ? 



Ex. 8 : What is the characteristic appearance of the kinematic graph of a trans- 

 formation whose states are all equihbrial? 



Ex. 9: (Continued.) What special name was such a transformation given in 

 an earlier chapter? 



Ex. 10: If the transformation is changed (the set of operands remaining the 

 same) are the states of equilibrium changed? 



Ex. 1 1 : If a machine's input is changed, do its states of equilibrium change ? 

 (Hint: See Ex.5.) 



5/4. Cycle. Related to the states of equilibrium is the cycle, a 

 sequence of states such that repeated application of the transforma- 

 tion takes the representative point repeatedly round the sequence. 

 Thus if T is 



J. \abcdefgh 

 c h b h a c c g 



then, from a, T generates the trajectory 



a c b h g c b h g c b . . . 



and the representative point repeatedly traverses the cycle 



c -^ b 



t I 



g ^ h 



Ex. 1 : Write down a transformation that contains two distinct cycles and 

 three states of equilibrium. 



Ex. 2: (Continued.) Draw its kinematic graph. 



Ex. 3 : Can a state of equilibrium occur in a cycle ? 



Ex. 4: Can an absolute system have more cycles than it has basins? 



Ex. 5: Can one basin contain two cycles? 



*Ex. 6: Has the system dx/dt = y, dy/dt = — x a cycle? 



*Ex. 7: If the transformation has a finite number of states and is closed and 

 single-valued, can a trajectory end in any way other than at a state of 

 equilibrium or in a cycle? 



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