Homework answers / question archive / 1) Investigate the stability of the equilibrium points of the system x = (x – a) (x2 – a) for all real values of the parameter a

1) Investigate the stability of the equilibrium points of the system x = (x – a) (x2 – a) for all real values of the parameter a

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1) Investigate the stability of the equilibrium points of the system x = (x – a) (x² – a) for all real values of the parameter a. (Hints: It might help to graph the right-hand side. An alternative is to rewrite the equation as x = -V’ (x)for a suitable potential energy function V and then use your intuition about particles moving in potentials.)

2) Consider the system x = x + e^y , y = -y First use qualitative arguments to obtain information about the phase portrait. Then, using a computer, plot the direction field. Finally, use the Runge-Kutta method to compute several trajectories, and plot them on the phase plane. Find all the fixed points of the system x = -x + x³, y = -2y, and use linearization to classify them. Then check your conclusions by deriving the phase portrait for the full nonlinear system.

3) (Out of touch with their own feelings) Suppose Romeo and Juliet react to each other, but not to themselves: R = aJ, J = bR. What happens? (Fire and water) Do opposites attract? Analyze R = aR + bJ, J = -bR – aJ. (Peas in a pod) If Romeo and Juliet are romantic clones R = aR + bJ, J = bR + aJ.  (should they expect boredom or bliss? (Romeo the robot) Nothing could ever change the way Romeo feels about Juliet: R = 0, J = aR + bJ. Does Juliet end up loving him or hating him?

4) (Name-calling) Suggest names for the four romantic styles, determined by the signs of a and b in R = aR + bJ. Consider the affair described by R = j, J = -R+J. a) Characterize the romantic styles of Romeo and Juliet. b) Classify the fixed point at the origin. What does this imply for the affair? c) Sketch R ( t ) and J ( t ) as functions of t, assuming R (0) = 1, J (0) = 0. In each of the following problems, predict the course of the love affair, depending on the signs and relative sizes of a and b.

5) (First-order phase transition) Consider the potential V ( x ) for the system x = rx +x³ – x⁵ . Calculate r _c, where r_c is defined by the condition that V has three equally deep wells, i.e., the values of V at the three local minima are equal. (Note: In equilibrium statistical mechanics, one says that a first-order phase transition occurs at r = rc  . For this value of r, there is equal probability of finding the system in the state corresponding to any of the three minima. The freezing of water into ice is the most familiar example of a first-order phase transition.)

6) Show that the first-order system x = r –x – e ^–x undergoes a saddle-node bifurcation as r is varied, and find the value of r at the bifurcation point. Show that the first-order system x = x(1 –x²) – a(1 –e ^–bx ) undergoes a transcritical bifurcation at x = 0 when the parameters a, b satisfy a certain equation, to be determined. (This equation defines a bifurcation curve in the ( a, b ) parameter space.) Then find an approximate formula for the fixed point that bifurcates from x = 0, assuming that the parameters are close to the bifurcation curve.

7) The research in this chapter indicates that humor used in lectures can help students retain information (to an extent), but that humor used in textbooks did not engender the same effect. Why might this be?

8) The authors mention in the section on humor and test anxiety that further research is necessary to determine if other mechanisms besides anxiety might mediate the relationship between humor and test performance. What other mechanisms might mediate this relationship? Design a study in order to explore this relationship. Include hypotheses, conditions, measures, participants, and expected results.

9) Given how humor may be used in the workplace, could humor usage be dependent on the job area (food service, retail, office work, etc.)? How might humor differ across different jobs and what outcomes might it have? How would these patterns tie in with current and past research?

10) Norman Cousins conducted informal research in an unusual way in that he was the subject of his own experiment. How do you think that might impact his research? Would it be more subjective or objective?

11) A) Do you believe that research thus far has supported humor alleviating physical symptoms in one area more than the other (e.g., in heart disease more than longevity)? Explain your reasoning. B) A speculative tone is present in much of this chapter. What do you think the reason for that tone is? Do you agree? Why or why not?

12) Under Risks of Humor in Psychotherapy, Kubie is quoted as saying that “[h]umor has its place in life. Let us keep it there by acknowledging that one place where it has a very limited role, if any, is in psychotherapy” (p. 866). Do you agree with this statement? Based on the research and views discussed in this chapter, do the benefits outweigh the costs? In agreeing or disagreeing with this statement, what argument and evidence would you provide?

13) As stated in this chapter, there has been a significant lack of research regarding the coping use of humor within social interaction and contexts. However, creating meaningful social interactions in a laboratory setting is a difficult task. How might you study the effects of coping humor within social relationships? How might this method add to the current literature?

14) Regarding the role of humor in shaping intergroup relations, Ford and Ferguson’s (2004) Prejudice Norm Theory proposes that disparagement humor functions as a “releaser” of prejudice, rather than as an initiator of prejudice. According to the theory, how does disparagement function as a releaser of prejudice? Describe the findings of relevant research to support your answer.

15) Sociologist William Martineau (1972) proposed that humor functions as either a “lubricant” or as an “abrasive” for social interaction in both intra- and intergroup settings. What is meant by intragroup humor and what are some ways that humor is used as a social lubricant and as a social abrasive in intragroup settings? From the framework of Social Identity Theory, how does out-group disparagement humor affect cohesion among members of an in-group?

16) Sense of humor enhances attraction for a potential romantic partner. Discuss the four explanations or hypotheses that psychologists have offered to explain this effect, describing research that either supports or fails to support each one.

17) As people age they lose the ability to comprehend humor but not the ability to appreciate humor. Extrapolating from research discussed in Chapter 6, what parts of the brain are likely to be affected by aging to greater or lesser degrees to account for this finding?

18) When sense of humor is defined as a tendency to frequently initiate humor and amuse one’s peers, it is associated with physical activity, social dominance, aggression, and precocious language abilities in the preschool years, and disruptive classroom behavior during elementary school, but also popularity among peers and creativity in high school. However, when sense of humor is defined as the ability to comprehend and produce humor, it is associated with social competence, cooperativeness, initiative, and leadership. How do you reconcile (make sense of) these apparent inconsistencies?

19) One contentious line of research discussed in this chapter focused on the stress and coping hypothesis and the modeling/reinforcement hypothesis. Describe each of these two hypotheses and discuss research supporting each. What could be done in future research to overcome some of the limitations of current studies designed to test these hypotheses?

20) Regarding the cognitive-perceptual processes in humor, what is reality assimilation? What is fantasy assimilation? How do these two perceptions affect the way in which a situation is interpreted? How would fantasy assimilation cause a situation to be perceived as humorous?

21) This chapter discusses humor from an evolutionary psychological perspective. Does this perspective adequately explain the adaptation of humor behavior in humans? Why or why not? Are there other evolutionary explanations, besides sexual selection, that may account for the use of humor?

22) Much of the research discussed in this chapter focuses on incongruity resolution through the presentation of canned jokes and cartoons. How might other forms of humor differentiate themselves in terms of brain processes? How might these different forms of humor impact the experience of mirth? What other factors might impact the physiology of humor differently?