Note for MATH 454 at UA
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Date Created: 02/06/15
TOPICS COVERED IN MATH 454 1 De nition and attributes of a dynamical system 0 A dynamical system is often characterized in terms of the following attributes lts dimension 7 Its linear or nonlinear nature 7 Its autonomous or non autonomous character 0 We only consider dynamical systems which are continuously differentiable This implies that solutions exist and are uniquely determined by initial conditions 0 We assume that the right hand side of the dynamical system has real coef cients and that the dependent variables are therefore always real 2 Special solutions 21 Fixed points 0 De nition of a xed point 0 How to nd all of the xed points of a dynamical system 22 Stability of a xed point 221 Onedimensional systems For one dimensional systems7 one can use the right hand side of the equation to determine the nonlinear stability of the xed point Linear stability is determined by the sign of the derivative of the right hand side calculated at the xed point 222 Twodimensional systems 0 For two dimensional systems7 we start by linearizing the system about a xed point The trace and the determinant of the Jacobian allows us to classify the xed point and to determine its linear stability properties 0 The Hartman Grobrnan theorem tells us that if a xed point is hyperbolic ie if all of the eigenvalues of the Jacobian of the linearization about the xed point have non zero real parts7 then the nature of the xed point does not change when nonlinear effects are taken into account 4 41 Bifurcations Different types of bifurcations Bifurcations that may occur in dynamical systems of dimension one or higher are The saddle node bifurcation The transcritical bifurcation The pitchfork bifurcation which can be supercritical or subcritical Hopf bifurcations may only occur in systems of dimension two or higher 42 Normal forms Normal forms are minimal equations that capture the nature of a particular bifurcation the word minimal is used in the sense that only the relevant linear and nonlinear terms appear in the normal form 421 Onedimensional systems For one dimensional systems identifying bifurcations and nding their normal forms in volves the following steps Plot the bifurcation diagram showing the branches of xed points as functions of the control parameter ldentify each bifurcation point bifurcations typically occur when two branches of xed points collide Determine the nature of each bifurcation based on the local characteristics of the bifurcation diagram For each bifurcation change variables so that the bifurcation occurs at z 0 and when the control parameter crosses zero Taylor expand the system in these new variables and only keep the relevant linear and nonlinear terms Re scale time and z to set most coef cients in the normal form to 1 or 71 Use the normal form to determinecon rm the stability of the xed points before and after the bifurcation and in the case of pitchfork bifurcations to decide if the bifurcation is supercritical or subcritical 422 Twodimensional systems For two dirnensional systerns7 saddle node7 transcritical and pitchfork bifurcations are de scribed and characterized by one dirnensional normal forms You should be able to recog nize a particular bifurcation based on the corresponding changes that occur in the phase plane A Hopf bifurcation occurs when the two cornplex conjugate eigenvalues of the Jacobian of the linearization about a xed point cross the imaginary axis The bifurcation is super critical if a limit cycle exists after the xed point has lost stability7 and subcritical if the limit cycle exists before the xed point has lost stability The normal form of the Hopf bifurcation is an equation for a single cornplex variable7 and has cornplex coef cients The steps to derive such a normal form are as follows Change variables in the original two dirnensional system so that the bifurcation occurs at z y 0 and when the control pararneter crosses zero Re write the resulting system in the basis of eigenvectors of the Jacobian of the linearization about the xed point7 evaluated at the bifurcation Use one cornplex coordinate variable z7 instead of two real variables Taylor expand the right hand side of the differential equation for 2 at least to order three in powers of z and 2 Make near identity changes of variables to remove the quadratic terms if any from the right hand side of the equation for z This operation generates cubic terrns Make near identity changes of variables to show that all of the cubic terms can be removed from the equation for 2 except the term in The resulting equation is the normal form of the Hopf bifurcation Re scale time and z to set some of the coef cients in the normal form to 1 or 71 Use the normal form to determine the amplitude of the limit cycle as a function of the control pararneter7 and to describe how the frequency of oscillations depends on the amplitude of the limit cycle The normal form can also be used to study the stability of the limit cycle
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