Sudden change in second order nonlinear systems: slow passage through bifurcation.

Dynamical systems modelling physical processes often evolve on several time- scales with different orders of magnitude. In modelling oscillating systems some simplifying assumptions have to be made. When the short-term behaviour of a natural system is considered the parameters that appear in mathematical models of such systems can be assumed constant. In the long term, however, these parameters will vary slowly because of gradual changes in the nature of the system. Moreover, system parameters can be varied deliberately by the experimenter. This slow change of the system parameter can produce an enormous effect on the state of the system at a certain moment, which can lead to undesirable responses; "small causes produce large effects". In this thesis we study sudden changes in systems that can be modeled by second order nonlinear differential equations. The model parameter slowly changes in a dynamical way and is a function of the (slow) time.Outside a certain transition region the system exhibits on a large time-scale a damped oscillation around a stable, slowly varying, parameter-dependent equilibrium solution. The dynamics of the regarded problems on a large time-scale are described with the aid of averaging methods. When the bifurcation parameter approaches a critical value, however, this asymptotic averaged approximation is not valid anymore. A sudden, rapid transition takes place, since the stability of an equilibrium changes or an equilibrium vanishes. In order to offer a quantitative analysis it is important to understand in what way the solutions of the differential equations behave in the vicinity of the bifurcation point. In order to describe a bifurcation or jump phenomenon a local approximation has to be made. Local analysis yields that Painlevé equations can play an important role in the bifurcation process. In specific cases the solutions of the nonlinear transition equations can exhibit algebraic growth or they can explode (via a singularity). In this study the validity of the local approximations has been proven for a large class of systems. With the aid of matching techniques and a thorough analysis of the transition equation an accurate prediction has been made of the behaviour of solutions after passage of the bifurcation point. Examples of nonlinear systems that are of the same type as the problems which are considered in this thesis can be found in mechanics, climatology, biology, astronomy, and space craft technology. The power of the mathematical analysis, that has been performed in this thesis, is that it can be applied to a large class of dynamical systems. The analysis in this thesis has been carried out with the use of singular perturbation techniques. The problems that we consider are related to physical systems. We distinguish between variables, which are related to the dynamics of the system, and parameters that exhibit a slow change on a large time-scale. The problem is treated dynamically, since we take into consideration a slowly varying parameter.In chapter 2 an elementary bistable system is considered that corresponds qualitatively to the Euler arc from mechanics. For this system a sudden moment of snap- through occurs, since the parameter that describes the "stiffness" of the problem slowly varies in time and causes a dynamical bifurcation. Solutions of the system, that originally oscillate around a slowly decreasing equilibrium solution, exhibit a sudden jump behaviour and a transition to an other equilibrium state occurs. From the point of view of the qualitative analysis it is important to make a good prediction of the moment in time at which this jump will take place. The significant degeneration, which describes the jump phenomenon, is a nonlinear differential equation that can not be solved in terms of known functions or combinations of known functions; the local transition behaviour is described by the first Painlevé transcendent. Although it is not possible to deduce an asymptotic expression for the exact moment of snap-through, we are able to obtain an expression for the upper and lower limit of the expected moment of snap-through. In order to achieve this goal we apply numerical methods. A specific solution of the first Painlevé equation can be derived that matches the parameter-dependent equilibrium solution which had existed before the jump took place. The zero of this specific solution can be considered as a natural constant for the first Painlevé equation. The limit expansion of this solution has an asymptotic series. The zero of the solution of the first Painlevé equation with certain matching conditions yields a better approximation for the moment that the system snaps through. The matching condition (and therefore also the approximation of the jump moment) depend on the amplitude and the phase of the original oscillation, and on the velocity at which the parameter slowly changes.In chapters 3 and 4 systems are analyzed which are modelled by second order nonlinear differential equations that pass a pitchfork bifurcation. In the vicinity of a certain critical value of the state parameter a transition occurs from a stable "straight" equilibrium to a parabolic equilibrium curve. The leading order transition equation, which describes the pitchfork bifurcation, is the second Painlevé transcendent. The transcendental solutions of this equation either algebraically grow, which corresponds with a transition from the linear equilibrium to one of the two stable branches of the parabolic equilibrium curve, or exponentially decay, which corresponds with a transition to the unstable, slowly varying equilibrium solution after bifurcation.In chapter 3 we analyze the validity of the different asymptotic expansions. In order to obtain a global picture of the system we apply matching techniques and approximation theorems, that are obtained by extending existing first and second order averaging methods, and we extend the local solutions. It is proven that the different local solutions overlap. The matching conditions depend on the initial conditions and on the values of the parameters. Analytical analysis of the transition equation provides information about the required matching procedures. Moreover, it is seen that the asymptotic approximation remains valid before, during, and after the pitchfork bifurcation. There is a connection between the slowly oscillating solutions before and after passage of the bifurcation point. It is possible to predict accurately which stable branch of the parabolic equilibrium curve win be followed after bifurcation dependent on the state of the system "far away" from the bifurcation point. In this thesis an interesting connection has been discovered between the recent theory of Painlevé equations and the applications of singular perturbation techniques.In chapter 4 we are concerned with the dynamics of a slowly varying Hamiltonian system for which the phase portrait for a fixed value of the forcing function qualitatively changes with time. This phase portrait periodically changes and a figure-eight separatrix periodically disappears and reappears. As the system parameter changes a double homclinic loop is born which grows to a maximum, shrinks back into the origin, lies dormant, and then is born again; the bifurcation parameter periodically crosses a critical value corresponding to a supercritical pitchfork bifurcation. Dependent on the initial state of the system and the values of the parameters the system can exhibit chaotic or (quasi) periodic behaviour. The sequence of stable upper and lower branches which a given trajectory follows after passage of the pitchfork bifurcation can be irregular and so the system can exhibit sensitive dependence on initial conditions. The attraction properties can be analyzed with the aid of a Poincaré map. Chaotic dynamics almost always occur in a system without friction. For a dissipative system it is more likely that it win exhibit a periodic behaviour from a certain moment in time. The validity of the approximating Poincaré map and of the matched asymptotic approximations can be proven on a large time- scale. The proof of this validity has been carried out with the aid of an approximation theorem that concerns the averaging of oscillating functions with a slowly varying frequency, an extension theorem, matching techniques, and connection formulas for the solutions of the second Painlevé equation. Numerical simulations confirm the results that are obtained with analytical methods. A mechanical example is the motion of a simple pendulum that is connected to a rotating, rigid body.Finally, in chapter 5 we study the general class of nonlinear second order problems with a slowly varying parameter that passes a critical value corresponding to a transcritical. bifurcation. The jump phenomenon and the pitchfork bifurcation can be generalized in the same way as the transcritical bifurcation problem has been generalized in chapter 5. In this case, in the vicinity of a certain critical value of the parameter a transition occurs from a stable "straight" equilibrium to an other "straight" equilibrium, whereas the originally stable equilibrium becomes unstable. Again, the local solution that is obtained with averaging methods is valid outside a certain nonlinear transition layer and yields matching conditions for the second order differential equation that is generic for this type of bifurcation. This significant degeneration, however, does not possess the Painlevé property. Solutions of the transition equation decrease exponentially, explode or exhibit algebraic growth, which corresponds to a transition from the one stable equilibrium to another. The chance of an explosion becomes larger when the amplitude of the original oscillation is larger. With the aid of local asymptotic approximations and an analysis of the transition equation it can be investigated whether or not the system win explode on a certain moment. The "explosion condition" depends on the initial conditions and the values of the parameters