Research On Bifurcation, Chaos And Control Of Three Types Of High Dimensional Systems

This dissertation considered three high dimensional systems. First as an example, a map was considered to study the bifurcations and chaos when the fixed points lost their stability. Then it focused on two kinds of period-doubling bifurcations and chaotic behaviors in a two-degree-of-freedom-vibro-impact system when the periodic solutions lost its stability. Finally a mechanical system with periodic coefficients was investigated. Its differential equations were established. The perturbed differential equations with periodic coefficients of its periodic motions were derived. Moreover stability, bifurcations, chaos and control were addressed. Numerical simulation results were presented. The main respects of the research are followings:1. It was surveyed in chapter1that some recent achievements and developments and of the theoretic research and engineering applications on the stability, bifurcations, chaos and control of maps, vibro-impact systems and dynamics systems with periodic coefficients. The main contents and results of the dissertation were introduced.2. A three-dimensional map was considered. Some kinds of bifurcations were investigated while the fixed points losing their stability. The corresponding analytic conditions which the variant parameters satisfy were derived when different kinds of bifurcations occur. A procedure to compute Hopf-Flip bifurcations of fixed points for3-dimensional maps was established by using projection technique. The normal form coefficients are computed. Finally, numerical simulation results demonstrated it. The typical period-doubling bifurcation leading to chaotic was found while the fixed points losing their stability. There was also a period-doubling bifurcation of tori leading to chaotic via curve doubling when the Hopf invariant circle lost its stability. As the control parameters were varied, it can vary from smooth invariant circle to tori doubling-tori breaking-smooth invariant circle. The fixed point would lock into one attractive invariant circle via four orbits while Hopf bifurcation occurs under strong resonant conditions (q=1). The attractive invariant circle would break into four quasi-periodic invariant circles as the control parameters vary. Moreover the four quasi-periodic invariant circles would lock into four fixed points as the parameters vary. Two chaotic attractor occured under non-resonant Hopf-Flip conditions. Six isolated chaotic attractors or six fixed points occured under weak-resonant Hopf-Flip conditions (λ06=1). Four connected or non-connected belt chaotic attractors occured under strong-resonant Hopf-Flip conditions (λ04＝1). 3. The typical period-doubling bifurcation leading to chaotic in a two-degree-of-freedom-vibro-impact system when the periodic solutions lost its stability was investigated theoretically and by means of numerical simulations. It was show that there was also non-typical period-doubling bifurcation leading to chaotic. Quasi-periodic motion occured while the periodic1-1impact motion losing its stability under non-resonant Hopf conditions (but near strong-resonant or weak-resonant zone, Hopf invariant circle occurs on the Poincare section while the fixed points losing their stability). If the system parameters varied near the resonant point, because of the existence of Arnold tongue, the fixed point would lock into one Hopf invariant circle via n orbits while losing its stability. As the control parameters vary across n’s order resonant parameters area (Arnold tongue), the fixed point will lock into n fixed point by sub-harmonic bifurcation. As the parameters vary further, the n fixed point will lock into periodic2n point by sub-harmonic bifurcation. And periodic4n point, periodic8n point, periodic16n point,..., chaotic will occur. Finally there was one period-doubling bifurcation leading to chaotic which had n branches.4. A two-degree-of-freedom mechanical system was investigated. The differential motion equations were established by making use of Lagrange’s equations. When the endpoint of the arm performs a prescribed motion, the undisturbed motion can be given by making use of the relations of edges and angles, and the differential variable denoting by the endpoint’s coordinates and their derivatives. The disturbed motion equations of implicit type can be derived while the prescribed motions were disturbed. Assuming the disturbance, the first-order approximate disturbed motion equations of explicit type can be derived by making use of asymptotic method. And the second-order approximate disturbed motion equations, the third-order one,..., arbitrary nth order approximate disturbed motion equations of explicit type can also be obtained. So the analysis of stability and bifurcation of motion was converted to the analysis of stability and bifurcation of the equilibrium points of nth order approximate disturbed motion equations. For the disturbed motion equations, the stability and bifurcation types of the equilibrium points are depended on the first three order nonlinear terms. In other words, the fourth-order term and higher order term do not affect essentially the analysis of dynamical behaviors. The first six order nonlinear approximate disturbed motion equations were derived in this dissertation. The difference of the numerical simulation results between the six-order approximate disturbed motion equations and the third-order one was compared. The simulation results show that the conclusion is correct.5. The relation of stability between linear differential equations with periodic coefficients and linear differential equations with constant coefficients was introduced. The stability criteria were given. The relation of stability between nonlinear systems with periodic coefficients and its corresponding linear systems was introduced. The stability criteria were given. A suitable map can be constructed making use of a series of integrating differential equations with periodic coefficients over ti=(i-1)xT->iT (i=1,…,n), which is the Poincare map on the Poincare section (σ={(φ10,φ10, φ20,φ20, t)∈R4×S t=T}). Several bifurcations conditions were given when the equilibrium points of the differential equations with periodic coefficients lost their stability. The analysis method studying of bifurcations of the differential equations with periodic coefficients is introduced. Numerical simulation results were presented. The Flip bifurcation, Hopf bifurcation under non-resonant conditions (λ0n(ε0)≠1, n≠2,3,4,…), Hopf bifurcation under strong-resonant conditions (λ3=1、λ2=1), Hopf-Flip bifurcation were obtained. Bifurcations behaviors of the differential systems with periodic coefficients are basically similar to the that of the map with constant coefficients. Two different results of Hopf-Flip bifurcation were given:One case is that it maybe form two quasi-periodic invariant circles or periodic n point when the equilibrium points lose their stability. Another case is that it can not any invariant circles or periodic n point. It is a particularcare when Hopf bifurcation occur under strong-resonant conditions (λ2=1). When the eigenvalues are close the unit circle, four negative real roots will be resolved into two negative real roots which magnitude are smaller then1and a pair of roots of complex is conjugate. Finally the roots of complex conjugate cross the unit circle and transform into a pair of-1root (the other two are negative real roots which magnitude are smaller then1). Because of the particularity of root crossing the unit circle, the bifurcation results are special:stable fixed point-periodic2focus points-stable Hopf invariant circle-periodic2node points-periodic4node points, periodic8node points, periodic16node points,..., periodic2n node points, and chaos. The phase diagram of period-doubling bifurcation leading to chaotic was given.6. Three methods of control chaos for the dynamical system with constant coefficients were introduced:the variable-parameter linear controller, the translation, the state variables feedback and parameter variation. The control of delaying Flip bifurcation and Hopf bifurcation has been studied. The bifurcation phase diagrams are obtained. For Flip bifurcation control:(1) Adopting the variable-parameter linear controller, periodic2points can be controlled to fixed point taking proper control parameters. Otherwise it may be controlled to Hopf circle.(2) Adopting the state variables feedback and parameter variation, periodic2points can be controlled to fixed point taking suitable control parameters. Otherwise it may be controlled to Hopf circle or chaos state.(3) Adopting the translation, it is failed to achieve the control of Flip bifurcation. For Hopf bifurcation control:(1) Adopting the variable-parameter linear controller, Hopf circle can be controlled to fixed point taking proper control parameters. Otherwise it may be controlled to other smooth Hopf circle or deformed Hopf circle.(2) Adopting the state variables feedback and parameter variation, Hopf circle can be controlled to fixed point taking suitable control parameters. Otherwise it may be controlled to other smooth Hopf circle or deformed Hopf circle or periodic7points or chaos state.(3) Adopting the translation, it is failed to achieve the control of Hopf bifurcation.