Center Manifold Theorem Technique Research Articles

Bifurcations of a harmonically excited system impacting between two moving rigid bodies near a 1:4 strong resonance point

A harmonically excited system with rigid body impacts is considered. The periodic motions and Poincaré mapping of the vibro-impact system are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré mapping to a two-dimensional one, and the normal form mapping associated with 1:4 strong resonance is obtained. Two-parameter bifurcations of the fixed points of 1:4 strong resonance in the vibro-impact system are analyzed. The simulation results illustrate some interesting dynamical features: in the vibro-impact system, there exist Neimark-Sacker bifurcations of periodic-impact motions and tangent and fold bifurcations of period-4 orbits near the bifurcation point of 1:4 strong resonance.

Vibro-impact dynamics near a strong resonance point

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincare map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincare map. A center manifold theorem technique is applied to reduce the Poincare map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, Tin, Ton and Tout type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasi-periodic impact motions.

Periodic-impact motions and bifurcations of vibro-impact systems near 1:4 strong resonance point

Two typical vibro-impact systems are considered. The periodic-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map associated with 1:4 strong resonance is obtained. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed. The results from simulation illustrate some interesting features of dynamics of the vibro-impact systems. Some complicated bifurcations, e.g., tangent, fold and Neimark–Sacker bifurcations of period-4 orbits are found to exist near the 1:4 strong resonance points of the vibro-impact systems.

Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops

A vibratory system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Local codimension two bifurcations of the vibratory system with symmetrical rigid stops, associated with double Hopf bifurcation and interaction of Hopf and pitchfork bifurcation, are analyzed by using the center manifold theorem technique and normal form method of maps. Dynamic behavior of the system, near the points of codimension two bifurcations, is investigated by using qualitative analysis and numerical simulation. Hopf-flip bifurcation of fixed points in the vibratory system with a single stop are briefly analyzed by comparison with unfoldings analyses of Hopf-pitchfork bifurcation of the vibratory system with symmetrical rigid stops. Near the value of double Hopf bifurcation there exist period-one double-impact symmetrical motion and quasi-periodic impact motions. The quasi-periodic impact motions are represented by the closed circle and “tire-like” attractor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via “tire-like” torus doubling.

Double Neimark–Sacker bifurcation and torus bifurcation of a class of vibratory systems with symmetrical rigid stops

A multidegree-of-freedom system having symmetrically placed rigid stops and subjected to periodic excitation is considered. The system consists of linear components, but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints. Such models play an important role in the studies of mechanical systems with clearances or gaps. Double Neimark–Sacker bifurcation of the system is analyzed by using the center manifold and normal form method of maps. The period-one double-impact symmetrical motion and homologous disturbed map of the system are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a four-dimensional one, and the normal form map associated with double Neimark–Sacker bifurcation is obtained. The bifurcation sets for the normal-form map are illustrated in detail. Local behavior of the vibratory systems with symmetrical rigid stops, near the points of double Neimark–Sacker bifurcations, is reported by the presentation of results for a three-degree-of-freedom vibratory system with symmetrical stops. The existence and stability of period-one double-impact symmetrical motion are analyzed explicitly. Also, local bifurcations at the points of change in stability are analyzed, thus giving some information on dynamical behavior near the points of double Neimark–Sacker bifurcations. Near the value of double Neimark–Sacker bifurcation there exist period-one double-impact symmetrical motion and quasi-periodic impact motions. The quasi-periodic impact motions are represented by the closed circle and “tire-like” attractor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via “tire-like” torus doubling.

Dynamical behavior of a class of vibratory systems with symmetrical rigid stops near the point of codimension two bifurcation

A multi-degree-of-freedom vibratory system having symmetrically placed rigid stops is considered. The system consists of linear components, but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints. Such models play an important role in the studies of mechanical systems with clearances or gaps. Local codimension two bifurcation of maps, involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously, is analyzed by using the center manifold theorem technique and normal form method for maps. Symmetrical double-impact periodic motion and Poincaré map of the system are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with the codimension two bifurcation is obtained. Local behaviors of the vibratory systems with symmetrical rigid stops, near the points of codimension two bifurcations, are reported by the presentation of results for a two-degree-of-freedom vibratory system with symmetrical stops. The existence and stability of symmetrical double-impact periodic motion are analyzed explicitly. Also, local bifurcations at the points of change in stability, are analyzed. Near the point of codimension two bifurcation, there exists not only Hopf bifurcation of period-one double-impact motion, but also pitchfork bifurcation of the motion. Pitchfork bifurcation of period-one double-impact symmetrical motion results in the period-one double-impact unsymmetrical motion. The unsymmetrical double-impact motion is of two antisymmetrical forms due to different initial conditions and symmetrical stops. With change of the forcing frequency, the unsymmetrical double-impact periodic motion will undergo Hopf bifurcation. Moreover the period-one double-impact symmetrical motion will undergo Hopf bifurcation directly as the forcing frequency is changed in the contrary direction. The routes of quasi-periodic impact motions to chaos are observed by results from simulation.

Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1:2 resonance case

A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincare section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincare map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1:2 resonance is obtained. Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed.

Hopf-flip bifurcations of vibratory systems with impacts

Two vibro-impact systems are considered. The period n single-impact motions and Poincaré maps of the vibro-impact systems are derived analytically. Stability and local bifurcations of single-impact periodic motions are analyzed by using the Poincaré maps. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. It is found that near the point of codim 2 bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Period doubling bifurcation of period one single-impact motion is commonly existent near the point of codim 2 bifurcation. However, no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of Hopf bifurcation associated with period two fixed points. The results from simulation shows that there exists an interest torus doubling bifurcation occurring near the value of Hopf-flip bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transit to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems.

Dynamical behavior of vibro-impact machinery near a point of codimension two bifurcation

A dual component system with vibro-impact is considered. Local codimension two bifurcation of maps, involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously, is analyzed by using the center manifold and normal form method for maps. The period n single-impact motion and Poincaré map of the vibro-impact system are derived analytically. Stability and local bifurcation of single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a three-dimensional one, and the normal form map associated with the codimension two bifurcation is obtained. Local behavior of the dual component system with vibro-impact, near the point of codimension two bifurcation, is analyzed. It is found that near the point of codimension two bifurcation, there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. Period doubling bifurcation of period one single-impact motion commonly exists near the point of codimension two bifurcation. However, no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of Hopf bifurcation associated with period two fixed points. The period two fixed points are symmetrical about the corresponding period one point. The results from simulation show that the attracting invariant circles associated with period two points are symmetrical about the corresponding period one point near the value of Hopf bifurcation of period two points. Two actual examples, the impact-forming machinery and inertial shaker, are chosen to analyze further the phenomena of codimension two bifurcation of maps, and local bifurcation analysis and numerical simulation are carried out to unfold dynamical behavior of these vibro-impact systems near the points of codimension two bifurcations.

Codimension two bifurcation of a vibro-bounce system

A three-degree-of-freedom vibro-bounce system is considered. The disturbed map of period one single-impact motion is derived analytically. A center manifold theorem technique is applied to reduce the Poincare map to a three-dimensional one, and the normal form map associated with Hopf-flip bifurcation is obtained. Dynamical behavior of the system, near the point of codimension two bifurcation, is investigated by using qualitative analysis and numerical simulation. It is found that near the point of Hopf-flip bifurcation there exists not only Hopf bifurcation of period one single-impact motion, but also Hopf bifurcation of period two double-impact motion. The results from simulation show that there exists an interesting torus doubling bifurcation near the codimension two bifurcation. The torus doubling bifurcation makes the quasi-periodic attractor associated with period one single-impact motion transform to the other quasi-periodic attractor represented by two attracting closed circles. The torus bifurcation is qualitatively different from the typical torus doubling bifurcation occurring in the vibro-impact systems. Different routes from period one single-impact motion to chaos are observed by numerical simulation.

Hopf bifurcations and chaos of a two-degree-of-freedom vibro-impact system in two strong resonance cases

A two-degree-of-freedom system with one-side amplitude constraint subjected to periodic excitation is considered. Dynamics of the system are studied with special attention to Hopf bifurcations of period motions in two kinds of strong resonance cases. The Poincaré map of the vibro-impact system with proportional damping property is established. The stability of a class of periodic motions with one impact is investigated by analytical methods. Hopf bifurcation values and intersecting conditions of the period motions with one impact, in two kinds of strong resonance cases, are determined. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, which is put into normal form by theory of normal forms. By theory of Hopf bifurcations of fixed points in R 2 -strong resonance, local dynamic behavior of the vibro-impact system, near points of resonance, is analyzed. The theoretical analyses are verified by numerical solutions. Routes of periodic impacts to chaos, in two kinds of strong resonance cases, are obtained by numerical simulations.

Bifurcations and chaos in a system with impacts

A vibro-impact system is considered. A body bounces on a flat horizontal surface of the vibro-bench. Hopf bifurcations of the vibro-impact system, in two kinds of strong resonance cases ( λ 0 3=1 and λ 0 4=1), are investigated. A Poincaré map of the system is established. The period 1 single-impact motion of the system and its stability are studied by analytical methods, and Hopf bifurcation values and intersecting conditions of the system in strong resonance cases are determined. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, which is put into normal form by theory of normal forms. By the theory of Hopf and subharmonic bifurcations of fixed points in R 2 -strong resonance, dynamic behavior of the vibro-impact system near points of resonance is analyzed. In the resonance case of λ 0 3=1, the system generally exhibits unstable period-3 three-impact motion; in the resonance case of λ 0 4=1, the system can exhibit stable period-4 four-impact motion and quasi-periodic motion. The theoretical analyses are verified by numerical solutions. Routes to chaos, in two kinds of strong resonance cases considered, are obtained by numerical simulations. Quasi-periodic impacts in non-resonance and weak resonance cases, doubling-periodic bifurcations and chaotic motions are also stated briefly by numerical analyses.