- Email: [email protected]
Stochastic bifurcation in duffing system subject to harmonic excitation and in presence of random noise
International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
Stochastic bifurcation in du!ng system subject to harmonic excitation and in presence of random noise Wei Xua;∗ , Qun Hea; b , Tong Fanga , Haiwu Rongc a Northwestern
Polytechnical University, Xian 710072, China College of Armed Police Force, Xian 710086, China c Foshan Institute of Science and Technology, Foshan 528000, China b Engineering
Received 15 November 2003; received in revised form 15 December 2003; accepted 2 February 2004
Abstract A global analysis of stochastic bifurcation in a special kind of Du!ng system, named as Ueda system, subject to a harmonic excitation and in presence of random noise disturbance is studied in detail by the generalized cell mapping method using digraph. It is found that for this dissipative system there exists a steady state random cell :ow restricted within a pipe-like manifold, the section of which forms one or two stable sets on the Poincare cell map. These stable sets are called stochastic attractors (stochastic nodes), each of which owns its attractive basin. Attractive basins are separated by a stochastic boundary, on which a stochastic saddle is located. Hence, in topological sense stochastic bifurcation can be de
1. Introduction The theory of stochastic bifurcation is still in its infancy. In fact, it is much harder to deal with stochastic bifurcation problems than deterministic bifurcation problems. The de
∗ Corresponding author. Tel.: +86-29-884-92393; fax: +86-29884-94404. E-mail address: [email protected] (W. Xu).
0020-7462/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2004.02.009
bifurcation can be based upon the sudden change of topological properties of the portrait of phase trajectories. At present, there are mainly two kinds of de
1474
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
variation of the random excitation is the di!culty encountered by P-bifurcation. On the other hand, lack of e!cient and accurate algorithm for calculating Lyapunov exponent is the di!culty encountered by D-bifurcation. Besides, several studies show that sometimes these two kinds of de
disappears; meanwhile, the stochastic attractor suddenly loses its attraction and its attractive basin. Both the former attractor and its attractive basin turn into a new expansion of attractive basin for the remaining stochastic attractor. Yet the former attractor still remains a marked brand (actually an unstable stochastic node) on the Poincare map where the lingering time of a cell is far longer than elsewhere in the attractive basin. As the level of the random disturbance further increases, the marked brand grows bigger and bigger, drawing towards the remaining attractor. When the level of the random disturbance further increases and passes through another critical value, the marked brand and the remaining attractor collide and merge into a much bigger attractor. These pictures clearly demonstrate the happening and evolution of stochastic bifurcations in a Ueda system. There comes the idea that a stochastic attractor may be taken as invariance for the randomly perturbed steady-state response. The shape, size, and stability of a stochastic attractor may be taken as its character. Whenever the character of an attractor changes radically, there happens the stochastic bifurcation. Hence, as an alternative, topological de
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
through the topological analysis of the cell mapping system. While generating a cell mapping system, the dynamical mechanism of the original system can be transplanted into it through the following ways: For a simple cell mapping system, the mechanism of its one step cell mapping is fully determined by the point mapping of the original system at a single representative point of the cell [5]. The simple cell mapping method is economic, simple and straightforward, but it may lose too much useful information about the original system. By introducing the information of the cell mappings of the adjacent cells, an interpolated cell mapping method was suggested by Tongue and Gu [6], somewhat improving the simple cell mapping method. Shortly after the simple cell mapping method was introduced, at
1475
including proposals for constituting algorithms for the generalized cell mapping and digraph. For topological analysis, the global generalized cell mapping can be identical with a digraph, with a cell identical to a vertex, and a cell mapping identical with a directional arc (or arcs) connecting two (or more) vertices. Then, the topological analysis of a cell mapping can be transformed to that of a digraph. In a digraph, there may exist some strongly connected sub-digraphs, in which every pair vertices are mutually reachable. A strongly connected sub-digraph is identical with a set of persistent self-cycling cells in a global cell map, namely an attractor of the cell mapping system. A strongly connected sub-digraph is a closed one, so is the persistent self-cycling cell set. If a vertex of a digraph does not belong to a close strongly connected sub-digraph, then its corresponding cell is called a transient cell. Transient self-cycling cell sets are usually associated with unstable solutions, such as saddles, unstable nodes, unstable chaotic sets, etc. Moreover, if a transient cell can reach a persistent self-cycling cell set, then the latter is called a domicile of the transient cell. A transient cell can own more than one domicile. Transient cells can be divided into single-domicile transient cells and multiple-domicile transient cells. The single-domicile transient cell sets and multiple-domicile transient cell sets usually associate with the attractive basin or the basin boundary. Thus, by topological sorting of a digraph, a clear topological structure will be presented on the generalized Poincare cell map. Hsu [11] further suggested that a generalized cell mapping system can be cast as a partially ordered set consisting of transient cells and persistent self-cycling cell sets. The global ordering of the transient cells and the persistent self-cycling sets then brings out certain numbers, characteristics of the topological property of the dynamical system. The advantage of the generalized cell mapping method using digraph lies in that it can determine not only the stable invariant mapping sets, but also the unstable ones (such as saddles, unstable nodes, etc.); moreover, it can provide a picture of global transient evolution process by using topological sorting algorithms. Based on [11], Xu and Hong suggested a new version of the generalized cell mapping digraph (GCMD) method [12], in which diMerent from [11] the Warshall algorithm in Boolean algebra was adopted. By using the GCMD method, Hong and Xu successfully
1476
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
clari
(1)
Fig. 1. The global cell map for system (1) with = 0:0.
where the over head dots indicate diMerentiation with respect to the time t, (t) is supposed to be a computer realizable ‘Gaussian unit white noise’, is a bifurcation parameter representing energy level of the random noise, and = 0:2, = 1:0, = 0:3, ! = 1:0. We study stochastic bifurcation of system (1) by the generalized cell mapping method using digraph through Monte-Carlo simulations. A cell structure of 200×200 cells is uniformly de
Fig. 2. The global cell map for system (1) with = 0:003.
two single-domicile transient cell sets (basins of attraction) and a multiple-domicile transient cell set (basin boundary, namely the incoming separatrix of saddle) in domain D, except for a sink cell (outside of D) of system (1). Now consider the eMect of random noise on the cell mapping system, especially on the evolution of global properties of the cell mapping system while the noise level is increased step by step. One thing we
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
Fig. 3. The global cell map for system (1) with = 0:007.
Fig. 4. The global cell map for system (1) with = 0:008.
are sure that in the presence of noise all the attractors, the saddle, and the boundary will be noisy, so we had better call them the stochastic ones. When the noise level is raised to = 0:003, in Fig. 2 one can see the two stochastic attractors and the stochastic saddle grow bigger and the associated stochastic boundary becomes wider, meanwhile, one of the attractors expands toward the saddle and its bottom closes with
1477
Fig. 5. The global cell map for system (1) with = 0:02.
Fig. 6. The global cell map for system (1) with = 0:029.
the stochastic boundary, which is actually a mixture of the separatrix and multiple-domicile transient cell mapping sets. This tendency will continue for a while, Fig. 3. When the value of is raised from 0.007 to 0.008, refer to Fig. 4, right after the bigger attractor and the saddle collide with each other, both the saddle and the associated boundary suddenly disappear. At the same time the attractor loses its attraction, then
1478
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
following Grebogi, Ott, and Yorke, we call it a kind of stochastic merging bifurcation. To have a quanti
Fig. 7. The global cell map for system (1) with = 0:03.
all of them, i.e. the former saddle, the boundary, and the former attractor, together with its basin of attraction, turn to be the new expansion of the attractive basin for the remaining stochastic attractor. However, the former attractor itself still remains a marked brand on the cell map as a transient non-attractive invariant set, which is actually an unstable stochastic node. By the way, in our former analysis we ever mistook this transient self-cycling set as a stochastic saddle. However, with the stochastic boundary gone, what can a stochastic saddle base on? So it must be an unstable stochastic node, where any transient cell map lingers far longer than on elsewhere in the basin of attraction, yet inevitably leaving this transient self-cycling set later on. Now since the number of attractors now decreases from 2 to 1, causing a sudden topological change of the global cell mapping in the referred region D, we call this a stochastic saddle-node bifurcation, which is typical in Ueda system. While is further raised, the unstable stochastic node grows bigger and draws towards the remaining attractor, namely the stable stochastic node, Fig. 5. While the value of passes from 0.029 to 0.03, the stable node and the unstable node collide and instantly merge into a much bigger stable stochastic node, Figs. 6 and 7. Since a sudden explosion in size of an attractor occurs, accompanied by a transition of an unstable node into a part of another stable one,
(A) Though the theory of stochastic bifurcation has been advanced to a new level in the last decade, there remain a lot of problems to be solved. Even the de
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
1479
Table 1 The number of cells in attractors, basins of attraction, unstable node, and saddle
The number of cells in attractor I
The number of cells in the basin for attractor I
The number of cells in the attractor II
The number of cells in the basin for attractor II
The number of cells in the saddle
The number of cells in the unstable node
0.000 0.003 0.005 0.007 0.008 0.009 0.01 0.02 0.029 0.03 0.04
12 70 156 266 308 369 388 942 1584 4024 5025
27 286 26 015 24 880 23 344 39 692 39 631 39 612 39 058 38 516 35 976 34 975
13 579 843 937 0 0 0 0 0 0 0
10436 4969 3705 2436 0 0 0 0 0 0 0
4 27 33 49 0 0 0 0 0 0 0
0 0 0 0 1040 1093 1134 1621 1961 0 0
is 1, while the happening probability of individual attractor depends upon the relative size of its attractive basin. Since for the same original non-linear dynamical system, the cell mapping system is supposed to be topologically similar to the point mapping system, the real dynamical :ow of the original system is most likely in a similar way. So it is reasonable to imagine that in this dissipative Ueda system the steady-state response (or responses) to the dominant harmonic excitation should play a leading role, closely around which are the randomly perturbed responses. Acknowledgements The authors are grateful to the support of National Science Foundation of China (Grant Nos. 10072049, 10332030) and also grateful to Prof. Xu Jianxue and Dr. Hong Ling for their helpful comments. References [1] L. Arnold, Random Dynamical Systems, Springer, New York, 1998. [2] P. Baxendale, Asymptotic behavior of stochastic :ows of diMeomorphisms, in: K. Ito, T. Hida (Eds.), Stochastic Processes and Their Applications, Lecture Notes in Mathematics, Vol. 1203, Springer, Berlin, Heidelberg, New York, 1986, pp. 1–19.
[3] C. Meunier, A.D. Verga, Noise and bifurcation, J. Stat. Phys. 50 (1988) 345–375. [4] H. Crauel, F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dyn. DiMerential Equations 10 (1998) 259–274. [5] C.S. Hsu, A theory of cell to cell mapping dynamical systems, ASME J. Appl. Mech. 47 (1980) 931–939. [6] B.H. Tongue, K.Q. Gu, A higher order method of interpolated cell mapping, J. Sound Vib. 125 (1988) 169–179. [7] C.S. Hsu, A generalized theory of cell to cell mapping for nonlinear dynamical systems, ASME J. Appl. Mech. 53 (1981) 695–701. [8] J.Q. Sun, C.S. Hsu, The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation, J. Appl. Mech. 57 (1990) 1018–1025. [9] J.Q. Sun, Random vibration analysis of a non-linear system with dry friction damping by the short-time Gaussian cell mapping method, J. Sound Vib. 180 (1995) 785–795. [10] R. Guder, E. Kreuzer, Using generalized cell mapping to approximate invariant measures on compact manifolds, Int. J. Bifurcation Chaos 7 (1999) 2487–2499. [11] C.S. Hsu, Global analysis of dynamical systems using posets and digraphs, Int. J. Bifurcation Chaos 5 (1995) 1085–1118. [12] Hong, Ling, Xu, Jianxue, Crises and chaotic transient by the generalized cell mapping digraph method, Phys. Lett. A 262 (1999) 361–375. [13] Hong, Ling, Xu, Jianxue, Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph (GCMD) method, Int. J. Bifurcation Chaos 11 (2001) 723–736. [14] W.Q. Zhu, M.Q. Lu, Q.T. Wu, Stochastic jump and bifurcation of a Du!ng oscillator under narrow-band excitation, J. Sound Vib. 165 (1993) 285–304.
Stochastic bifurcation in du!ng system subject to harmonic excitation and in presence of random noise Wei Xua;∗ , Qun Hea; b , Tong Fanga , Haiwu Rongc a Northwestern
Polytechnical University, Xian 710072, China College of Armed Police Force, Xian 710086, China c Foshan Institute of Science and Technology, Foshan 528000, China b Engineering
Received 15 November 2003; received in revised form 15 December 2003; accepted 2 February 2004
Abstract A global analysis of stochastic bifurcation in a special kind of Du!ng system, named as Ueda system, subject to a harmonic excitation and in presence of random noise disturbance is studied in detail by the generalized cell mapping method using digraph. It is found that for this dissipative system there exists a steady state random cell :ow restricted within a pipe-like manifold, the section of which forms one or two stable sets on the Poincare cell map. These stable sets are called stochastic attractors (stochastic nodes), each of which owns its attractive basin. Attractive basins are separated by a stochastic boundary, on which a stochastic saddle is located. Hence, in topological sense stochastic bifurcation can be de
1. Introduction The theory of stochastic bifurcation is still in its infancy. In fact, it is much harder to deal with stochastic bifurcation problems than deterministic bifurcation problems. The de
∗ Corresponding author. Tel.: +86-29-884-92393; fax: +86-29884-94404. E-mail address: [email protected] (W. Xu).
0020-7462/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2004.02.009
bifurcation can be based upon the sudden change of topological properties of the portrait of phase trajectories. At present, there are mainly two kinds of de
1474
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
variation of the random excitation is the di!culty encountered by P-bifurcation. On the other hand, lack of e!cient and accurate algorithm for calculating Lyapunov exponent is the di!culty encountered by D-bifurcation. Besides, several studies show that sometimes these two kinds of de
disappears; meanwhile, the stochastic attractor suddenly loses its attraction and its attractive basin. Both the former attractor and its attractive basin turn into a new expansion of attractive basin for the remaining stochastic attractor. Yet the former attractor still remains a marked brand (actually an unstable stochastic node) on the Poincare map where the lingering time of a cell is far longer than elsewhere in the attractive basin. As the level of the random disturbance further increases, the marked brand grows bigger and bigger, drawing towards the remaining attractor. When the level of the random disturbance further increases and passes through another critical value, the marked brand and the remaining attractor collide and merge into a much bigger attractor. These pictures clearly demonstrate the happening and evolution of stochastic bifurcations in a Ueda system. There comes the idea that a stochastic attractor may be taken as invariance for the randomly perturbed steady-state response. The shape, size, and stability of a stochastic attractor may be taken as its character. Whenever the character of an attractor changes radically, there happens the stochastic bifurcation. Hence, as an alternative, topological de
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
through the topological analysis of the cell mapping system. While generating a cell mapping system, the dynamical mechanism of the original system can be transplanted into it through the following ways: For a simple cell mapping system, the mechanism of its one step cell mapping is fully determined by the point mapping of the original system at a single representative point of the cell [5]. The simple cell mapping method is economic, simple and straightforward, but it may lose too much useful information about the original system. By introducing the information of the cell mappings of the adjacent cells, an interpolated cell mapping method was suggested by Tongue and Gu [6], somewhat improving the simple cell mapping method. Shortly after the simple cell mapping method was introduced, at
1475
including proposals for constituting algorithms for the generalized cell mapping and digraph. For topological analysis, the global generalized cell mapping can be identical with a digraph, with a cell identical to a vertex, and a cell mapping identical with a directional arc (or arcs) connecting two (or more) vertices. Then, the topological analysis of a cell mapping can be transformed to that of a digraph. In a digraph, there may exist some strongly connected sub-digraphs, in which every pair vertices are mutually reachable. A strongly connected sub-digraph is identical with a set of persistent self-cycling cells in a global cell map, namely an attractor of the cell mapping system. A strongly connected sub-digraph is a closed one, so is the persistent self-cycling cell set. If a vertex of a digraph does not belong to a close strongly connected sub-digraph, then its corresponding cell is called a transient cell. Transient self-cycling cell sets are usually associated with unstable solutions, such as saddles, unstable nodes, unstable chaotic sets, etc. Moreover, if a transient cell can reach a persistent self-cycling cell set, then the latter is called a domicile of the transient cell. A transient cell can own more than one domicile. Transient cells can be divided into single-domicile transient cells and multiple-domicile transient cells. The single-domicile transient cell sets and multiple-domicile transient cell sets usually associate with the attractive basin or the basin boundary. Thus, by topological sorting of a digraph, a clear topological structure will be presented on the generalized Poincare cell map. Hsu [11] further suggested that a generalized cell mapping system can be cast as a partially ordered set consisting of transient cells and persistent self-cycling cell sets. The global ordering of the transient cells and the persistent self-cycling sets then brings out certain numbers, characteristics of the topological property of the dynamical system. The advantage of the generalized cell mapping method using digraph lies in that it can determine not only the stable invariant mapping sets, but also the unstable ones (such as saddles, unstable nodes, etc.); moreover, it can provide a picture of global transient evolution process by using topological sorting algorithms. Based on [11], Xu and Hong suggested a new version of the generalized cell mapping digraph (GCMD) method [12], in which diMerent from [11] the Warshall algorithm in Boolean algebra was adopted. By using the GCMD method, Hong and Xu successfully
1476
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
clari
(1)
Fig. 1. The global cell map for system (1) with = 0:0.
where the over head dots indicate diMerentiation with respect to the time t, (t) is supposed to be a computer realizable ‘Gaussian unit white noise’, is a bifurcation parameter representing energy level of the random noise, and = 0:2, = 1:0, = 0:3, ! = 1:0. We study stochastic bifurcation of system (1) by the generalized cell mapping method using digraph through Monte-Carlo simulations. A cell structure of 200×200 cells is uniformly de
Fig. 2. The global cell map for system (1) with = 0:003.
two single-domicile transient cell sets (basins of attraction) and a multiple-domicile transient cell set (basin boundary, namely the incoming separatrix of saddle) in domain D, except for a sink cell (outside of D) of system (1). Now consider the eMect of random noise on the cell mapping system, especially on the evolution of global properties of the cell mapping system while the noise level is increased step by step. One thing we
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
Fig. 3. The global cell map for system (1) with = 0:007.
Fig. 4. The global cell map for system (1) with = 0:008.
are sure that in the presence of noise all the attractors, the saddle, and the boundary will be noisy, so we had better call them the stochastic ones. When the noise level is raised to = 0:003, in Fig. 2 one can see the two stochastic attractors and the stochastic saddle grow bigger and the associated stochastic boundary becomes wider, meanwhile, one of the attractors expands toward the saddle and its bottom closes with
1477
Fig. 5. The global cell map for system (1) with = 0:02.
Fig. 6. The global cell map for system (1) with = 0:029.
the stochastic boundary, which is actually a mixture of the separatrix and multiple-domicile transient cell mapping sets. This tendency will continue for a while, Fig. 3. When the value of is raised from 0.007 to 0.008, refer to Fig. 4, right after the bigger attractor and the saddle collide with each other, both the saddle and the associated boundary suddenly disappear. At the same time the attractor loses its attraction, then
1478
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
following Grebogi, Ott, and Yorke, we call it a kind of stochastic merging bifurcation. To have a quanti
Fig. 7. The global cell map for system (1) with = 0:03.
all of them, i.e. the former saddle, the boundary, and the former attractor, together with its basin of attraction, turn to be the new expansion of the attractive basin for the remaining stochastic attractor. However, the former attractor itself still remains a marked brand on the cell map as a transient non-attractive invariant set, which is actually an unstable stochastic node. By the way, in our former analysis we ever mistook this transient self-cycling set as a stochastic saddle. However, with the stochastic boundary gone, what can a stochastic saddle base on? So it must be an unstable stochastic node, where any transient cell map lingers far longer than on elsewhere in the basin of attraction, yet inevitably leaving this transient self-cycling set later on. Now since the number of attractors now decreases from 2 to 1, causing a sudden topological change of the global cell mapping in the referred region D, we call this a stochastic saddle-node bifurcation, which is typical in Ueda system. While is further raised, the unstable stochastic node grows bigger and draws towards the remaining attractor, namely the stable stochastic node, Fig. 5. While the value of passes from 0.029 to 0.03, the stable node and the unstable node collide and instantly merge into a much bigger stable stochastic node, Figs. 6 and 7. Since a sudden explosion in size of an attractor occurs, accompanied by a transition of an unstable node into a part of another stable one,
(A) Though the theory of stochastic bifurcation has been advanced to a new level in the last decade, there remain a lot of problems to be solved. Even the de
W. Xu et al. / International Journal of Non-Linear Mechanics 39 (2004) 1473 – 1479
1479
Table 1 The number of cells in attractors, basins of attraction, unstable node, and saddle
The number of cells in attractor I
The number of cells in the basin for attractor I
The number of cells in the attractor II
The number of cells in the basin for attractor II
The number of cells in the saddle
The number of cells in the unstable node
0.000 0.003 0.005 0.007 0.008 0.009 0.01 0.02 0.029 0.03 0.04
12 70 156 266 308 369 388 942 1584 4024 5025
27 286 26 015 24 880 23 344 39 692 39 631 39 612 39 058 38 516 35 976 34 975
13 579 843 937 0 0 0 0 0 0 0
10436 4969 3705 2436 0 0 0 0 0 0 0
4 27 33 49 0 0 0 0 0 0 0
0 0 0 0 1040 1093 1134 1621 1961 0 0
is 1, while the happening probability of individual attractor depends upon the relative size of its attractive basin. Since for the same original non-linear dynamical system, the cell mapping system is supposed to be topologically similar to the point mapping system, the real dynamical :ow of the original system is most likely in a similar way. So it is reasonable to imagine that in this dissipative Ueda system the steady-state response (or responses) to the dominant harmonic excitation should play a leading role, closely around which are the randomly perturbed responses. Acknowledgements The authors are grateful to the support of National Science Foundation of China (Grant Nos. 10072049, 10332030) and also grateful to Prof. Xu Jianxue and Dr. Hong Ling for their helpful comments. References [1] L. Arnold, Random Dynamical Systems, Springer, New York, 1998. [2] P. Baxendale, Asymptotic behavior of stochastic :ows of diMeomorphisms, in: K. Ito, T. Hida (Eds.), Stochastic Processes and Their Applications, Lecture Notes in Mathematics, Vol. 1203, Springer, Berlin, Heidelberg, New York, 1986, pp. 1–19.
[3] C. Meunier, A.D. Verga, Noise and bifurcation, J. Stat. Phys. 50 (1988) 345–375. [4] H. Crauel, F. Flandoli, Additive noise destroys a pitchfork bifurcation, J. Dyn. DiMerential Equations 10 (1998) 259–274. [5] C.S. Hsu, A theory of cell to cell mapping dynamical systems, ASME J. Appl. Mech. 47 (1980) 931–939. [6] B.H. Tongue, K.Q. Gu, A higher order method of interpolated cell mapping, J. Sound Vib. 125 (1988) 169–179. [7] C.S. Hsu, A generalized theory of cell to cell mapping for nonlinear dynamical systems, ASME J. Appl. Mech. 53 (1981) 695–701. [8] J.Q. Sun, C.S. Hsu, The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation, J. Appl. Mech. 57 (1990) 1018–1025. [9] J.Q. Sun, Random vibration analysis of a non-linear system with dry friction damping by the short-time Gaussian cell mapping method, J. Sound Vib. 180 (1995) 785–795. [10] R. Guder, E. Kreuzer, Using generalized cell mapping to approximate invariant measures on compact manifolds, Int. J. Bifurcation Chaos 7 (1999) 2487–2499. [11] C.S. Hsu, Global analysis of dynamical systems using posets and digraphs, Int. J. Bifurcation Chaos 5 (1995) 1085–1118. [12] Hong, Ling, Xu, Jianxue, Crises and chaotic transient by the generalized cell mapping digraph method, Phys. Lett. A 262 (1999) 361–375. [13] Hong, Ling, Xu, Jianxue, Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph (GCMD) method, Int. J. Bifurcation Chaos 11 (2001) 723–736. [14] W.Q. Zhu, M.Q. Lu, Q.T. Wu, Stochastic jump and bifurcation of a Du!ng oscillator under narrow-band excitation, J. Sound Vib. 165 (1993) 285–304.