Bifurcation analysis of the logistic map via two periodic impulsive forces

Jiang, Haibo; Li, Tao; Xiao-Liang, None Zeng; Zhang, Liping

doi:10.1088/1674-1056/23/1/010501

“…The boundary crisis appears in the dynamic evolution of the system when θ > 6.773 (the explanation of the boundary crisis can be referred to Lim and Kim [31], and Jiang et al [32]). Taking θ = 9 as an example, for ρ = 0.3, the variation of the bifurcation of flow of Route 1 with ϕ is shown in Figure 5a.…”

Section: Evolution Characteristicsmentioning

confidence: 99%

Analysis of Stability-To-Chaos in the Dynamic Evolution of Network Traffic Flows under a Dual Updating Mechanism

Liu¹,

Yan²,

Easa³

et al. 2021

Preprint

0

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This paper proposes a traffic-flow evolutionary model under a dual updating mechanism that describes the day-to-day (DTD) dynamics of traffic flow and travel cost. To illustrate the concept, a simple two-route network is considered. Based on the nonlinear dynamic theory, the equilibrium stability condition of the system is derived and the condition for the division between the bifurcation and chaotic states of the system is determined. The characteristics of the DTD dynamic evolution of network traffic flow are investigated using numerical experiments. The results show that the system is absolutely stable when the sensitivity of travelers toward the route cost parameter (θ) is equal to or less than 0.923. The bifurcation appears in the system when θ is larger than 0.923. For values of θ equal to or larger than 4.402, the chaos appears in the evolution of the system. The results also show that with the appearance of chaos, the boundary and interior crises begin to appear in the system when θ is larger than 6.773 and 10.403, respectively. The evolution of network traffic flow is always stable when the proportion of travelers who do not change the route is 84% or greater.

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“…The interior crisis appears in the dynamic evolution of the system when 403 . 10 > θ (an explanation of the interior crisis can be referred to Lim and Kim [31] and Jiang et al [32]). Taking The interior crisis appears in the dynamic evolution of the system when θ > 10.403 (an explanation of the interior crisis can be referred to Lim and Kim [31] and Jiang et al [32]).…”

Section: Evolution Characteristicsmentioning

confidence: 99%

“…10 > θ (an explanation of the interior crisis can be referred to Lim and Kim [31] and Jiang et al [32]). Taking The interior crisis appears in the dynamic evolution of the system when θ > 10.403 (an explanation of the interior crisis can be referred to Lim and Kim [31] and Jiang et al [32]). Taking θ = 11 as an example, for ρ = 0.4, the variation of the bifurcation of flow of Route 1 with ϕ is shown in Figure 6a.…”

Section: Evolution Characteristicsmentioning

confidence: 99%

“…Variation of the flow bifurcation diagram and Lyapunov exponent of the system when 9 = θ .The interior crisis appears in the dynamic evolution of the system when of the interior crisis can be referred to Lim and Kim[31] and Jiang et al[32]). Taking…”

mentioning

confidence: 99%

“…Time series and power spectrum when 5 = θ .The boundary crisis appears in the dynamic evolution of the system when 773 . 6 > θ (the explanation of the boundary crisis can be referred to Lim and Kim[31], and Jiang et al[32]). Taking of the bifurcation of flow of Route 1 with ϕ is shown inFigure 5a.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Analysis of Stability-To-Chaos in the Dynamic Evolution of Network Traffic Flows under a Dual Updating Mechanism

Liu

¹

,

Yan

²

,

Easa

³

et al. 2018

View full text Add to dashboard Cite

This paper proposes a traffic-flow evolutionary model under a dual updating mechanism that describes the day-to-day (DTD) dynamics of traffic flow and travel cost. To illustrate the concept, a simple two-route network is considered. Based on the nonlinear dynamic theory, the equilibrium stability condition of the system is derived and the condition for the division between the bifurcation and chaotic states of the system is determined. The characteristics of the DTD dynamic evolution of network traffic flow are investigated using numerical experiments. The results show that the system is absolutely stable when the sensitivity of travelers toward the route cost parameter (θ) is equal to or less than 0.923. The bifurcation appears in the system when θ is larger than 0.923. For values of θ equal to or larger than 4.402, the chaos appears in the evolution of the system. The results also show that with the appearance of chaos, the boundary and interior crises begin to appear in the system when θ is larger than 6.773 and 10.403, respectively. The evolution of network traffic flow is always stable when the proportion of travelers who do not change the route is 84% or greater.

show abstract