Transforming to Chaos by Normal Forms

Hacınlıyan, A.; Perdahci, N. Ziya; Şahin, Gökhan; Yildirim, Haci Ahmet

doi:10.1007/978-1-4020-2316-3_4

Cited by 2 publications

(4 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The normal form expansion about this equilibrium point is not expected to yield a resonance condition [1,14].…”

Section: Towards Oscillatory Solutionsmentioning

confidence: 94%

“…Chaotic behavior is not ordinarily expected, although for strongly coupled AC components, trajectories that are dense in the phase space can result in seemingly chaotic regions [1]. Interaction of the electromagnetic field with matter is one way to add nonlinearity by coupling the Bloch system to the Maxwell equations.…”

Section: Introductionmentioning

confidence: 99%

“…Equilibrium manifolds of the MB system as g is varied against nances are possible but have complicated normal forms[1,14]. (c) An eigenvalue spectrum {±βi, μ} where for all equilibrium points we have μ = −(k + γ ⊥ + γ ).The eigenvalue pair ±βi gives rise to a resonant normal form in all orders 2n + 1 (n = 1, 2, .…”

mentioning

confidence: 99%

See 2 more Smart Citations

Approximate solutions of Maxwell Bloch equations and possible Lotka Volterra type behavior

2010

View full text Add to dashboard Cite

show abstract

“…The normal form expansion about this equilibrium point is not expected to yield a resonance condition [1,14].…”

Section: Towards Oscillatory Solutionsmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Approximate solutions of Maxwell Bloch equations and possible Lotka Volterra type behavior

2010

View full text Add to dashboard Cite

show abstract

“…The latter point when linearized has eigenvalues −Aγ±γ √ 4−4A+A 2 γ 2 so that when 4 − 4A + A 2 γ changes sign, a transition of domain indicating a Hopf bifurcation can occur. The normal form expansion about this equilibrium point is not expected to yield a resonance condition [5].…”

mentioning

confidence: 95%

Maxwell-Bloch Equations as Predator-Prey System

et al. 2010

View full text Add to dashboard Cite

Abstract. Regions of the full parameter space for which chaotic behavior in laser models based on the Maxwell-Bloch equation occurs are studied [1]. The range in the parameter space have been charted, all possibilities for the value of the maximal Lyapunov exponent are shown to exist, positive maximal Lyapunov exponents characterizing chaotic behavior occurs for parameter values that correspond to the range of parameters for Helium-Neon lasers. The Maxwell-Bloch equations as given by [2] and [3] involve the coupling of the fundamental cavity mode, E with the collective variables P and ∆, that represent the atomic polarization and the population inversion. They are represented by the following equations.For the parameter values k = σ, γ ⊥ = g 2 /k = 1, g 2 ∆o/k = r, γ = b, the system can be transformed into the Lorenz system about the equilibrium point ∆ = ∆ o by setting x = E, y = gP/k,z = ∆ o − ∆. The meaning of the parameters in the original equations as given by Arrechi[2], while σ, r, b are the Lorenz parameters.Maxwell-Bloch equations can be transformed into a system proposed by [4], that resembles the Lotka-Volterra problem with I representing the intensity of the laser field and N denoting population density, the dimensionless rate equations for I and N are obtained by assuming I = E 2 and the subsidary condition gP = kE∆. Starting with the first Maxwell-Bloch Equation multiplying both sides by E gives EĖ = −kE 2 + gP E,Ṗ = −γ ⊥ P + gE∆ = −γ ⊥ P + g 2 P/k The subsidary condition enables one to solve for the polarization as P = Po exp (−(γ ⊥ P − g 2 /k)t) and the third equation becomes∆ = −γ (∆ − ∆o) − 4kE

show abstract

Transforming to Chaos by Normal Forms

Cited by 2 publications

References 6 publications

Approximate solutions of Maxwell Bloch equations and possible Lotka Volterra type behavior

Approximate solutions of Maxwell Bloch equations and possible Lotka Volterra type behavior

Maxwell-Bloch Equations as Predator-Prey System

Contact Info

Product

Resources

About