Stability of equilibrium and bifurcation analysis in delay di⁄erential … · 2020-04-01 · Hopf...

Stability of equilibrium and bifurcation analysisin delay differential equations

Griselda Itovich1, Franco Gentile2,3 and Jorge Moiola2,4

1Sede Alto Valle y Valle Medio, Universidad Nacional de Río Negro,2 IIIE (UNS-CONICET), 3Dpto. de Matemática y 4Dpto. de Ing. Eléctrica y de Comp.,

Universidad Nacional del Sur, Argentina

XV Congreso A. MonteiroJunio 2019 - Bahía Blanca

Itovich - Gentile - Moiola (1Sede Alto Valle y Valle Medio, Universidad Nacional de Río Negro, 2 IIIE (UNS-CONICET), 3Dpto. de Matemática y 4Dpto. de Ing. Eléctrica y de Comp., Universidad Nacional del Sur, Argentina)DDE: Stability and bifurcationsXV Congreso A. Monteiro Junio 2019 - Bahía Blanca 1

Delay and Neutral Delay differential equations

Delay differential equations (Ddes):x = f (x, xτ , µ), xτ = x(t − τ), τ > 0.Neutral Ddes (NDdes): x = f (x, xτ , xτ , µ), xτ = x(t − τ),τ > 0.

Time Domain Approach (TDA).

Characteristic equations.

Equilibrium: stability and bifurcations.

Frequency Domain Approach (FDA).

Limit cycles: stability and bifurcations.

Characteristic equations

Ddes already analyzed

x + γx = f (u),

f (u) = αu + βu2,u = xτ or u = xτ

Model 1: u = xτP(s,γ, α, τ) = es(s2 + γτ 2)− ατ 2

Model 2: u = xτP(s,γ, α, τ) = es(s2 + γτ 2)− ατs

Characteristic equation:exponential polynomialP(s,γ, α, τ) = 0

mStability analysis(trivial equilibrium)

mWhen all the roots of Phave negative real part?

Pontryagin (1955)

Static stability areas for DdesTwo theorems set with Time Domain Approach (TDA)

Model 1

0 2.5 5 7.5 10 12.5 15 17.5 202

Model 1: Stability of equilibria, = 1

Model 2

0 2.5 5 7.5 10 12.5 15 17.5 202

1 3 5 7

Model 2: Stability of equilibria, = 1

Hopf bifurcation curves and dynamic stabilityResults combining Frequency and Time Domain Approaches

Model 1

0 2 4 6 8 10 12 14 16 18 202

Model 1: Stability of cycles, = 1

Model 2

0 2 4 6 8 10 12 14 16 18 202

1 3 5 7

Model 2: Stability of cycles, = 1

NDdes: Model A

x + γx = f (u)f (u) = αu + βu2,

u = xτ = x(t − τ),γ > 0, α 6= 0.

Ddes analyzed asModels 1 and 2x + γx = f (u),

f (u) = αu + βu2,u = xτ or u = xτ

Hopf bifurcation curves

αk = (−1)k (1− τ 2/(kπ)2)

0 2 4 6 8 10 12 14 16 18 202

Static stability theorem for Model A

TheoremP(s) = ess2 + esγτ 2 − αs2 = 0, γ, τ > 0, α 6= 0. Let γ = 1 andrk = y−1k = (kπ)−1 , k ∈ N and other restrictive conditions on theparameters. Then, all the roots of P lie on the left half plane iff theseconditions are fulfilled:I) For 0 < τ < y1 ⇒ α < 0 and α > −1+ r21 τ 2.II) For yk < τ < yk+1, k ∈ N, it is required:a) If k is odd ⇒ α > 0 , α < −1+ r2k τ 2 and α < 1− r2k+1τ 2.b) If k is even ⇒ α < 0 , α > 1− r2k τ 2 and α > −1+ r2k+1τ 2.

CorollaryLet x + γx = αxτ + βx2τ . Then x = 0 is asymptotically stable underthe parameter conditions set in the Theorem.

Static stability areas for Model A

0 2,5 5 7,5 10 12.5 15 17.5 202

Stability of equilibria, = 1

NDdes: Model BTime Domain Approach (TDA)

Zhang & Stépán (2018)

x = av + bv + cv3

x = dxdt , v = x(t − 1)a, b, c ∈ R

Hopf curves{a = cos yb = −y sin y , y ≥ 0

a1 0.5 0 0.5 1

Stability of equilibria, =1

Stability condition

0 > b > −√1− a2 arccos a

Static stability theorem for Model B

TheoremP(s) = ess− as− bτ = 0, τ > 0, a, b ∈ R. Letτ = 1, |a| < 1, y0 = arccos a. Then, all the roots of P lie on the lefthalf plane iff holds 0 > b > −

√1− a2y0.

CorollaryLet x = ax + bv + cv3. Then x = 0 is asymptotically stable underthe parameter conditions set in the Theorem.

Feedback control theory for Ddes and NDdes

Extended Graphical Hopf Bifurcation theorem (Mees, Chua andAllwright, 1979b, 1979a).

1 Local existence of a branch of periodic solutions.2 Approximation of amplitude θ and frequency ω of each periodicsolution.

3 Stability of each periodic solution.4 Stability along the Hopf bifurcations curves.

Outcomes about Hopf degeneracies (up to now with NDdes)1 Determination of fold of cycles bifurcations.

Model BFrequency Domain Approach (FDA)

x = av + bv + cv3

x = dxdt , v = x(t − 1)a, b, c ∈ Rm

(y = −x(t − 1))L(−y) = G ∗(s)L(g(y)),

G ∗(s) = se−s

s−be−s ,

g(y) = −ay − cy3,−y = G ∗(0)g(y) =⇒ y = 0

λ(s) = G ∗(s) dgdy

∣∣∣y=0

= −ase−ss−be−s

Re( )1 0.5 0 0.5 1

1.5a= 0.9, b= 0.5

Re( )2 1.5 1 0.5 0 0.5 1

1.5a= 0.9, b= 0.5

Model BStability of cycles, Hopf degeneracy condition and cycles approximations

Curvature coeffi cient: σ1

sgnσ1={−1⇒ HB stable1⇒ HB unstable

Hopf degeneracy condition{sgnσ1 = − sgn(cb(1− a2 − ab))b = − arccos a

√1− a2,

Approximations of ω and θb = −ω sinωθ = 2√

√cosω − a

0 .56 0.54 0.52 0.5 0.48 0.46 0.440

1 .8 1.75 1.7 1.650

Conclusions and future work

Conclusions

1 Static and dynamicanalyses of Ndes have beenperformed .

2 Static analysis: TDA andstudying the roots ofcharacteristic equations.

3 Dynamic analysis: FDMand setting Graphical Hopfbifurcation Theorem(GHT) outcomes.

Future work

1 Continue exploring NDdes.2 Get higher orderapproximations of periodicsolutions via FDA and GHT.

3 Understand the dynamicsclose to diverse Hopfdegeneracies.

4 Contrast results with otheranalytical and numericaltechniques.

References

Bellman, R. and Cooke, K. [1963] Differential-Difference Equations, Academic Press, New York.

Bellman, R. and Danskin, J. Jr.[1954] A Survey of the Matematical Theory of Time-Lag, Retarded Control and Hereditary

Processes, Report 256, The Rand Corporation, California.

Mees, A. I. and Allwright, D. J. [1979a]. "Using characteristic loci in the Hopf bifurcation", Proceedings Instn. Electrical

Engrs., 126:628-632.

Mees, A.I. and Chua, L. [1979b] "The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and

systems", IEEE Transactions on Circuits and Systems, 26(4), 235-254.

Pontryagin, L. S. [1955] "On the zeros of some elementary trascendental functions", American Mathematical Society

Tanslations 2(1), 95-110.

Stépán, G. [1989] Retarded Dynamical Systems: Stability and Characteristic Functions, Pitman Research Notes in Mathematics

Series, Vol. 210, Longman, UK.

Zhang, L. and Stépán, G. [2018] "Hopf bifurcation analysis of scalar implicit neutral delay differential equation", Electronic J.

of Qualitative Theory of Diff. Eqns., 62, pp. 1-9.

Thank you for your attention!

Stability of equilibrium and bifurcation analysis in delay di⁄erential … · 2020-04-01 · Hopf...

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