QUALITATIVE BEHAVIOR OF SOLUTIONS: Equilibrium Points and Stability - - PowerPoint PPT Presentation

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QUALITATIVE BEHAVIOR OF SOLUTIONS: Equilibrium Points and Stability - - PowerPoint PPT Presentation

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QUALITATIVE BEHAVIOR OF SOLUTIONS: Equilibrium Points and Stability Linear Systems Pasquale Palumbo IASI-CNR Viale Manzoni 30, 00185 Roma, Italy Summer School on Parameter Estimation in Physiological Models Lipari, September 2009

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QUALITATIVE BEHAVIOR OF SOLUTIONS: Equilibrium Points and Stability Linear Systems

Pasquale Palumbo IASI-CNR Viale Manzoni 30, 00185 Roma, Italy Summer School on Parameter Estimation in Physiological Models Lipari, September 2009

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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Equilibrium points

Time-invariant, nonlinear systems (ODE models): ˙ x(t) = f

  • x(t)
  • ,

x(0) = x0, x(t) ∈ I Rn, f : I Rn → I Rn. The unique solution to the Chauchy problem will be denoted by: x(t) = ϕ(t, x0) where ϕ : I R+ × I Rn → I Rn is the state-transition map. x0 is the initial state: it univocally determines the evolution of the state variables x Equilibrium points are such that no motion occurs if they are chosen as initial state: ϕ(t, xe) = xe, ∀t ≥ 0

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 3

Equilibrium points

Time-invariant, nonlinear systems (ODE models): ˙ x(t) = f

  • x(t)
  • ,

x(0) = x0, x(t) ∈ I Rn, f : I Rn → I Rn. The unique solution to the Chauchy problem will be denoted by: x(t) = ϕ(t, x0) where ϕ : I R+ × I Rn → I Rn is the state-transition map. x0 is the initial state: it univocally determines the evolution of the state variables x Equilibrium points are such that no motion occurs if they are chosen as initial state: ϕ(t, xe) = xe, ∀t ≥ 0

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 4

Equilibrium points

Time-invariant, nonlinear systems (ODE models): ˙ x(t) = f

  • x(t)
  • ,

x(0) = x0, x(t) ∈ I Rn, f : I Rn → I Rn. By definition, the equilibrium points vanish the time-derivative: f(xe) = 0 In case of a generic nonlinear system we can have:

– no equilibrium points at all: f(x) = 0 has no solutions – a unique equilibrium point: f(x) = 0 admits a unique solution – isolated equilibrium points: f(x) = 0 admits a discrete number of solutions – infinite equilibrium points: f(x) = 0 admits infinite solutions

In case of linear systems, f(x) = Ax, we can have:

– the origin is always an equilibrium point – if rank(A) = n, the origin is the unique equilibrium point – if rank(A) = r < n, there exist ∞n−r (uncountable) equilibrium points – there can never be isolated equilibrium points, unless the origin

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 5

Equilibrium points

Time-invariant, nonlinear systems (ODE models): ˙ x(t) = f

  • x(t)
  • ,

x(0) = x0, x(t) ∈ I Rn, f : I Rn → I Rn. By definition, the equilibrium points vanish the time-derivative: f(xe) = 0 In case of a generic nonlinear system we can have:

– no equilibrium points at all: f(x) = 0 has no solutions – a unique equilibrium point: f(x) = 0 admits a unique solution – isolated equilibrium points: f(x) = 0 admits a discrete number of solutions – infinite equilibrium points: f(x) = 0 admits infinite solutions

In case of linear systems, f(x) = Ax, we can have:

– the origin is always an equilibrium point – if rank(A) = n, the origin is the unique equilibrium point – if rank(A) = r < n, there exist ∞n−r (uncountable) equilibrium points – there can never be isolated equilibrium points, unless the origin

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 6

Stability

Time-invariant, nonlinear systems (ODE models): ˙ x(t) = f

  • x(t)
  • ,

x(0) = x0, x(t) ∈ I Rn, f : I Rn → I Rn.

  • Stability. The equilibrium point xe is stable if:

∀ε > 0, ∃δ > 0 : x0 − xe < δ = ⇒ x(t) − xe < ε, ∀t ≥ 0 In case of linear systems, f(x) = Ax, the stability of a given equilibrium point implies and is implied by the stability of the origin

  • Attractivity. The equilibrium point xe is:

– locally attractive if: ∃η > 0 : x0 − xe < η = ⇒ x(t) − xe → 0 – globally attractive if: ∀x0 ∈ I Rn, it is : x(t) − xe → 0

Attractivity can occur only if the equilibrium point is isolated In case of linear systems, f(x) = Ax, only the origin can be attractive

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 7

Stability

Time-invariant, nonlinear systems (ODE models): ˙ x(t) = f

  • x(t)
  • ,

x(0) = x0, x(t) ∈ I Rn, f : I Rn → I Rn.

  • Stability. The equilibrium point xe is stable if:

∀ε > 0, ∃δ > 0 : x0 − xe < δ = ⇒ x(t) − xe < ε, ∀t ≥ 0 In case of linear systems, f(x) = Ax, the stability of a given equilibrium point implies and is implied by the stability of the origin

  • Attractivity. The equilibrium point xe is:

– locally attractive if: ∃η > 0 : x0 − xe < η = ⇒ x(t) − xe → 0 – globally attractive if: ∀x0 ∈ I Rn, it is : x(t) − xe → 0

Attractivity can occur only if the equilibrium point is isolated In case of linear systems, f(x) = Ax, only the origin can be attractive

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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Stability

Time-invariant, nonlinear systems (ODE models): ˙ x(t) = f

  • x(t)
  • ,

x(0) = x0, x(t) ∈ I Rn, f : I Rn → I Rn. Asymptotic Stability. The equilibrium point xe is locally/globally asymptotically stable if it is stable and locally/globally attractive Global asymptotic stability can occur only when the equilibrium point is unique Exponential Stability. The equilibrium point xe is exponentially stable if: ∃α > 0 : ∀ε > 0, ∃δ > 0 : x0−xe < δ = ⇒ x(t)−xe < ε·e−αt In case of linear systems, f(x) = Ax, asymptotical stability is always global and exponential

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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Stability

Time-invariant, nonlinear systems (ODE models): ˙ x(t) = f

  • x(t)
  • ,

x(0) = x0, x(t) ∈ I Rn, f : I Rn → I Rn. Asymptotic Stability. The equilibrium point xe is locally/globally asymptotically stable if it is stable and locally/globally attractive Global asymptotic stability can occur only when the equilibrium point is unique Exponential Stability. The equilibrium point xe is exponentially stable if: ∃α > 0 : ∀ε > 0, ∃δ > 0 : x0−xe < δ = ⇒ x(t)−xe < ε·e−αt In case of linear systems, f(x) = Ax, asymptotical stability is always global and exponential

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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Stability: example of attractivity without stability

Time-invariant, nonlinear systems (ODE models): ˙ x1 = x2

1 (x2 − x1) + x5 2

(x2

1 + x2 2 )

  • 1 + (x2

1 + x2 2 )

  • ˙

x2 = x2

2 (x2 − 2x1)

(x2

1 + x2 2 )

  • 1 + (x2

1 + x2 2)

  • pasquale.palumbo@iasi.cnr.it

Lipari 2009, Summer School on BioMathematics

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Linear systems evolution

Time-invariant, linear systems (ODE models): ˙ x(t) = Ax(t), x(0) = x0 x(t) ∈ I Rn, A ∈ I Rn×n The solution is a linear transformation of the initial state: x(t) = Φ(t)x0 Φ(t) is the state transition matrix Φ(t) obeys the following matricial Chauchy problem ˙ Φ(t) = AΦ(t), Φ(0) = In The solution is the exponential matrix: Φ(t) = eAt = I + At + A2t2 2 + · · · =

∞

  • k=0

Aktk k!

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 12

Linear systems evolution

Time-invariant, linear systems (ODE models): ˙ x(t) = Ax(t), x(0) = x0 x(t) ∈ I Rn, A ∈ I Rn×n The solution is a linear transformation of the initial state: x(t) = Φ(t)x0 Φ(t) is the state transition matrix Φ(t) obeys the following matricial Chauchy problem ˙ Φ(t) = AΦ(t), Φ(0) = In The solution is the exponential matrix: Φ(t) = eAt = I + At + A2t2 2 + · · · =

∞

  • k=0

Aktk k!

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

slide-13
SLIDE 13

Linear systems evolution

Time-invariant, linear systems (ODE models): ˙ x(t) = Ax(t), x(0) = x0 x(t) ∈ I Rn, A ∈ I Rn×n The solution is a linear transformation of the initial state: x(t) = Φ(t)x0 Φ(t) is the state transition matrix Φ(t) obeys the following matricial Chauchy problem ˙ Φ(t) = AΦ(t), Φ(0) = In The solution is the exponential matrix: Φ(t) = eAt = I + At + A2t2 2 + · · · =

∞

  • k=0

Aktk k!

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 14

Linear systems evolution

Time-invariant, linear systems (ODE models): ˙ x(t) = Ax(t), x(0) = x0 x(t) ∈ I Rn, A ∈ I Rn×n Explicit solution to: ˙ Φ(t) = AΦ(t), Φ(0) = In Φ(t) = eAt = I + At + A2t2 2 + · · · =

∞

  • k=0

Aktk k! eAt

  • t=0 =
  • In + At + A2t2

2

+ · · ·

  • t=0 = In

d dt

  • eAt

= d dt ∞

  • k=0

Aktk k!

  • =

∞

  • k=1

Aktk−1 (k − 1)! = A

∞

  • k=1

Ak−1tk−1 (k − 1)! = AeAt Further property (semigroup): Φ(t1 + t2) = Φ(t1) · Φ(t2)

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 15

Linear systems evolution

Time-invariant, linear systems (ODE models): ˙ x(t) = Ax(t), x(0) = x0 x(t) ∈ I Rn, A ∈ I Rn×n Explicit solution to: ˙ Φ(t) = AΦ(t), Φ(0) = In Φ(t) = eAt = I + At + A2t2 2 + · · · =

∞

  • k=0

Aktk k! eAt

  • t=0 =
  • In + At + A2t2

2

+ · · ·

  • t=0 = In

d dt

  • eAt

= d dt ∞

  • k=0

Aktk k!

  • =

∞

  • k=1

Aktk−1 (k − 1)! = A

∞

  • k=1

Ak−1tk−1 (k − 1)! = AeAt Further property (semigroup): Φ(t1 + t2) = Φ(t1) · Φ(t2)

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 16

Linear systems evolution

Time-invariant, linear systems (ODE models): ˙ x(t) = Ax(t), x(0) = x0 x(t) ∈ I Rn, A ∈ I Rn×n Explicit solution to: ˙ Φ(t) = AΦ(t), Φ(0) = In Φ(t) = eAt = I + At + A2t2 2 + · · · =

∞

  • k=0

Aktk k! eAt

  • t=0 =
  • In + At + A2t2

2

+ · · ·

  • t=0 = In

d dt

  • eAt

= d dt ∞

  • k=0

Aktk k!

  • =

∞

  • k=1

Aktk−1 (k − 1)! = A

∞

  • k=1

Ak−1tk−1 (k − 1)! = AeAt Further property (semigroup): Φ(t1 + t2) = Φ(t1) · Φ(t2)

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 17

Linear systems evolution: spectral decomposition

Time-invariant, linear systems (ODE models): ˙ x(t) = Ax(t), x(0) = x0 x(t) ∈ I Rn, A ∈ I Rn×n Spectrum of matrix A: {λ1, . . . , λn} n distinct eigenvalues The associated eigenvectors {u1, . . . , un} are a basis for the state space: U =

  • u1 · · · un
  • =

⇒ A = UΛU−1 Λ =    λ1 O ... O λn    Spectral decomposition of matrix A: A = UΛV =

n

  • i=1

λiuiv T

i ,

v T

i

are the rows of matrix U−1

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics

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SLIDE 18

Linear systems evolution: spectral decomposition

Time-invariant, linear systems (ODE models): ˙ x(t) = Ax(t), x(0) = x0 x(t) ∈ I Rn, A ∈ I Rn×n Spectrum of matrix A: {λ1, . . . , λn} n distinct eigenvalues The associated eigenvectors {u1, . . . , un} are a basis for the state space: U =

  • u1 · · · un
  • =

⇒ A = UΛU−1 Λ =    λ1 O ... O λn    Spectral decomposition of matrix A: A = UΛV =

n

  • i=1

λiuiv T

i ,

v T

i

are the rows of matrix U−1

pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics