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
Equilibrium points Time-invariant, nonlinear systems (ODE models): R n �→ I R n , R n . ˙ � � x ( t ) = f x ( t ) , x ( 0 ) = x 0 , x ( t ) ∈ I f : I The unique solution to the Chauchy problem will be denoted by: x ( t ) = ϕ ( t , x 0 ) R + × I R n �→ I R n is the state-transition map . where ϕ : I x 0 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 , x e ) = x e , ∀ t ≥ 0 pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics
Equilibrium points Time-invariant, nonlinear systems (ODE models): R n �→ I R n , R n . ˙ � � x ( t ) = f x ( t ) , x ( 0 ) = x 0 , x ( t ) ∈ I f : I The unique solution to the Chauchy problem will be denoted by: x ( t ) = ϕ ( t , x 0 ) R + × I R n �→ I R n is the state-transition map . where ϕ : I x 0 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 , x e ) = x e , ∀ t ≥ 0 pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics
Equilibrium points Time-invariant, nonlinear systems (ODE models): R n �→ I R n , R n . ˙ � � x ( t ) = f x ( t ) x ( 0 ) = x 0 , x ( t ) ∈ I f : I , By definition, the equilibrium points vanish the time-derivative: f ( x e ) = 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
Equilibrium points Time-invariant, nonlinear systems (ODE models): R n �→ I R n , R n . ˙ � � x ( t ) = f x ( t ) x ( 0 ) = x 0 , x ( t ) ∈ I f : I , By definition, the equilibrium points vanish the time-derivative: f ( x e ) = 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
Stability Time-invariant, nonlinear systems (ODE models): R n �→ I R n , R n . x ( t ) = f ˙ � x ( t ) � x ( 0 ) = x 0 , x ( t ) ∈ I f : I , Stability. The equilibrium point x e is stable if: ∀ ε > 0 , ∃ δ > 0 : � x 0 − x e � < δ = ⇒ � x ( t ) − x e � < 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 x e is: – locally attractive if: ∃ η > 0 : � x 0 − x e � < η = ⇒ � x ( t ) − x e � �→ 0 R n , – globally attractive if: ∀ x 0 ∈ I it is : � x ( t ) − x e � �→ 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
Stability Time-invariant, nonlinear systems (ODE models): R n �→ I R n , R n . x ( t ) = f ˙ � x ( t ) � x ( 0 ) = x 0 , x ( t ) ∈ I f : I , Stability. The equilibrium point x e is stable if: ∀ ε > 0 , ∃ δ > 0 : � x 0 − x e � < δ = ⇒ � x ( t ) − x e � < 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 x e is: – locally attractive if: ∃ η > 0 : � x 0 − x e � < η = ⇒ � x ( t ) − x e � �→ 0 R n , – globally attractive if: ∀ x 0 ∈ I it is : � x ( t ) − x e � �→ 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
Stability Time-invariant, nonlinear systems (ODE models): R n �→ I R n , R n . x ( t ) = f ˙ � x ( t ) � x ( 0 ) = x 0 , x ( t ) ∈ I f : I , Asymptotic Stability. The equilibrium point x e 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 x e is exponentially stable if: � x ( t ) − x e � < ε · e − α t ∃ α > 0 : ∀ ε > 0 , ∃ δ > 0 : � x 0 − x e � < δ = ⇒ 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
Stability Time-invariant, nonlinear systems (ODE models): R n �→ I R n , R n . x ( t ) = f ˙ � x ( t ) � x ( 0 ) = x 0 , x ( t ) ∈ I f : I , Asymptotic Stability. The equilibrium point x e 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 x e is exponentially stable if: � x ( t ) − x e � < ε · e − α t ∃ α > 0 : ∀ ε > 0 , ∃ δ > 0 : � x 0 − x e � < δ = ⇒ 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
Stability: example of attractivity without stability Time-invariant, nonlinear systems (ODE models): x 2 1 ( x 2 − x 1 ) + x 5 x 2 2 ( x 2 − 2 x 1 ) ˙ 2 ˙ x 1 = x 2 = ( x 2 1 + x 2 � 1 + ( x 2 1 + x 2 � ( x 2 1 + x 2 � 1 + ( x 2 1 + x 2 � 2 ) 2 ) 2 ) 2 ) pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics
Linear systems evolution Time-invariant, linear systems (ODE models): R n , R n × n ˙ x ( t ) = Ax ( t ) , x ( 0 ) = x 0 x ( t ) ∈ I A ∈ I The solution is a linear transformation of the initial state: x ( t ) = Φ( t ) x 0 Φ( t ) is the state transition matrix Φ( t ) obeys the following matricial Chauchy problem ˙ Φ( t ) = A Φ( t ) , Φ( 0 ) = I n The solution is the exponential matrix : ∞ Φ( t ) = e At = I + At + A 2 t 2 A k t k � + · · · = 2 k ! k = 0 pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics
Linear systems evolution Time-invariant, linear systems (ODE models): R n , R n × n ˙ x ( t ) = Ax ( t ) , x ( 0 ) = x 0 x ( t ) ∈ I A ∈ I The solution is a linear transformation of the initial state: x ( t ) = Φ( t ) x 0 Φ( t ) is the state transition matrix Φ( t ) obeys the following matricial Chauchy problem ˙ Φ( t ) = A Φ( t ) , Φ( 0 ) = I n The solution is the exponential matrix : ∞ Φ( t ) = e At = I + At + A 2 t 2 A k t k � + · · · = 2 k ! k = 0 pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics
Linear systems evolution Time-invariant, linear systems (ODE models): R n , R n × n ˙ x ( t ) = Ax ( t ) , x ( 0 ) = x 0 x ( t ) ∈ I A ∈ I The solution is a linear transformation of the initial state: x ( t ) = Φ( t ) x 0 Φ( t ) is the state transition matrix Φ( t ) obeys the following matricial Chauchy problem ˙ Φ( t ) = A Φ( t ) , Φ( 0 ) = I n The solution is the exponential matrix : ∞ Φ( t ) = e At = I + At + A 2 t 2 A k t k � + · · · = 2 k ! k = 0 pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics
Linear systems evolution Time-invariant, linear systems (ODE models): R n , R n × n x ( t ) = Ax ( t ) , ˙ x ( 0 ) = x 0 x ( t ) ∈ I A ∈ I ˙ Explicit solution to: Φ( t ) = A Φ( t ) , Φ( 0 ) = I n Φ( t ) = e At = I + At + A 2 t 2 ∞ A k t k � + · · · = 2 k ! k = 0 e At � � I n + At + A 2 t 2 � t = 0 = + · · · t = 0 = I n � 2 � � ∞ � A k t k ∞ A k t k − 1 ∞ A k − 1 t k − 1 d = d e At � � � � ( k − 1 )! = Ae At � = ( k − 1 )! = A dt dt k ! k = 0 k = 1 k = 1 Further property ( semigroup ): Φ( t 1 + t 2 ) = Φ( t 1 ) · Φ( t 2 ) pasquale.palumbo@iasi.cnr.it Lipari 2009, Summer School on BioMathematics
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