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Dynamical bifurcations and singularly perturbed systems of differential equations Jacek Banasiak Konferencja ,,XXX Lat Instytutu Matematyki Stosowanej i Mechaniki Uniwersytetu Warszawskiego” Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Part I: Singular Perturbations Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Motivating example – a dengue fever model . Assumptions: a) Host population: susceptible S h , infectives I h , recovered with immunity R h , Malthusian demography, b) Vector population: susceptible S v , infective I v , balanced population: S v ( t ) + I v ( t ) = M 0 , c) Vector population smaller that the host population, d) Non-lethal. Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Then I v S h S ′ = (Ψ h − µ 1 h ) S h + Ψ h I h + (Ψ h + ρ h ) R h − σ v β hv , h N h I v S h I ′ = σ v β hv − ( γ h + µ 1 h ) I h , h N h R ′ = γ h I h − ( ρ h + µ 1 h ) R h , h I h S v S ′ = µ 1 v S v − σ v β vh , v N h I h S v I ′ = − µ 1 v I v + σ v β vh . (1) v N h Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Table: Parameter values day − 1 year − 1 Parameters 7 . 666 × 10 − 5 2 . 8 × 10 − 2 Ψ h 3 . 704 × 10 − 3 1 . 352 × 10 0 γ h 3 . 454 × 10 − 4 1 . 261 × 10 − 1 δ h 1 . 460 × 10 − 2 5 . 33 × 10 0 ρ h 4 . 212 × 10 − 5 1 . 5 × 10 − 2 µ 1 h 2 . 19 × 10 2 σ v 0 . 6 5 . 2 × 10 1 0 . 1429 µ 1 v Dimensionless parameters β vh 0 . 8333 2 × 10 − 2 β hv Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Problem: The original models are too complex for a robust analysis and may yield redundant information for particular applications. Aim: to build a simpler (macro) model for the evolution of macro-variables relevant for a chosen time scale which, for these variables, retains the main features of the dynamics of the detailed (micro) model. The process often is referred to as the aggregation, or lumping, of states. Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Solution Micro Micromodel solution Micromodel Microsolution aggregation aggregation Macro Macromodel solution Solution Figure: Commutativity of the aggregation diagram Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Solution Microsolution Micromodel + comp. errors Micromodel Microsolution aggregation aggregation Approximate Macromodel + aggreg. errors macrosolution Solution Figure: Approximate commutativity of the aggregation diagram Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Tikhonov theorem — aggregation in systems of ODEs We are concerned with models in which the existence of two characteristic time scales leads to singularly perturbed systems x ′ = f ( t , x , y , ǫ ) , x (0) = ˚ x , ǫ y ′ = g ( t , x , y , ǫ ) , y (0) = ˚ y , (2) where ′ denotes differentiation with respect to t and f and g are sufficiently regular functions from open subsets of R + × R n × R m × R + to, respectively, R n and R m , for some n , m ∈ N . Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Tikhonov theorem gives conditions ensuring that the solutions ( x ǫ ( t ) , y ǫ ( t )) of (2) converge to ( ¯ x ( t ) , ¯ y ( t , ¯ x )), where ¯ y ( t , x ) is the solution to the equation 0 = g ( t , x , y , 0) , (3) called the quasi steady state , and ¯ x ( t ) is the solution of x ′ = f ( t , x , ¯ ◦ y ( t , x ) , 0) , x (0) = x , (4) obtained from the first equation of (2) by substituting the unknown y by the known quasi steady state ¯ y . Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Main assumptions: the quasi-steady states are isolated in some set [0 , T ] × ¯ U ; for each fixed t and x , the quasi steady state solution ¯ y ( t , x ) of (3) is an asymptotically stable equilibrium of d ˜ y d τ = g ( t , x , ˜ y , 0); (5) x ∈ ¯ ◦ ¯ x ( t ) ∈ U for t ∈ [0 , T ] provided U ; ◦ ◦ y belongs to the basin of attraction of ¯ y (0 , x ). Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Then the following theorem is true. Theorem Let the above assumptions be satisfied. Then there exists ε 0 > 0 such that for any ε ∈ ] 0 , ε 0 ] there exists a unique solution ( x ε ( t ) , y ε ( t )) of Problem (2) on [0 , T ] and ε → 0 x ε ( t ) lim = ¯ x ( t ) , t ∈ [0 , T ] , ε → 0 y ε ( t ) lim = ¯ y ( t ) , t ∈ ] 0 , T ] , (6) where ¯ x ( t ) is the solution of (4) and ¯ y ( t ) = ¯ y ( t , ¯ x ( t )) is the solution of (3). Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Back to the model. 1 With ǫ = 1000 , (1) can be written as 0 . 013 S h + 0 . 028 I h + 5 . 358 R h − 4 . 38 I v S h S ′ = , h N h 4 . 38 I v S h I ′ = − 1 . 367 I h , h N h R ′ = 1 . 352 I h − 5 . 345 R h , h 0 . 052 S v − 0 . 182 I h S v ǫ S ′ = , v N h − 0 . 052 I v + 0 . 182 I h S v ǫ I ′ = . (7) v N h Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

The Tikhonov theorem allows for the reduction of (1) to a SIR system ¯ (Ψ h − µ 1 h ) ¯ S h + Ψ h ¯ I h + (Ψ h + ρ h ) ¯ R h − λ ( t ) ¯ S ′ = S h , h ¯ λ ( t ) ¯ S h − ( γ h + µ 1 h )¯ I ′ = I h , h ¯ γ h ¯ I h − ( ρ h + µ 1 h ) ¯ R ′ = R h , (8) h with modified infection force σ v β vh M 0 ¯ λ = σ v β hv I v = σ v β hv I h ¯ , (9) ¯ ¯ µ 1 v ¯ N h + σ v β vh ¯ N h N h I h where N h ( t ) = N h (0) e (Ψ h − µ 1 h ) t . ¯ Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Part II: Dynamic bifurcations Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

In the classical bifurcation theory we consider the differential equation y = g ( x , y ) , ˙ (10) where x is a parameter, and investigate the character of the equilibrium y ∗ = y ∗ ( x ); that is, the solution to 0 = g ( x , y ) , when x passes through some exceptional values, called the bifurcation points. Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

y g ( x , y ) = 0 ( x 2 , y 2 ) x ( x 1 , y 1 ) ( x 3 , y 3 ) Figure: Dynamics described by Eqn (14) if g ( x , y ) = 0 is attractive. Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

If we move x according to some rule τ → x ( τ ), then modified (14): y ( τ ) = g ( x ( τ ) , y ( τ )) , ˙ (11) will generate a ‘long-term’ dynamics on the manifold g ( x , y ) = 0 . Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

y g ( x , y ) = 0 y ( τ ) ¯ ( x 2 , y 2 ) x ( τ ) x ( x 1 , y 1 ) ( x 3 , y 3 ) Figure: Moving x generates a dynamics on the manifold g ( x , y ) = 0 . Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

In general, the bifurcation parameter can be coupled with the main equation: x ˙ = ǫ f ( x , y ) , y ˙ = g ( x , y ) . (12) Changing time as ǫτ = t we obtain (2), x ′ = f ( x , y ) , ǫ y ′ = g ( x , y ); (13) that is, a singularly perturbed system in the Tikhonov form. Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Hence, long term dynamics of (12) is equivalent to small ǫ dynamics of (2). Both problems are equivalent for ǫ > 0. On the other hand, we may ask how well the solutions of (14) and (15) with ǫ = 0: x ˙ = 0 , y ˙ = g ( x , y ) , (14) (fast dynamics) and x ′ = f ( x , y ) , 0 = g ( x , y ) , (15) (slow dynamics) approximate the true solution? Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

y y ( x ) ¯ (¯ x ( t ) , ¯ y (¯ x ( t ))) ˚ x x ( t ) ¯ x ˚ y ( x ǫ ( t ) , y ǫ ( t )) Figure: Dynamics described by Eqns (12) and (15) by the Tikhonov theorem. Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

Quite often, however, g ( x , y ) = 0 has branching solutions. y stable branch stable branch unstable branch x b x unstable branch Figure: Transcritical bifurcation at the bifurcation point x b Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

y stable branch stable branch unstable branch x b x stable branch Figure: Hopf bifurcation at the bifurcation point x b Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

y stable branch unstable branch stable branch unstable branch x b x Figure: Backward bifurcation at the bifurcation point x b Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe

By classical bifurcation theory, it is expected that the solutions to (12) should converge to the equilibria y ∗ ( x ) of y ′ = g ( x , y ) whenever they are attracting. In terms of (15), the solutions y ǫ ( t ) should converge to the quasi steady states; that is, to solutions to g ( x , y ) = 0 , whenever they are attracting. Jacek Banasiak Dynamical bifurcations and singularly perturbed systems of diffe