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Differential Equations with Time Delay

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Differential Equations with Time Delay Differential equations with time delay Marek Bodnar Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, University of Warsaw MIM Colloquium December 8th, 2016 Corollary A proper phase space is a space of functions. Adding time delay obtaining x0(t) = a x(t τ); x(0) = x − − 0 and we cannot solve it since the value of x( τ) is not known. − In general, there are no solutions of this linear equation that can be expressed in terms of elementary functions. Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method What are delay differential equations? (Almost) the simplest differential equation a ) t x0(t) = a x(t); x(0) = x0; ( − x has a solution t x(t) = a + (x a)e− : 0 − t Marek Bodnar (MIM) Delay differential equations December 8th, 2016 1 / 39 Corollary A proper phase space is a space of functions. In general, there are no solutions of this linear equation that can be expressed in terms of elementary functions. Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method What are delay differential equations? (Almost) the simplest differential equation a ) t x0(t) = a x(t); x(0) = x0; ( − x has a solution t x(t) = a + (x a)e− : 0 − t Adding time delay obtaining x0(t) = a x(t τ); x(0) = x − − 0 and we cannot solve it since the value of x( τ) is not known. − Marek Bodnar (MIM) Delay differential equations December 8th, 2016 1 / 39 Corollary A proper phase space is a space of functions. Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method What are delay differential equations? (Almost) the simplest differential equation a ) t x0(t) = a x(t); x(0) = x0; ( − x has a solution t x(t) = a + (x a)e− : 0 − t Adding time delay obtaining x0(t) = a x(t τ); x(t) = φ(t); t [ τ; 0]; − − 2 − and we need to pose initial condition on the whole interval [ τ; 0]. − In general, there are no solutions of this linear equation that can be expressed in terms of elementary functions. Marek Bodnar (MIM) Delay differential equations December 8th, 2016 1 / 39 Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method What are delay differential equations? (Almost) the simplest differential equation a ) t x0(t) = a x(t); x(0) = x0; ( − x has a solution t x(t) = a + (x a)e− : 0 − t Adding time delay obtaining x0(t) = a x(t τ); x(t) = φ(t); t [ τ; 0]; − − 2 − and we need to pose initial condition on the whole interval [ τ; 0]. − In general, there are no solutions of this linear equation that can be expressed in terms of elementary functions. Corollary A proper phase space is a space of functions. Marek Bodnar (MIM) Delay differential equations December 8th, 2016 1 / 39 Solution does not converge to a steady state. If β > a, then solution takes negative values. Surprises x(t) The solution is (an easy exercise) a + β x(t) = a + β cos(t): a t π π π 3π 2 2 − a β 2 − Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Continuation of a simple example For some particular parameters (and initial functions) we can find π a solution. Set τ = 2 and consider π h π i x0(t) = a x t ; x(t) = a + β cos(t); for t ; 0 : − − 2 2 −2 Marek Bodnar (MIM) Delay differential equations December 8th, 2016 2 / 39 Solution does not converge to a steady state. If β > a, then solution takes negative values. Surprises Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Continuation of a simple example For some particular parameters (and initial functions) we can find π a solution. Set τ = 2 and consider π h π i x0(t) = a x t ; x(t) = a + β cos(t); for t ; 0 : − − 2 2 −2 x(t) The solution is (an easy exercise) a + β x(t) = a + β cos(t): a t π π π 3π 2 2 − a β 2 − Marek Bodnar (MIM) Delay differential equations December 8th, 2016 2 / 39 If β > a, then solution takes negative values. Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Continuation of a simple example For some particular parameters (and initial functions) we can find π a solution. Set τ = 2 and consider π h π i x0(t) = a x t ; x(t) = a + β cos(t); for t ; 0 : − − 2 2 −2 x(t) The solution is (an easy exercise) a + β x(t) = a + β cos(t): a t π π π 3π 2 2 − a β 2 − Surprises Solution does not converge to a steady state. Marek Bodnar (MIM) Delay differential equations December 8th, 2016 2 / 39 Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Continuation of a simple example For some particular parameters (and initial functions) we can find π a solution. Set τ = 2 and consider π h π i x0(t) = a x t ; x(t) = a + β cos(t); for t ; 0 : − − 2 2 −2 x(t) The solution is (an easy exercise) a + β x(t) = a + β cos(t): a t π π π 3π 2 2 − a β 2 − Surprises Solution does not converge to a steady state. If β > a, then solution takes negative values. Marek Bodnar (MIM) Delay differential equations December 8th, 2016 2 / 39 Proof: It is enough to take ' such that '( τ) > a; − '(0) = 0. Corollary Although delay differential equations look very similar to ordinary differential equations, they are different and intuitions from ODE sometimes do not work. Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Negativity of the solution x0(t) = a x(t τ); x(t) = φ(t); t [ τ; 0] (?) − − 2 − Proposition For any parameters a > 0, τ > 0 there exists an initial function ' such that solution to (?) takes negative values. Marek Bodnar (MIM) Delay differential equations December 8th, 2016 3 / 39 Corollary Although delay differential equations look very similar to ordinary differential equations, they are different and intuitions from ODE sometimes do not work. Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Negativity of the solution x0(t) = a x(t τ); x(t) = φ(t); t [ τ; 0] (?) − − 2 − Proposition For any parameters a > 0, τ > 0 there exists an initial function ' such that solution to (?) takes negative values. Proof: It is enough to take ' such that '( τ) > a; − '(0) = 0. Marek Bodnar (MIM) Delay differential equations December 8th, 2016 3 / 39 Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Negativity of the solution x0(t) = a x(t τ); x(t) = φ(t); t [ τ; 0] (?) − − 2 − Proposition For any parameters a > 0, τ > 0 there exists an initial function ' such that solution to (?) takes negative values. Proof: It is enough to take ' such that '( τ) > a; − '(0) = 0. Corollary Although delay differential equations look very similar to ordinary differential equations, they are different and intuitions from ODE sometimes do not work. Marek Bodnar (MIM) Delay differential equations December 8th, 2016 3 / 39 The case τ > 0 All above statements become false: Solutions converges to 1 only if ατ < π/2; Solutions do not need to be monotonic (even for small ατ); Solutions need not to be non-negative. Even if '(t) (0; 1], then solution to ( ) are non-negative only 2 } for small ατ. For ατ > 3:06 they can take negative values. M.B. Appl. Math. Lett. (2000) Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Logistic equation with delay x0(t) = αx(t τ) 1 x(t τ) ; x(t) = '(t) 0; t [ τ; 0]: ( ) − − − ≥ 2 − } Solution to ( ) without delay (i.e. τ = 0) } All solutions, except the trivial one, converges to 1. All solutions are monotonic. All solutions are non-negative. Marek Bodnar (MIM) Delay differential equations December 8th, 2016 4 / 39 Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Logistic equation with delay x0(t) = αx(t τ) 1 x(t τ) ; x(t) = '(t) 0; t [ τ; 0]: ( ) − − − ≥ 2 − } Solution to ( ) without delay (i.e. τ = 0) } All solutions, except the trivial one, converges to 1. All solutions are monotonic. All solutions are non-negative. The case τ > 0 All above statements become false: Solutions converges to 1 only if ατ < π/2; Solutions do not need to be monotonic (even for small ατ); Solutions need not to be non-negative. Even if '(t) (0; 1], then solution to ( ) are non-negative only 2 } for small ατ. For ατ > 3:06 they can take negative values. M.B. Appl. Math. Lett. (2000) Marek Bodnar (MIM) Delay differential equations December 8th, 2016 4 / 39 Gallery of properties Stability Models Linear equation Non-negativity Phase space Continuation Step method Logistic equation with delay — graph of the solutions x˙(t) = αx(t τ) 1 x(t τ) : − − − 1:5 τ = 1 α = 1 α = π/2 1 α = 2:46 τ = 0 ) t ( α = 1 x 0:5 0 0:5 − 0 2 4 6 8 10 t Marek Bodnar (MIM) Delay differential equations December 8th, 2016 5 / 39 Wright’s hypothesis: The steady state is globally stable for all ατ < π/2.
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