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Continuous dependence on initial conditions means that if for every $\epsilon > 0$ there exists a $\delta(\epsilon) > 0 $ such that $$\Vert x_0 - \widetilde{x_0} \Vert \leq \delta(\epsilon) \Rightarrow \Vert x(t, x_0) - x(t, \widetilde{x_0}) \Vert \leq \epsilon \: \forall \: t_0 = 0 \leq t \leq T$$ Where I used the definition given here with $t_0 = 0$.

The classic example why this matters is the Lorenz System/Attractor which shows at least sensitive dependence on initial conditions. A nice intro to the topic is this blog post where it is shown that the Lorenz ODEs show very different behaviour for seemingly small deviations in the initial condition.

Speaking of chaotic dynamical systems, you can also have chaotic behaviour due to changes in the parameters $\mu$ of the system. Thus, your definition of well-posedness for parametric systems $f\big(x(t), t ; \mu\big)$ might be extended to requiring also continuous dependence on the parameters.

Why are these continuous dependence criteria required for a well-posed problem? From a computational science standpoint, you want to trust the solutions you obtain from simulations. The issue is that every computer can represent initial conditions only up to finite accuracy - any differences below the threshold of the datatype can not be realized. However, since you can also measure actual physical quantities only up to certain accuracy, you will in practice never exactly hit the exact physical outset. For systems with no continuous dependence on initial data, you can thus basically not trust your simulation! This is why for instance in weather simulations people run multiple simulations with different initial conditions and obtain an average to predict whats likely going to happen.

Continuous dependence on initial conditions means that if for every $\epsilon > 0$ there exists a $\delta(\epsilon) > 0 $ such that $$\Vert x_0 - \widetilde{x_0} \Vert \leq \delta(\epsilon) \Rightarrow \Vert x(t, x_0) - x(t, \widetilde{x_0}) \Vert \leq \epsilon \: \forall \: t_0 = 0 \leq t \leq T$$ Where I used the definition given here with $t_0 = 0$.

The classic example why this matters is the Lorenz System/Attractor which shows at least sensitive dependence on initial conditions. A nice intro to the topic is this blog post where it is shown that the Lorenz ODEs show very different behaviour for seemingly small deviations in the initial condition.

Speaking of chaotic dynamical systems, you can also have chaotic behaviour due to changes in the parameters $\mu$ of the system. Thus, your definition of well-posedness for parametric systems $f\big(x(t), t ; \mu\big)$ might be extended to requiring also continuous dependence on the parameters.

Why are these continuous dependence criteria required for a well-posed problem? From a computational science standpoint, you want to trust the solutions you obtain from simulations. The issue is that every computer can represent initial conditions only up to finite accuracy - any differences below the threshold of the datatype can not be realized. However, since you can also measure actual physical quantities only up to certain accuracy, you will in practice never exactly hit the true physical outset. For systems with no continuous dependence on initial data, you can thus basically not trust your simulation! This is why for instance in weather simulations people run multiple simulations with different initial conditions and obtain an average to predict whats likely going to happen.

Continuous dependence on initial conditions means that if for every $\epsilon > 0$ there exists a $\delta(\epsilon) > 0 $ such that $$\Vert x_0 - \widetilde{x_0} \Vert \leq \delta(\epsilon) \Rightarrow \Vert x(t, x_0) - x(t, \widetilde{x_0}) \Vert \leq \epsilon \: \forall \: t_0 = 0 \leq t \leq T$$ Where I used the definition given here with $t_0 = 0$.

The classic example violating this (and back in the days sparking interest in chaotic dynamics) is the Lorenz System/Attractor. A nice intro to the topic is this blog post where it is shown that the Lorenz ODEs do not continuosly depend on the initial data.

Speaking of chaotic dynamical systems, you can also have chaotic behaviour due to changes in the parameters $\mu$ of the system. Thus, your definition of well-posedness for parametric systems $f\big(x(t), t ; \mu\big)$ might be extended to requiring also continuous dependence on the parameters.

Why are these continuous dependence criteria required for a well-posed problem? From a computational science standpoint, you want to trust the solutions you obtain from simulations. The issue is that every computer can represent initial conditions only up to finite accuracy - any differences below the threshold of the datatype can not be realized. However, since you can also measure actual physical quantities only up to certain accuracy, you will in practice never exactly hit the exact physical outset. For systems with no continuous dependence on initial data, you can thus basically not trust your simulation! This is why for instance in weather simulations people run multiple simulations with different initial conditions and obtain an average to predict whats likely going to happen.

Continuous dependence on initial conditions means that if for every $\epsilon > 0$ there exists a $\delta(\epsilon) > 0 $ such that $$\Vert x_0 - \widetilde{x_0} \Vert \leq \delta(\epsilon) \Rightarrow \Vert x(t, x_0) - x(t, \widetilde{x_0}) \Vert \leq \epsilon \: \forall \: t_0 = 0 \leq t \leq T$$ Where I used the definition given here with $t_0 = 0$.

The classic example why this matters is the Lorenz System/Attractor which shows at least sensitive dependence on initial conditions. A nice intro to the topic is this blog post where it is shown that the Lorenz ODEs show very different behaviour for seemingly small deviations in the initial condition.

Speaking of chaotic dynamical systems, you can also have chaotic behaviour due to changes in the parameters $\mu$ of the system. Thus, your definition of well-posedness for parametric systems $f\big(x(t), t ; \mu\big)$ might be extended to requiring also continuous dependence on the parameters.

Why are these continuous dependence criteria required for a well-posed problem? From a computational science standpoint, you want to trust the solutions you obtain from simulations. The issue is that every computer can represent initial conditions only up to finite accuracy - any differences below the threshold of the datatype can not be realized. However, since you can also measure actual physical quantities only up to certain accuracy, you will in practice never exactly hit the exact physical outset. For systems with no continuous dependence on initial data, you can thus basically not trust your simulation! This is why for instance in weather simulations people run multiple simulations with different initial conditions and obtain an average to predict whats likely going to happen.

Continuous dependence on initial conditions means that if for every $\epsilon > 0$ there exists a $\delta(\epsilon) > 0 $ such that $$\Vert x_0 - \widetilde{x_0} \Vert \leq \delta(\epsilon) \Rightarrow \Vert x(t, x_0) - x(t, \widetilde{x_0}) \Vert \leq \epsilon \: \forall \: t_0 = 0 \leq t \leq T$$ Where I used the definition given here with $t_0 = 0$.

The classic example violating this (and back in the days sparking interest in chaotic dynamics) is the Lorenz System/Attractor. A nice intro to the topic is this blog post where it is shown that the Lorenz ODEs do not continuosly depend on the initial data.

Speaking of chaotic dynamical systems, you can also have chaotic behaviour due to changes in the parameters $\mu$ of the system. Thus, your definition of well-posedness for parametric systems $f\big(x(t), t ; \mu\big)$ might be extended to requiring also continuous dependence on the parameters.

Why are these continuous dependence criteria required for a well-posed problem? From a computational science standpoint, you want to trust the solutions you obtain from simulations. The issue is that every computer can represent initial conditions only up to finite accuracy - any differences below the threshold of the datatype can not be realized. However, since you can also measure actual physical quantities also only up to certain accuracy, you will in practice never exactly hit the exact physical outset. For systems with no continuous dependence on initial data, you can thus basically not trust your simulation! This is why for instance in weather simulations people run multiple simulations with different initial conditions and obtain an average to predict whats likely to happen.

Continuous dependence on initial conditions means that if for every $\epsilon > 0$ there exists a $\delta(\epsilon) > 0 $ such that $$\Vert x_0 - \widetilde{x_0} \Vert \leq \delta(\epsilon) \Rightarrow \Vert x(t, x_0) - x(t, \widetilde{x_0}) \Vert \leq \epsilon \: \forall \: t_0 = 0 \leq t \leq T$$ Where I used the definition given here with $t_0 = 0$.

The classic example violating this (and back in the days sparking interest in chaotic dynamics) is the Lorenz System/Attractor. A nice intro to the topic is this blog post where it is shown that the Lorenz ODEs do not continuosly depend on the initial data.

Speaking of chaotic dynamical systems, you can also have chaotic behaviour due to changes in the parameters $\mu$ of the system. Thus, your definition of well-posedness for parametric systems $f\big(x(t), t ; \mu\big)$ might be extended to requiring also continuous dependence on the parameters.

Why are these continuous dependence criteria required for a well-posed problem? From a computational science standpoint, you want to trust the solutions you obtain from simulations. The issue is that every computer can represent initial conditions only up to finite accuracy - any differences below the threshold of the datatype can not be realized. However, since you can also measure actual physical quantities only up to certain accuracy, you will in practice never exactly hit the exact physical outset. For systems with no continuous dependence on initial data, you can thus basically not trust your simulation! This is why for instance in weather simulations people run multiple simulations with different initial conditions and obtain an average to predict whats likely going to happen.